Simplicial model structures on pro-categories

We describe a method for constructing simplicial model structures on ind- and pro-categories. Our method is particularly useful for constructing"profinite"analogues of known model categories. Our construction quickly recovers Morel's model structure for pro-p spaces and Quick's model structure for profinite spaces, but we will show that it can also be applied to construct many interesting new model structures. In addition, we study some general properties of our method, such as its functorial behaviour and its relation to Bousfield localization. We compare our construction to the infinity-categorical approach to ind- and pro-categories in an appendix.


Introduction
In [Qui08;Qui11], Quick constructed a fibrantly generated Quillen model structure on the category of simplicial profinite sets that models the homotopy theory of "profinite spaces".This can be seen as a continuation of Morel's work in [Mor96], where, for a given prime p, he presented a model structure on the same category that models the homotopy theory of "pro-p spaces".
The purpose of this paper is to present a new and uniform method that immediately gives these two model structures, as well as many others.For example, while Quick's model structure is in a sense derived from the classical homotopy theory of simplicial sets, our method also applies to the Joyal model structure, thus providing a homotopy theory of profinite ∞-categories.Our construction can also be used to obtain a model category of profinite P-stratified spaces, where P is a finite poset, whose underlying ∞category is the ∞-category of profinite P-stratified spaces defined in [BGH20].
One general form that our results take is the following version of pro-completion of model categories: Theorem 1.1.Let E be a simplicial model category in which every object is cofibrant and let C be an (essentially) small full subcategory of E closed under finite limits and cotensors by finite simplicial sets.Then for any collection T of fibrant objects in C, the pro-completion Pro(C) carries a fibrantly generated simplicial model structure with the following properties: (1) The weak equivalences are the T-local equivalences; that is, f : C → D is a weak equivalence if and only if f * : Map(D, t) → Map(C, t) is a weak equivalence of simplicial sets for any t ∈ T.
(3) The inclusion C → E induces a simplicial Quillen adjunction E Pro(C).
(4) If T ⊆ C is closed under pullbacks along fibrations and cotensors by finite simplicial sets, then the underlying ∞-category of this model structure on Pro(C) is equivalent to Pro(N(T)), where N(T) denotes the homotopy coherent nerve of the full simplicial subcategory of E spanned by the objects of T.
The model structures of Quick and Morel mentioned above can be obtained from this theorem by appropriately choosing a full subcategory C of sSet and a collection T of fibrant objects.Another known model structure that can be recovered from the above theorem is the model structure for "profinite groupoids" constructed by Horel in [Hor17,§4].
The new model category Pro(C) is a kind of pro-completion of E with respect to the pair (C, T), and could be denoted E or E ∧ (C,T) .The left adjoint E → Pro(C) of the Quillen adjunction mentioned in item (3) can be seen as a "pro-C completion" functor.For the model structures of Morel, Quick and Horel mentioned above, this functor agrees with the profinite completion functor.
We would like to point out that the above formulation is slightly incomplete since there are multiple ways of choosing sets of generating (trivial) fibrations, which theoretically could lead to different model structures on Pro(C), though always with the weak equivalences as described above.A noteworthy fact is that the above theorem also holds for model categories enriched over the Joyal model structure on simplicial sets, so in particular it applies to the Joyal model structure itself.In this case, the model structure obtained on Pro(C) is enriched over the Joyal model structure, but not necessarily over the classical Kan-Quillen model structure on sSet.Another fact worth mentioning is that there exist many simplicial model categories satisfying the hypotheses of the above theorem; that is, all objects being cofibrant.Indeed, by a result of Dugger (Corollary 1.2 of [Dug01]), any combinatorial model category is Quillen equivalent to such a simplicial model category.
Even though we are mostly interested in model structures on pro-categories, we will first describe our construction in the context of ind-categories, and then dualize those results.We have chosen this approach since in the case of ind-categories our construction produces cofibrantly generated model categories, which to most readers will be more familiar territory than that of fibrantly generated model categories.In addition, this will make it clear that the core of our argument, which is contained in Section 3, only takes a few pages.Another reason for describing our construction in the context of ind-categories is that an interesting example occurs there: if we apply our construction to a well-chosen full subcategory of the category of topological spaces, then we obtain a model category that is Quillen equivalent to the usual Quillen model structure on Top, but that has many favourable properties, such as being combinatorial.
Our original motivation partly came from the desire to have a full fledged Quillen style homotopy theory of profinite ∞-operads, by using the category of dendroidal Stone spaces (i.e.dendroidal profinite sets).However, not every object is cofibrant in the operadic model structure for dendroidal sets, so the methods from the current paper do not apply directly to this case.The extra work needed to deal with objects that are not cofibrant is of a technical nature, and very specific to the example of dendroidal sets.For this reason, we have decided to present this case separately, see [BM21].

Relation to the construction by Barnea-Schlank.
There are several results in the literature that describe general methods for constructing model structures on ind-or procategories.The construction in the current paper is quite close in spirit to that by Barnea and Schlank in [BS16a].They show that if C is a category endowed with the structure of a "weak fibration category", then there exists an "induced" model structure on Pro(C) provided some additional technical requirements are satisfied.However, there are important examples of model structures on pro-categories that are not of this form.For example, Quick's model structure is not of this kind, as explained just above Proposition 7.4.2 in [BHH17].In the present paper, we prove the existence of a certain model structure on the pro-category of a simplicial category endowed with the extra structure of a so-called "fibration test category" (defined in Definition 5.1).While the definition of a fibration test category given here seems less general than that of a weak fibration category, there are many interesting examples where it is easy to prove that a category is a fibration test category while it is not clear whether this category is a weak fibration category in the sense of [BS16a].In particular, Quick's model structure can be obtained through our construction, see Example 5.5 and Corollary 6.6.Another advantage is that we do not have to check the technical requirement of "pro-admissibility" (see Definition 4.4 of [BS16a]) to obtain a model structure on Pro(C), which is generally not an easy task.We also believe that our description of the weak equivalences in Pro(C), namely as the T-local equivalences for some collection of objects T, is often more natural and flexible than the one given in [BS16a].It is worth pointing out that if both our model structure and that of [BS16a] on Pro(C) exist, then they agree by Remark 5.12 below.
Overview of the paper.In Section 2, we will establish some terminology and mention a few facts on simplicial model categories and ind-and pro-categories.We will then describe our general construction of the model structure for ind-categories in Section 3. We illustrate our construction with an example in Section 4, where we construct a convenient model category of spaces.In Section 5, we dualize our results to the context of the homotopy theory of ∞-categories [Lur09].A simplicial model category is a model category E that is enriched, tensored and cotensored over simplicial sets, and that satisfies the additional axiom SM7 phrased in terms of pullback-power maps, or dually in terms of pushout-product maps (see e.g.Definition 9.1.6and Proposition 9.3.7 of [Hir03] or Definition II.2.2 of [Qui67]).We emphasize that we will use this terminology in a somewhat non-standard way.Namely, by a simplicial model category, we will either mean that the axiom SM7 holds with respect to the Kan-Quillen model structure or the Joyal model structure.Whenever it is necessary to emphasize the distinction, we will call a simplicial model category of the former kind a sSet KQ -enriched model category and the latter a sSet J -enriched model category.Note that any sSet KQ -enriched model category is sSet J -enriched, since sSet KQ is a left Bousfield localization of sSet J .
We will make use of the following fact about the (categorical) fibrations in sSet J .
Lemma 2.1.There exists a set M of maps between finite simplicial sets such that a map between quasi-categories X → Y is a fibration in sSet J if and only if it has the right lifting property with respect to all maps in M.
Proof.Let H denote the simplicial set obtained by gluing two 2-simplices to each other along the edges opposite to the 0th and 2nd vertex, respectively, and then collapsing the edges opposite to the 1st vertex to a point in both of these 2-simplices.This means that H looks as follows, where the dashed lines represent the collapsed edges: A map from H into a quasicategory X consists of an arrow f ∈ X 1 , a left and right homotopy inverse g, h ∈ X 1 and homotopies g f ∼ id and f h ∼ id.Let {0} → H denote the inclusion of the leftmost vertex into H.It follows from Corollary 2.4.6.5 of [Lur09] that if X → Y is an inner fibration between quasicategories that has the right lifting property with respect to {0} → H, then it is an categorical fibration.The converse is also true.To see this, note that for any quasicategory Z, a map H → Z lands in the largest Kan complex k(Z) contained in Z. Since {0} → H is a weak homotopy equivalence, we see that Map(H, Z) = Map(H, k(Z)) Map({0}, k(Z)) = Map({0}, Z), so the inclusion {0} → H is a categorical equivalence.In particular, any categorical fibration has the right lifting property with respect to {0} → H.We conclude that the set M = {Λ n k → ∆ n | 0 < k < n} ∪ {{0} → H} has the desired properties.

Ind-and pro-categories
In this section we recall some basic definitions concerning ind-and pro-categories.Most of these will be familiar to the reader, with the possible exception of Theorem 2.3 below.
For details, we refer the reader to [GAV72, Exposé 1], [EH76, §2.1], [AM86, Appendix] and [Isa02].In the discussion below, all (co)limits are asssumed to be small.For a category C, its ind-completion Ind(C) is obtained by freely adjoining filtered (or directed) colimits to C. Dually, the free completion under cofiltered limits is denoted Pro(C).This in particular means that Pro(C) op = Ind(C op ), so any statement about indcategories dualizes to a statement about pro-categories and vice versa.We will therefore mainly discuss ind-categories here and leave it to the reader to dualize the discussion.
One way to make the above precise, is to define the objects in Ind(C) to be all diagrams I → C for all filtered categories I.Such objects are called ind-objects and denoted C = {c i } i∈I .The morphisms between two such objects C = {c i } i∈I and D = {d j } j∈J are defined by If C is a simplicial category, then Ind(C) can be seen as a simplicial category as well.The enrichment is expressed by a formula similar to (1), namely One can define the pro-category Pro(C) of a (simplicial) category C as the category of all diagrams I → C for all cofiltered I, and with (simplicial) hom sets dual to the ones above.An object in Pro(C) is called a pro-object.One could also simply define Pro(C) as Ind(C op ) op .It can be shown that any object in Ind(C) is isomorphic to one where the indexing category I is a directed poset, and dually that any object in Pro(C) is isomorphic to one that is indexed by a codirected poset (see Proposition 8.1.6 of [GAV72, Exposé 1], or Theorem 2.1.6 of [EH76] with a correction just after Corollary 3.11 of [BS15]).
There is a fully faithful embedding C → Ind(C) sending an object c to the constant diagram with value c, again denoted c.We will generally identify C with its image in Ind(C) under this embedding.This embedding preserves all limits and all finite colimits that exist in C. The universal property of Ind(C) states that Ind(C) has all filtered colimits and that any functor F : C → E, where E is a category that has all filtered colimits, has an essentially unique extension to a functor F : Ind(C) → E that preserves filtered colimits.This extension can be defined explicitly by F({c i }) = colim i F(c i ).
Recall that if E is a category that has all filtered colimits, then an object c in E is called compact if Hom E (c, −) commutes with filtered colimits.The dual notion is called cocompact.One can deduce from the definition of the morphisms in Ind(C) that any object in the image of C → Ind(C) is compact.Dually, the objects of C are cocompact in Pro(C).
There is the following recognition principle for ind-completions, of which we leave the proof to the reader.Lemma 2.2 (Recognition principle).Let E be a category closed under filtered colimits and let C → E be a full subcategory.If (i) any object in C is compact in E , and (ii) any object in E is a filtered colimit of objects in C, then the canonical extension Ind(C) → E, coming from the universal property of Ind(C), is an equivalence of categories.
To avoid size issues, we assume from now on that C is an (essentially) small category.The fact that the presheaf category Set C op is the free cocompletion of C leads to an alternative description of Ind(C) that is sometimes easier to work with.Namely, we can think of Ind(C) as the full subcategory of Set C op consisting of those presheaves which are filtered colimits of representables.If C is small and has finite colimits, as will be the case in all of our examples, then these are exactly the functors C op → Set that send the finite colimits of C to limits in Set (see Théorème where the right-hand side stands for the category of left exact functors.From this description, one sees immediately that Ind(C) has all small limits and that the inclusion Ind(C) → Set C op preserves these.The category Ind(C) also has all colimits in this case.Namely, finite coproducts and pushouts can be computed "levelwise" in C as described in [AM86, Appendix 4], while filtered colimits exist as mentioned above.Note however, that the inclusion Ind(C) → Set C op does not preserve all colimits, but only filtered ones.One sees dually that if C is small and has all finite limits, then Pro(C) lex(C, Set) op .
As above, it follows that Pro(C) is complete and cocomplete in this case.
Another consequence of the fact that finite coproducts and pushouts in Ind(C) are computed "levelwise" is the following: if F : C → E , with E cocomplete, preserves finite colimits, then its extension F : Ind(C) → E given by the universal property also preserves finite colimits.Since it also preserves filtered colimits, we conclude that it preserves all colimits.In fact, more is true.The above description of Ind(C) as lex(C op , Set) allows us to construct a right adjoint R of F. Namely, if we define R(E)(c) := Hom(Fc, E), then R(E) : C op → Set is left exact, hence R defines a functor E → Ind(C).Adjointness follows from the Yoneda lemma.We therefore see that, up to unique natural isomorphism, there is a 1-1 correspondence between finite colimit perserving functors C → E and functors Ind(C) → E that have a right adjoint.
There are two important examples of adjunctions obtained in this way that we would like to mention here.The first one is the ind-completion functor.If E is a cocomplete category and C a full subcategory closed under finite colimits, then the inclusion C ⊆ E induces an adjunction U : Ind(C) E : (•) Ind whose right adjoint we call ind-completion (relative to C) or ind-C completion.Dually, if E is complete and C is a full subcategory closed under finite limits, then we obtain an adjunction (•) Pro : E Pro(C) : U whose left adjoint we call pro-completion (relative to C) or pro-C completion.In many examples, C is the full subcategory of E consisting of objects that are "finite" in some sense, and this left adjoint is better known as the profinite completion functor.For instance, in the case of groups, this functor (•) Pro : Grp → Pro(FinGrp) is the well-known profinite completion functor for groups.
The other important example is about cotensors in ind-categories.Suppose C is a small simplicial category that has all finite colimits and tensors with finite simplicial sets, and that furthermore these tensors commute with these finite colimits.We will call C finitely tensored if this is the case (cf.Definition 3.1 for a precise definition).If X is a simplicial set, then we can write it as colim i X i where i ranges over all finite simplicial subsets X i ⊆ X. Define − ⊗ X : C → Ind(C) by c ⊗ X = {c ⊗ X i } i .This functor preserves finite colimits since these are computed "levelwise" in Ind(C), hence it extends to a functor − ⊗ X : Ind(C) → Ind(C) that has a right adjoint (−) X .These define tensors and cotensors by arbitrary simplicial sets on Ind(C).In particular, Ind(C) is a simplicial category that is complete, cocomplete, tensored and cotensored (note the similarity with Proposition 4.10 of [BS16b]).The dual of this statement says that for any small simplicial category C that has finite limits and cotensors with finite simplicial sets, and in which these finite cotensors commute with finite limits in C, the pro-category Pro(C) is a simplicial category that is complete, cocomplete, tensored and cotensored.We call C finitely cotensored in this case.
Let us return to the basic definition (1) of morphisms in Ind(C).If C = {c i } and D = {d i } are objects indexed by the same filtered category I, then any natural transformation with components f i : c i → d i represents a morphism in Ind(C).Morphisms of this type (or more precisely, morphisms represented in this way) will be called level maps or strict maps.Up to isomorphism, any morphism in Ind(C) has such a strict representation (see Corollary 3.2 of [AM86,Appendix]).One can define the notion of a "level" diagram or "strict" diagram in a similar way.Given an indexing category K, a conceptual way of thinking about these is through the canonical functor A strict diagram can be thought of as an object in the image of this functor.If K is a finite category and C has all finite colimits, then the above functor is an equivalence of categories ( [Mey80,§4]).This shows in particular that, up to isomorphism, any finite diagram in Ind(C) is a strict diagram if C is small and has finite colimits.
In our context, the following extension of Meyer's result is important.Suppose that K is a category which can be written as a union of a sequence of finite full subcategories Let C be a small category that has finite colimits.Then any functor f : K n → C has a left Kan extension g : K → C defined in terms of finite colimits as in (the dual to) Theorem X.3.1 of [Mac71].For X : K → C, write sk n X for the left Kan extension of the restriction of X to K n .We call X n-skeletal if the canonical map sk n X → X is an isomorphism, and skeletal if this is the case for some n.The full subcategory sk(C K ) ⊆ C K spanned by the skeletal functors K → C can be viewed as a full subcategory of Ind(C) K via the inclusion C → Ind(C).Note that for any X in Ind(C) K , we have X = colim n sk n X. Exactly as in (the dual of) the proof of Proposition 7.4.1 of [BHH17], the result of [Mey80, §4] mentioned above can be used to show that the hypotheses of the recognition principle for ind-categories are satisfied, hence that the induced functor Ind(sk(C K )) → Ind(C) K is an equivalence of categories.In fact, the assumption that K is a union of a sequence of finite full subcategories is irrelevant, and the following more general result, which we write down for future reference, can be proved by the same argument.Note that a category K can be written as a union of finite full subcategories if and only if for any k, k ∈ K, the set Hom K (k, k ) is finite.
Theorem 2.3.Let C be a small category that has finite colimits, and let K be a small category that can be written as a union of finite full subcategories.Write sk(C K ) for the full subcategory of C K of those functors K → C that are isomorphic to the left Kan extension of a functor K → C for some finite full subcategory K ⊆ K. Then Ind(sk(C K )) Ind(C) K .
This theorem recovers the well-known equivalence Ind(sSet fin ) sSet when applied to ∆ op = ∪ n ∆ op ≤n and C = FinSet.Note that we already (implicitly) used this equivalence when we defined tensors by simplicial sets for ind-categories above.
We can also apply the dual of this theorem to the same categories K = ∆ op and C = FinSet.Write Set = Pro(FinSet) for the category of profinite sets, which is well known to be equivalent to the category of Stone spaces Stone.Since we want to apply the dual of Theorem 2.3, we need to work with right Kan extensions instead of left Kan extensions.In particular, we obtain the full subcategory of FinSet ∆ op on those simplicial sets that are the right Kan extension of some functor ∆ op ≤n → FinSet.These are exactly the coskeletal degreewise finite simplicial sets, i.e. the lean simplicial sets.In particular, the theorem above recovers the equivalence Pro(L) s Set proved in Proposition 7.4.1 of [BHH17].
An example that plays an important role in Section 9 is that of bisimplicial (profinite) sets.The dual of the above theorem shows that the category of bisimplicial profinite is canonically equivalent to the category Pro(L (2) ) for a certain full subcategory L (2) of the category of bisimplicial sets bisSet = sSet ∆ op .This category L (2) consists of those bisimplicial sets that are isomorphic to the right Kan extension of a functor for some t, n ∈ N. We will refer to such bisimplicial sets as doubly lean.
3 The completed model structure on Ind(C) In this section, we will describe our construction of the model structure on Ind(C), where C is what we call a "cofibration test category".In Section 5, we will dualize this construction to the context of pro-categories.After that, we will study the functorial behaviour of the construction in Section 7 and discuss Bousfield localizations in Section 8.
Throughout these sections, the terms "weak equivalence" and "fibration" of simplicial sets refer to either the classical Kan-Quillen model structure or to the Joyal model structure.When we say that a model category is simplicial, then this can either mean that it is enriched over the Kan-Quillen model structure or over the Joyal model structure.
We wish to single out the definition of being finitely tensored, since it occurs many times throughout this paper.Definition 3.1.Let C be simplicial category.Then C is called finitely tensored if (i) it admits finite colimits, (ii) it admits tensors by finite simplicial sets, and (iii) these commute with each other, meaning that the canonical map is an isomorphism for any finite diagram {c i } in C and any finite simplicial set X.
Remark 3.2.It is worth pointing out that condition (iii) is equivalent to asking that the finite colimits of (i) are enriched colimits; that is, for any finite diagram {c i } in C and any object d in C, the canonical map colim As explained in Section 2.2, if C is finitely tensored, then the category Ind(C) is a tensored and cotensored simplicial category that is both complete and cocomplete.We will endow C with some additional structure, that of a "cofibration test category", and show that it induces a simplicial model structure on Ind(C) in Theorem 3.9 below.Definition 3.3.A cofibration test category (C, T) consists of a small finitely tensored simplicial category C, a full subcategory T of test objects and two classes of maps in T called cofibrations, denoted , and trivial cofibrations, denoted ∼ , both containing all isomorphisms, that satisfy the following properties: (1) The initial object ∅ is a test object, and for every test object t ∈ T, the map ∅ → t is a cofibration.
(2) For every cofibration between test objects s t and cofibration between finite simplicial set U V, the pushout-product map t ⊗ U ∪ s⊗U s ⊗ V → t ⊗ V is a cofibration between test objects which is trivial if either s t or U V is so.
(3) A morphism r → s in T is a trivial cofibration if and only if it is a cofibration and Map(t, r) → Map(t, s) is a weak equivalence of simplicial sets for every t ∈ T.
(4) Any object c ∈ C has the right lifting property with respect to trivial cofibrations.2) and (4) also hold with respect to sSet J .For item (3), note that the map Map(t, r) → Map(t, s) is a map between Kan complexes by Remark 3.4, hence that it is a weak equivalence in sSet J if and only if it is in sSet KQ .
We will often write C for a cofibration test category (C, T), omitting the full subcategory of test objects T from the notation.We will write cof(C) for the set of cofibrations.Note that this is a subset of the morphisms of T.
The role of the test objects t ∈ T is to detect the weak equivalences in Ind(C) "from the left".More precisely, the weak equivalences in Ind(C) will be those arrows C → D for which Map(t, C) → Map(t, D) is a weak equivalence for every t ∈ T. For this reason, we will call an arrow c → d in C for which Map(t, c) → Map(t, d) is a weak equivalence for every t ∈ T a weak equivalence, and denote such arrows by ∼ −→.We write we(C) for the set of weak equivalences in C. Using this terminology, item (3) of the above definition can be rephrased as saying that the trivial cofibrations are precisely the cofibrations that are weak equivalences.In particular, the set of trivial cofibrations in a cofibration test category C is fully determined by the full subcategory T and the set cof(C).
Let us look at a few examples.Note that we will discuss more interesting examples in Section 5, where we consider fibration test categories, the dual of cofibration test categories.
Example 3.6.Suppose E is a simplicial model category in which every object is fibrant and let C ⊆ E be a (small) full subcategory closed under finite colimits and finite tensors.If we define T to be the full subcategory on the cofibrant objects, then (C, T) forms a cofibration test category where the (trivial) cofibrations are the (trivial) cofibrations of E between objects of T. We say that C inherits this structure of a cofibration test category from E .Properties (1), (2) and (4) of Definition 3.3 follow directly from the fact that E is a (simplicial) model category and the fact that any object in E is fibrant.For one direction of property (3), note that since all object in E are fibrant, the functor Map(t, −) preserves weak equivalences for any cofibrant object t.For the converse direction, note that a cofibration r s is trivial if and only if it is mapped to an isomorphism in the homotopy category Ho(E ).By the Yoneda lemma applied to the full subcategory Ho(T) ⊆ Ho(E ) spanned by the objects of T, this is equivalent to Hom Ho(E ) (t, r) → Hom Ho(E ) (t, s) being a weak equivalence for every t.Since Map(t, r) → Map(t, s) is a weak equivalence by assumption and Hom Ho(E ) (t, -) equals the set of path components of (the maximal Kan complex contained in) Map(t, -), this is indeed the case.
Example 3.7.Suppose that a cofibration test category (C, T) is given, and let T ⊆ T be a full subcategory such that ∅ ∈ T and such that for any cofibration s t between objects of T and any cofibration U V in sSet fin , the object t ⊗ U ∪ s⊗U s ⊗ V is again in T .We will call such a full subcategory T ⊆ T closed under finite pushout-products.Then (C, T ) is again a cofibration test category if we define the (trivial) cofibrations to be those of (C, T) between objects of T .All items of Definition 3.3 are straightforward to show except possibly property (3).The "only if" direction follows immediately.For the "if" direction of (3), suppose r s is a map in T that is a cofibration with the property that Map(t, r) → Map(t, s) is a weak equivalence for any t ∈ T .Applying this to t = r and t = s and using that these mapping spaces are fibrant, we obtain left and right homotopy inverses of r s, where homotopies in T are defined using the tensor − ⊗ ∆ 1 (in the case of sSet KQ ) or − ⊗ H (in the case sSet J , where H is as in the proof of Lemma 2.1).Since Map(t, -) is a simplicial functor it preserves these homotopies, showing that Map(t, r) → Map(t, s) is homotopy equivalence for every t ∈ T. We conclude that r s is a trivial cofibration in T and hence a trivial cofibration in T by definition.
is straightforward to verify that (C, T) is a cofibration test category in the sense of Definition 3.3 (with respect to sSet KQ ).This example will be studied further in Section 4.
For a cofibration test category C, we will write I for the image of the set of cofibrations of C in Ind(C), and J for the image of the set of trivial cofibrations of C in Ind(C).Identifying C with its image in Ind(C), we can write Recall that the sets of (trivial) cofibrations cof(C) and cof(C) ∩ we(C) in (C, T) are both contained in T; that is, any (trivial) cofibration is a map between test objects.The sets I and J are generating (trivial) cofibrations for a model structure on Ind(C) in which the weak equivalences are as above.
Theorem 3.9.Let C be a cofibration test category.Then Ind(C) carries a cofibrantly generated (hence combinatorial) simplicial model structure, the completed model structure, where a map C → D is a weak equivalence if and only if Map(t, C) → Map(t, D) is a weak equivalence for every t ∈ T. A set of generating cofibrations (generating trivial cofibrations) is given by I (J, respectively).Every object is fibrant in this model structure.
Remark 3.10.As mentioned in Remark 3.5, the definition of a cofibration test category depends on whether we work with the Joyal model structure or the Kan-Quillen model structure on sSet.In the first case, the model structure on Ind(C) will be sSet J -enriched, while in the latter case, it will be sSet KQ -enriched.
The proof uses the following lemmas.
Lemma 3.11.Let C be a cofibration test category.The weak equivalences of Ind(C) as defined in Theorem 3.9 are stable under filtered colimits.
, which are weak equivalences by assumption.The proof therefore reduces to the statement in sSet that a filtered colimit of weak equivalences, indexed by some filtered category I, is again a weak equivalence.This can be proved for the Kan-Quillen and Joyal model structure in exactly the same way.Namely, this is equivalent to the statement that the functor colim : sSet I → sSet, where sSet I is endowed with the projective model structure, preserves weak equivalences.To see that this is the case, factor {X i } ∼ −→ {Y i } in sSet I as a projective trivial cofibration {X i } ∼ {Z i } followed by a pointwise trivial fibration {Z i } ∼ {Y i }.Then colim X i → colim Z i is again a trivial cofibration, so in particular a weak equivalence.Furthermore, since the generating cofibrations ∂∆ n → ∆ n in sSet are maps between compact objects, we see that colim Z i → colim Y i must have the right lifting property with respect to these maps, i.e. it is a trivial fibration.We conclude that colim : sSet I → sSet preserves weak equivalences.Lemma 3.12.Let C be a cofibration test category, let s t be a cofibration in C, i.e. a map in I, and let C → D be an arrow in Ind(C) which has the right lifting property with respect to all maps in J. Then Map(t, C) → Map(s, C) × Map(s,D) Map(t, D) is a fibration, which is trivial if either s t is trivial or if C → D is a weak equivalence in the sense of Theorem 3.9.
Proof.Let M be a set of trivial cofibrations in sSet fin such that a map between fibrant objects in sSet is a fibration if and only if it has the right lifting property with respect to the maps in M. For the Kan-Quillen model structure, one can take the set of horn inclusions, while for sSet J , the set M from Lemma 2.1 works.By Remark 3.4, for any test object t ∈ T and any C ∈ Ind(C) the simplicial set Map(t, C) is fibrant.For any t ∈ T and any map   i.e. a retract r of i.By (2) and (4) of Definition 3.3, there exists a lift F in be given, where s → C is any map in Ind(C).The maps f r : t → C and id C : C → C give, by the universal property of the pushout, a retract r of j : C → D. Since tensors preserve colimits, we see that Now let u ∈ T be any test object.We deduce from the existence of the deformation retract G that Map(u, C) → Map(u, D) is the inclusion of a deformation retract, hence a weak equivalence.
Proof of Theorem 3.9.We check all the four assumptions of Kan's recognition theorem as spelled out in [Hir03, Theorem 11.3.1].The weak equivalences satisfy the two out of three property and are closed under retracts since this holds for the weak equivalences in sSet.
(1) Since all objects of C are compact in Ind(C), the sets I and J permit the small object argument.
(2) It suffices to prove that any transfinite composition of pushouts of maps in J is a weak equivalence.This follows immediately from Lemma 3.11 and Lemma 3.13.
(3) We need to show that any map having the right lifting property with respect to maps in I has the right lifting property with respect to maps in J and is a weak equivalence.The first of these follows since J ⊆ I. To see that any map that has the right lifting property with respect to maps in I is a weak equivalence, let such a map C → D be given.Note that t ⊗ ∂∆[n] → t ⊗ ∆[n] is in I for any t ∈ T and n ≥ 0 by items (1) and (2) of Definition 3.3.This implies that Map(t, C) → Map(t, D) is a trivial fibration for any t ∈ T and in particular that C → D is a weak equivalence.
(4) We need to show that if C → D has the right lifting property with respect to maps in J and is a weak equivalence, then it has the right lifting property with respect to maps in I. Let s t in I be given.Then Map(t, C) ∼ Map(s, C) × Map(s,D) Map(t, D) is a trivial fibration by Lemma 3.12, and in particular surjective on 0-simplices.In particular, C → D has the right lifting property with respect to s t.The fact that this model structure is simplicial follows from Lemma 3.12.By (4), all objects in C ⊆ Ind(C) are fibrant.Since the generating trivial cofibrations are maps between compact objects and any C ∈ Ind(C) is a filtered colimit of objects in C, it follows that all objects in Ind(C) are fibrant.
Example 3.14.Let (C, T) be the cofibration test category from Example 3.8.The model structure on Ind(C) obtained by applying Theorem 3.9 turns out to be Quillen equivalent to the Kan-Quillen model structure on sSet and the Quillen model structure on Top.More precisely, there is a canonical way to factor the geometric realization functor for simplicial sets | • | : sSet → Top as a composite sSet → Ind(C) → Top, where both of these functors are left Quillen equivalences.This will be proved in Proposition 4.2.
Example 3.15.For this example, Top is again a convenient category of spaces as in Example 3.8.If P is a topological operad, then the category P-Alg of P-algebras admits a model structure, obtained through transfer along the free-forgetful adjunction F : Top P-Alg : U.In particular, any object is fibrant in this model structure.This category is Top-enriched, since one can view Hom P-Alg (S, T) as a subspace of Hom Top (US, UT).For any topological space X and any P-algebra S, one can endow the space S X with the "pointwise" structure of a P-algebra.By restricting the usual homeomorphism coming from the cartesian closed structure on Top, we obtain a natural homeomorphism Hom P-Alg (S, T X ) ∼ = Hom Top (X, Hom P-Alg (S, T)).One can furthermore show that − X : P-Alg → P-Alg has a left adjoint that makes P-Alg into a tensored and cotensored topological category.In particular, it can be viewed as a tensored and cotensored simplicial category.Since the cotensors, fibrations and weak equivalences are defined underlying in Top, we see that P-Alg is a sSet KQ -enriched model category with respect to this enrichment.By Example 3.6, any small full subcategory closed under finite colimits and tensors with finite simplicial sets inherits the structure of a cofibration test category.
Example 3.16.One can modify the previous example in a way that is similar to Example 3.8.Namely, suppose that C ⊆ P-Alg is a small full subcategory closed under finite colimits and tensors by finite simplicial sets, and suppose that F|X| is contained in C for any finite simplicial set X, where F : Top → P-Alg is the left adjoint of the free-forgetful adjunction.Define the full subcategory of test objects T ⊆ C to be the category of objects of the form F|X| for X a finite simplicial set, and define the (trivial) cofibrations to be the maps of the form F|i| : F|X| → F|Y| where i is a (trivial) cofibration between finite simplicial sets in sSet KQ .Then (C, T) is a cofibration test category, hence we obtain a model structure on Ind(C) by Theorem 3.9.Since the inclusion C → P-Alg preserves finite colimits, it induces an adjunction Ind(C) P-Alg.One can show that this adjunction is a Quillen equivalence.

Example: a convenient model category of topological spaces
Throughout this section, let Top be a convenient category of spaces, such as k-spaces, compactly generated weak Hausdorff spaces or compactly generated Hausdorff spaces.Suppose that a small full subcategory C ⊆ Top is given that is closed under finite colimits and tensors with finite simplicial sets, and that contains the space |X| for any finite simplicial set X.As explained in Example 3.8, if we define T to be the collection of spaces of the form |X|, where X is any finite simplicial set, and if we define a map to be a (trivial) cofibration if and only if it is the geometric realization of a (trivial) cofibration in sSet KQ between finite simplicial sets, then (C, T) is a cofibration test category.In this section, we will study this example in more detail.
We begin by characterizing the weak equivalences of Ind(C).For the second statement, note that Map( * , The inclusion C → Top induces an adjunction L : Ind(C) Top : (•) Ind as explained in Section 2.2, where L is defined by L({c i }) = colim i c i for any {c i } in Ind(C).Since geometric realization commutes with colimits, we see that the geometric realization functor  One can show that the model category Ind(C), with (C, T) a cofibration test category of the type considered above, is very similar to Top.We mention a few similarities.We first note that it is possible define homotopy groups for objects of Ind(C), and that they detect weak equivalences.If C is an object in Ind(C), then by a basepoint of C we mean a map * → C. It follows directly from this definition that π n (C, c 0 ) = π n (Sing C, c 0 ) for any C ∈ Ind(C) and c 0 ∈ C. We conclude the following: ) is a bijection for any c 0 ∈ C and n ≥ 0.Moreover, the homotopy groups for objects in Ind(C) commute with filtered colimits.
Proof.The first statement follows since f is a weak equivalence if and only if Sing f is so.The second part follows since both the functor Sing and the homotopy groups of simplicial sets commute with filtered colimits.
It can also be shown that one can take the same generating (trivial) cofibrations in Ind(C) as in the usual Quillen model structure on Top.Define Proposition 4.5.The sets I and J are sets of generating cofibrations and generating trivial cofibrations for Ind(C), respectively.
Proof.We need to show that the geometric realization of any cofibration (or trivial cofibration) between finite simplicial sets lies in the saturation of I (or J, respectively).This follows from the fact that | • | : sSet → Ind(C) preserves colimits and that all maps of the form are isomorphic to a map in I (or J, respectively).For any two objects C = {c i } i and D = {d j } j in Ind(C), one can compute their product levelwise by C × D = {c i × d j } (i,j)∈I×J .Since the finite colimits of C are computed in Top, and finite colimits in Ind(C) can be computed levelwise, we see that the functor − × D : C → Ind(C) preserves finite colimits for any D in Ind(C).As explained in Section 2.2, it follows from this that the product functor − × D : Ind(C) → Ind(C) has a right adjoint.In particular, Ind(C) is cartesian closed.This cartesian closed structure interacts well with the model structure defined above.Proposition 4.6.Ind(C) is a cartesian closed model category.

Proof. It suffices to show that for any pair of generating cofibrations
One can furthermore show that the full subcategory of Top on the CW-complexes embeds fully faithfully into Ind(C).Note that any finite CW-complex X is (homeomorphic to) an object in C. Proposition 4.7.There is a fully faithful functor from the category of CW-complexes into Ind(C) that preserves and detects weak equivalences.
Proof.If X is a CW-complex, then one can always choose a CW-decomposition.The finite CW-subcomplexes in this decomposition together with their inclusions form a directed diagram {X i } for which colim i X i ∼ = X.Suppose that we have chosen a CW- decomposition for any CW-complex X and denote the associated directed diagram of finite CW-subcomplexes by {X i X }.Since a map from a compact space into a CW-complex (with a given CW-decomposition) always lands in a finite CW-subcomplex, we see that the canonical map lim is an isomorphism for any pair of CW-complexes.By definition of the morphisms in Ind(C), this implies that the functor that sends a CW-complex X to the ind-object {X i X } in Ind(C) is well-defined and fully faithful.Preservation and detection of weak equivalences follows directly form the fact that Sing detects weak equivalences and that colim i X Sing(X i X ) ∼ = Sing(X) for any CW-complex X.
We end this section by discussing a specific example of such a full subcategory C, namely the category CM of compact metrizable spaces.Under Gelfand-Naimark duality, this category corresponds to the category of separable commutative unital C * -algebras.If we let Top be the category of compactly generated Hausdorff spaces, then CM as a full subcategory is closed under all finite colimits and tensors by finite simplicial sets.In particular, by the above we obtain a model structure on Ind(CM) that is equivalent to the Quillen model structure on Top.In [Bar17], Barnea also proposes a model structure on Ind(CM).However, this model structure does not agree with the one constructed above, so we will briefly describe his model structure and the difference with ours.Let us denote our model structure by Ind(CM) Q .
Barnea shows in [Bar17] that CM is a "special weak cofibration category", and hence that there exists an induced model structure on Ind(CM), which we will denote by Ind(CM) B .This model structure is cofibrantly generated and one can take the set of Hurewicz cofibrations in CM as a set of generating cofibrations, while one can take the Hurewicz cofibrations that are also homotopy equivalences as a set of generating trivial cofibrations.If we define T = CM and if we define a map in T to be a (trivial) cofibration if it is in the set of generating (trivial) cofibrations just mentioned, then (CM, T) is a cofibration test category and the completed model structure on Ind(CM) coincides with the one that Barnea constructed.Since Barnea's model structure Ind(CM) B has strictly more generating (trivial) cofibrations than our model structure Ind(CM) Q , we see that the identity functor is a left Quillen functor Ind(CM) Q → Ind(CM) B .To see that the model structures do not coincide, we will show that Ind(CM) Q has strictly more weak equivalences than Ind(CM) B .Let C be any metrizable infinite Stone space, such as a Cantor space.Then, for Sing and | • | as defined just above Lemma 4.1, the counit | Sing C| → C is a weak equivalence in Ind(CM) Q .However, this map is not a weak equivalence in Ind(CM) B , since Map(C, | Sing C|) → Map(C, C) is not a weak equivalence of simplicial sets: Since these mapping spaces are discrete, this would imply that the map is an isomorphism.However, it is not surjective since there is no map C → | Sing C| that gets mapped to id C .The model structure Ind(CM) Q defined here is similar to the Quillen model structure on Top, while Barnea's model structure Ind(CM) B bears some similarity to the Strøm model structure on Top.

The dual model structure on Pro(C)
A model structure on E also gives rise to a model structure on E op , where the fibrations (cofibrations) of E op are the cofibrations (fibrations, respectively) of E .In particular, E is cofibrantly generated if and only if E op is fibrantly generated.Since Pro(C) Ind(C op ) op , this implies that if C is the dual of a cofibration test category, then Pro(C) admits a fibrantly generated simplicial model structure.We explicitly dualize the main definition and result of Section 3 in this section, and then discuss a few examples of such fibrantly generated simplicial model structures on pro-categories.Again, we work with sSet endowed with either the Joyal or the Kan-Quillen model structure.
We say that a simplicial category C is finitely cotensored if C op is finitely tensored in the sense of Definition 3.1.Explicitly, this means that C admits finite limits and cotensors by finite simplicial sets, and that these commute with each other.As explained in Section 2.2, if C is a small simplicial category that is finitely cotensored, then the simplicial category Pro(C) is tensored, cotensored, complete and cocomplete.
Definition 5.1.A fibration test category (C, T) consists of a small finitely cotensored simplicial category C, a full subcategory T ⊆ C of test objects and two classes of maps in T called fibrations, denoted , and trivial fibrations, denoted ∼ , both containing all isomorphisms, that satisfy the following properties: (1) The terminal object * is a test object, and for every test object t ∈ T, the map t → * is a fibration.
(2) For every fibration between test objects s t and cofibration between finite simplicial set U V, the pullback-power map s V → s U × t U t V is a fibration between test objects which is trivial if either s t or U V is so.
(3) A morphism c → d in T is a trivial fibration if and only if it is a fibration and Map(d, t) → Map(c, t) is a weak equivalence of simplicial sets for every t ∈ T.
(4) Any object c ∈ C has the left lifting property with respect to trivial fibrations.
For a fibration test category C, we write fib(C) for the set of fibrations and we(C) for the set of maps c → d that induce a weak equivalence Map(d, t) → Map(c, t) for every t ∈ T. By property (3), the set of trivial fibrations is fib(C) ∩ we(C).Note that the definition of a fibration test category is formally dual to that of a cofibration test category.More precisely, (C, T) is a fibration test category if and only if (C op , T op ) is a cofibration test category in the sense of Definition 3.3, where the (trivial) cofibrations of (C op , T op ) are defined as the (trivial) fibrations of (C, T).
We let P ⊆ Ar(Pro(C)) denote the image of the set fib(C) along the inclusion C → Pro(C), and Q ⊆ Ar(Pro(C)) the image of the set of trivial fibrations.The sets P and Q are the generating (trivial) fibrations of the completed model structure on Pro(C).The following theorem is formally dual to Theorem 3.9.
Theorem 5.2.Let (C, T) be a fibration test category.Then Pro(C) carries a fibrantly generated (hence cocombinatorial) simplicial model structure, the completed model structure, where a map C → D is a weak equivalence if and only if Map(D, t) → Map(C, t) is a weak equivalence for every t ∈ T. A set of generating fibrations (generating trivial fibrations) is given by P (Q, respectively).Every object is cofibrant in this model structure.
Example 5.3.Dualizing Example 3.6, we see that if E is a simplicial model category in which every object is cofibrant, then any small full subcategory C ⊆ E closed under finite limits and finite cotensors admits the structure of a fibration test category.Namely, defining T to be the full subcategory of fibrant objects of C, and defining the (trivial) fibrations to be those of E between objects in T, then (C, T) is a fibration test category.As in Example 3.6, will say that C inherits the structure of a fibration test category from E .Remark 5.4.Observe that for the fibration test category (C, T) from the previous example, the completed model structure on Pro(C) is a special case of Theorem 1.1, namely the case where T is the collection of all fibrant objects in C. By (the dual of) Example 3.7, it follows that we can take T to be any collection of fibrant objects in C that is closed under "finite pullback-powers".The general case, where we let T be any collection of fibrant objects in C, is discussed in Section 8.
Example 5.5.Recall that we call a simplicial set lean if it is degreewise finite and coskeletal.The full subcategory of sSet spanned by all lean simplicial sets L is closed under finite limits and finite cotensors.By Example 5.3, it inherits the structure of a fibration test category from sSet KQ , which we will denote by L KQ .By Theorem 5.2 we obtain a model structure on Pro(L).Since this category is equivalent to the category of simplicial profinite sets s Set by (the dual of) Theorem 2.3, we in particular obtain a simplicial model structure on s Set.This model structure coincides with Quick's model structure for profinite spaces [Qui08], as explained in Corollary 6.6 below.We denote it by s Set Q .
Example 5.6.Consider the full simplicial subcategory T p of sSet whose objects are those lean Kan complexes that have finite p-groups as homotopy groups.One can show that T p is closed under "finite pullback-powers", so by the previous example and the dual of Example 3.7, we obtain a fibration test category L p = (L, T p ) in which the (trivial) fibrations are the (trivial) Kan fibrations between objects of T p .It is proved in Corollary 6.7 that the completed model structure on Pro(L p ) agrees with Morel's model structure for pro-p spaces [Mor96].
Example 5.7.The category of lean simplicial sets also inherits the structure of a fibration test category from the Joyal model structure sSet J , which we will denote by L J .The corresponding model structure on s Set obtained from Theorem 5.2 will be called the profinite Joyal model structure and its fibrant objects will be called profinite quasi-categories.We will come back to this model category in Section 9, and we will describe its underlying ∞-category in Remark A.11.
Example 5.8.In [Hai19], Haine defines the Joyal-Kan model structure on sSet /P , where P is (the nerve of) a poset.This model category describes the homotopy theory of Pstratified spaces.Since it is a left Bousfield localization of the Joyal model structure on sSet /P , any object is cofibrant and it is a sSet J -enriched model category.Actually, this model structure can be shown to be sSet KQ -enriched [Hai19, § §2.4-5].In particular, any small full subcategory C closed under finite limits and cotensors by finite simplicial sets inherits the structure of a fibration test category.If P is a finite poset and C = L /P is the full subcategory of lean simplicial sets over (the nerve of) P, then one can show that Pro(L /P ) ∼ = s Set /P .In particular, by Theorem 5.2, we obtain a model structure on s Set /P that is sSet KQ -enriched.It is shown in Example A.8 that the underlying ∞-category of this model category is the ∞-category of profinite P-stratified spaces defined in [BGH20].
Example 5.9.We call a groupoid finite if it has finitely many arrows (including the identity arrows).The category of finite groupoids FinGrpd inherits the structure of a fibration test category from the canonical model structure on Grpd [And78,§5].(Note that Grpd can be viewed as a sSet KQ -enriched model structure by defining Map(A, B) = N(Fun(A, B)) for any A, B ∈ Grpd.)The completed model structure on the category of profinite groupoids Grpd = Pro(FinGrpd) obtained from Theorem 5.2 coincides with the model structure for profinite groupoids defined by Horel in [Hor17, §4].To see this, note that Horel shows in [Hor17, §4] that the Barnea-Schlank model structure on Grpd exists and coincides with his model structure.By Remark 5.12 below, the Barnea-Schlank model structure on Grpd must coincide with our model structure.In particular, Horel's model structure agrees with the one that we construct in this example.
Example 5.10.Similarly, we call a category finite if it has finitely many arrows.The category of all categories admits the canonical model structure, defined for example in [Rez00].Since this model structure is sSet J -enriched, the category of finite categories FinCat inherits the structure of a fibration test category.By Theorem 5.2, we obtain a sSet J -enriched model structure on Cat = Pro(FinCat) which we will call the model structure for profinite categories.
Example 5.11.Let bisSet be endowed with the Reedy model structure with respect to the Kan-Quillen model structure on sSet.Recall that the category of bisimplicial profinite sets bis Set is equivalent to Pro(L (2) ), where L (2) denotes the category of doubly lean bisimplicial sets defined at the end of Section 2.2.Since any object in bisSet is cofibrant, L (2) inherits the structure of a fibration test category from the Reedy model structure on bisSet.By applying Theorem 5.2, we obtain a model structure on Pro(L (2) ) bis Set.This model structure coincides with the Reedy model structure on bis Set with respect to the Quick model structure on s Set, as will be shown in Proposition 6.9.
Remark 5.12.As discussed in the introduction, there are similarities between our construction of a model structure on Pro(C) and the construction of Barnea-Schlank in [BS16a].Suppose C is a fibration test category in the sense of Definition 5.1.Then C comes with a set fib(C) of fibrations and a set of we(C) of weak equivalences.It is very unlikely that the triple (C, fib(C), we(C)) is a "weak fibration category" in the sense of Definition 1.2 of [BS16a].Namely, that definition asks that fib(C) contains all isomorphisms of C, that it is closed under composition, and that a pushout of a map in fib(C) is again in fib(C).However, if we define fib (C) to be the smallest set that contains fib(C) and that satisfies these properties, then (C, fib (C), we(C)) might be a weak fibration category.If this is the case, then the "induced" model structure on Pro(C), in the sense of Theorem 1.8 of [BS16a], could exist.The cofibrations of this model structure are defined as the maps that have the left lifting property with respect to fib (C) ∩ we(C), while the trivial cofibrations are the maps that have the left lifting property with respect to fib (C).
Since the maps in fib (C) are clearly fibrations in our construction of the "completed model structure" on Pro(C) (see Theorem 5.2), we conclude that the (trivial) cofibrations for both model structures must agree.In particular, if both our model structure and the Barnea-Schlank model structure of [BS16a] exist on Pro(C), then they must coincide.An example where this happens is when C = FinGrpd.(See Example 5.9 above.)

Comparison to some known model structures
As stated in Theorem 5.2, for any fibration test category (C, T), all objects in the completed model structure on Pro(C) are cofibrant.We will now show that, in the case that C is the category L of lean simplicial sets, this statement can often be strengthened to say that the cofibrations are exactly the monomorphisms.We show how this can be used to prove that the model structures on s Set obtained in Example 5.5 and Example 5.6 agree with Quick's model structure and Morel's model structure, respectively.It will also follow that the cofibrations in the profinite Joyal model structure from Example 5.7 are exactly the monomorphisms.We conclude this section by showing that the model structure on bis Set from Example 5.11 agrees with the Reedy model structure on bis Set with respect to Quick's model structure on s Set.The main result about cofibrations in s Set is the following: Proposition 6.1.Let L be the category of lean simplicial sets endowed with the structure of a fibration test category.Suppose that for any contractible lean Kan complex K, the map K → * is a trivial fibration in L, and further that any trivial fibration L ∼ K in L is a trivial Kan fibration.Then the cofibrations in the completed model structure on Pro(L) s Set are the monomorphisms.
This proposition clearly applies to the fibration test categories L KQ , L p and L J of Examples 5.5 to 5.7.The following lemmas will be used to prove this result.Recall that the category of profinite sets Set is equivalent to the category of Stone spaces Stone.Lemma 6.2.A map of profinite sets (or simplicial profinite sets) S → T is a monomorphism if and only if it is (isomorphic to) the limit of a cofiltered diagram {S i T i } i∈I consisting of monomorphisms between finite sets (or degreewise finite simplicial sets, respectively).
Proof.Note that in the category of Stone spaces, the monomorphisms are precisely the injective continuous maps.Since a cofiltered limit of injective maps is again injective, we see that if S → T is an inverse limit of monomorphisms S i T i , then S → T is itself a monomorphism.
Conversely, suppose that S → T is a monomorphism of profinite sets (or simplicial profinite sets).Write T = lim i T i as a cofiltered limit of finite sets (or lean simplicial sets, respectively), and, for every i, write S i for the image of the composition S → T → T i .Then {S i } i∈I is a cofiltered diagram since the structure maps T i → T j restrict to maps S i → S j for any i → j in I. Since {S i → T i } i∈I is levelwise a monomorphism, the proof is complete if we can show that S → lim i S i is an isomorphism.Since isomorphisms of Stone spaces are detected on the underlying sets, it suffices to show that this map is both injective and surjective.It is injective since the composition S → lim i S i → T is, while it is surjective by [RZ10, Corollary 1.1.6] We will denote the two-element set {0, 1} by 2. Lemma 6.3.A map of (profinite) sets S → T is a monomorphism if and only if it has the left lifting property with respect to 2 → * .
Proof.We leave the case where S → T is a map of sets to the reader.For the "if" direction in the profinite case, suppose that f : S → T has the left lifting property with respect to 2 → * , but is not a monomorphism.Regarding S and T as Stone spaces, there must exist distinct s, s ∈ S such that f (s) = f (s ).Choose some clopen U ⊆ S such that s ∈ U and s ∈ U. Then the indicator function 1 U : S → 2 is continuous but does not extend to a map T → 2. We conclude that S → T must be a monomorphism.
For the converse, note that by Lemma 6.2 we may assume without loss of generality that S → T can be represented by levelwise monomorphisms {S i → T i }.Since 2 is cocompact in Set, any map S → 2 factors through S i for some i.Since S i → T i is a monomorphism of sets, the result follows.− → ∆ op along * 2 − → Set and along * 2 − → Set.In particular, a map of simplicial (profinite) sets X → Y has the left lifting property with respect to R n 2 → * if and only if X n → Y n has the left lifting property with respect to 2 → * , hence the result follows from Lemma 6.3.
Proof of Proposition 6.1.We first show that any cofibration in the model category Pro(L) is a monomorphism.Since R n 2 → * has the right lifting property with respect to all monomorphisms in sSet, we see that it is a trivial Kan fibration, hence by assumption a trivial fibration in the fibration test category L and a generating trivial fibration in Pro(L).By Lemma 6.4, any cofibration in Pro(L) s Set is a monomorphism.
For the converse, suppose X → Y is a monomorphism in Pro(L).By Lemma 6.2, we may assume that X → Y is a cofiltered limit of monomorphisms between degreewise finite simplicial sets {X i → Y i } i∈I .We see that for every i, the map X i → Y i has the left lifting property with respect to the generating trivial fibrations of Pro(L) since these are trivial Kan fibrations between lean simplicial sets.Since any generating trivial fibration is a map between cocompact objects, it follows that X → Y also has the left lifting property with respect to the generating trivial fibrations.Proposition 6.1 shows that, for L KQ and L p the fibration test categories of Examples 5.5 and 5.6, the cofibrations of the model categories Pro(L KQ ) and Pro(L p ) are the monomorphisms.This means that the cofibrations coincide with those of Quick's model structure [Qui08] and Morel's model structure [Mor96], respectively.The same is true for the weak equivalences.This follows from the results in §7 of [BHH17]  The proof of Proposition 6.1 admits an analogue for bisimplicial sets (in fact, for the category of presheaves on K for any small category K that can be written as a union of finite full subcategories), which we leave as an exercise to the reader.Proposition 6.8.Let bisSet be endowed with a simplicial model structure in which the cofibrations are the monomorphisms, and let L (2) be the full subcategory of doubly lean bisimplicial sets, which inherits the structure of a fibration test category in the sense of Example 5.3.Then the cofibrations in Pro(L (2) ) bis Set are the monomorphisms.
Note that this proposition implies that the cofibrations in the model structure on bis Set from Example 5.11 are exactly the monomorphisms.We will show that, in fact, this model structure coincides with the Reedy model structure on bis Set with respect to s Set Q .We do this by inspecting the generating (trivial) fibrations of the Reedy model structure.For the following proof, note that Quick's model structure coincides with the completed model structure on Pro(L KQ ) by Corollary 6.6, hence that any (trivial) Kan fibration between lean Kan complexes is a (trivial) fibration in Quick's model structure.
Proposition 6.9.The model structure on Pro(L (2) ) of Example 5.11 coincides with the Reedy model structure on bis Set (with respect to s Set Q ).
Proof.Note that if L → K is a (trivial) Reedy fibration between Reedy fibrant doubly lean bisimplicial sets, then L n and M n L × M n K K n are lean Kan complexes for every n.In particular, the map L n → M n L × M n K K n is a (trivial) fibration between lean Kan complexes for every n.This shows that any generating (trivial) fibration in Pro(L (2) ) is a (trivial) fibration in the Reedy model structure on bis Set.
For the converse, note that the Reedy model structure on bis Set is fibrantly generated.Its generating (trivial) fibrations are maps of the form for any n ≥ 0, where G n is the right adjoint to the functor X → X n , where ∂G n is the right adjoint to latching object functor X → L n X, and where L → K is a generating (trivial) fibration in s Set.It can be shown using the right adjointness of G n and ∂G n that these functors restrict to functors L → L (2) .One can furthermore deduce from the adjointness that if L and K are fibrant in sSet, then both the domain and codomain of the map (3) are Reedy fibrant in bis Set and hence in bisSet.This shows that any map of the form (3), with L → K a (trivial) fibration in L, is a (trivial) fibration in L (2) .In particular, any generating (trivial) fibration in the Reedy model structure on bis Set is a (trivial) fibration in Pro(L (2) ).We conclude that both model structures coincide.

Quillen pairs
As explained in Section 2.2, there is an easy criterion for constructing adjunctions between ind-categories: if C is a small category that admits finite colimits and if E is any cocomplete category, then a functor F : Ind(C) → E has a right adjoint if and only it preserves all colimits.Furthermore, these functors correspond to functors C → E that preserve all finite colimits.There is a dual criterion for pro-categories.In the simplicial case, this can be strengthened as in the following lemma.
If E is a tensored cocomplete simplicial category, then we say that colimits and tensors commute in E if the analogue of item (iii) of Definition 3.1 holds for all diagrams in E and all simplicial sets.Lemma 7.1.Let C be a small finitely tensored simplicial category and let E be a tensored cocomplete simplicial category in which colimits and tensors commute.Then any simplicial functor F : C → E that preserves finite colimits and tensors with finite simplicial sets extends to a functor F : Ind(C) → E that admits a right adjoint.Moreover, this adjunction is an enriched adjunction.
Proof.The simplicial functor F : Ind(C) → E is defined on objects by F({c i }) = colim i F(c i ) and on the internal homs by Map({c We saw in the preliminaries that F preserves all colimits and has a right adjoint (as functor of unenriched categories).In particular, it is part of an enriched adjunction if and only if it preserves tensors.To see that this is the case, let X = colim j X j be a simplicial set written as a filtered colimit of finite simplicial sets.Then {c using the hypothesis that F preserves tensors with finite simplicial sets.
In this section we give some assumptions under which an adjunction of the type above is a Quillen adjunction, and give a further criterion for this adjunction to be a Quillen equivalence.This gives a straightforward way of constructing "profinite" versions of certain classical Quillen adjunctions, as illustrated in Example 7.7.At the end of this section, we show that if C ⊆ E inherits the structure of a (co)fibration test category in the sense of Example 3.6, then the ind-or pro-completion functor (relative to C) is a Quillen functor.
Example 7.3.The nerve functor N : FinGrpd → L KQ is a morphism of fibration test categories.Similarly, taking the nerve of a category gives a morphism of fibration test categories N : FinCat → L J .
Remark 7.6.One could weaken the definition of a morphism of (co)fibration test categories φ : C 1 → C 2 by only asking it to be an (unenriched) functor of underlying categories and not asking it to preserve (co)tensors.In this case, one would still obtain a Quillen adjunction between the completed model structures, but it would merely be a Quillen adjunction between the underlying model categories, and not a simplicial one.Moreover, the proof of Proposition 7.8 below would not go through in this case.
Example 7.7.The nerve functors from Example 7.3 induce simplicial Quillen adjunctions Π 1 : s Set Q Grpd : N and h : s Set J Cat : N.These left adjoints are profinite versions of the fundamental groupoid and the homotopy category, respectively.
We call the restriction φ : T 1 → T 2 of a morphism of cofibration test categories homotopically essentially surjective if for any t ∈ T 2 , there exists a t ∈ T 1 together with a weak equivalence φ(t) ∼ −→ t in T 2 .
(b) In the case of a morphism of cofibration test categories, if moreover for any t ∈ T 1 and c ∈ C 1 the map is a weak equivalence, then the induced Quillen adjunction of Proposition 7.5 is a Quillen equivalence.
(b') In the case of fibration test categories, if φ is homotopically essentially surjective and for any t ∈ T 1 and c ∈ C 1 , the map is a weak equivalence, then the induced Quillen adjunction of Proposition 7.5 is a Quillen equivalence.
Proof.We again only include a proof for cofibration test categories, as the case of a morphism of fibration test categories is dual.For item (a), let f : C → D be a map in Ind(C 2 ) and suppose that φ * ( f ) is a weak equivalence in Ind(C 1 ).
For item (b), since the right adjoint φ * detects weak equivalences by part (a), it suffices to show that the unit C → φ * φ !C is a weak equivalence for every cofibrant C in Ind(C 1 ).Since C is an cofiltered limit of objects in C 1 , by Remark 7.4 it is enough to show that c → φ * φ !c is a weak equivalence for every c in C 1 .By definition of the weak equivalences in Ind(C 1 ) and by the simplicial adjunction φ !φ * , this is equivalent to being a weak equivalence, which holds by assumption.
An interesting consequence of Proposition 7.8 is that if, for a (co)fibration test category (C, T), one "enlarges" C to a bigger category C but keeps T the same, then one obtains Quillen equivalent model structures on Ind(C) and Ind(C ) (or Pro(C) and Pro(C )).The next example gives an illustration of this.
Example 7.9.Recall from Example 5.5 that the category of lean simplicial sets L inherits the structure of a fibration test category from sSet KQ .We could give the category of degreewise finite simplicial sets sFinSet a similar structure of a fibration test category, namely by defining the test objects to be the lean Kan complexes and the (trivial) fibrations to be those of L KQ .That is, the test objects and the (trivial) fibrations of sFinSet and of L KQ are identical.It is well known that the pro-categories Pro(sFinSet) and Pro(L) s Set are not equivalent.However, the inclusion ι : L KQ → sFinSet is a morphism of fibration test categories that satisfies item (b) of Proposition 7.8, hence the induced adjunction The hypotheses for item (b) of Proposition 7.8 can usually be weakened, namely if T is "large enough" in the following sense.Definition 7.10.Let (C, T) be a cofibration test category.We say that T is closed under pushouts along cofibrations if, for any cofibration r s in T and any map r → t in T, the pushout s ∪ r t is again contained in T.
Dually, for a fibration test category (C, T), we say that T is closed under pullbacks along fibrations if, for any fibration s r and any map t → r in T, the pullback s × r t is again contained in T.
This definition can be seen as ensuring that T has all finite homotopy (co)limits.If T is closed under pushouts along cofibrations, then it is enough to assume in item (b) that the restriction φ : T 1 → T 2 is homotopically fully faithful, i.e. that Map(s, t) → Map(φ(s), φ(t)) is a weak equivalence for all s, t ∈ T 1 .The main ingredient is the following useful lemma.
Lemma 7.11.Let (C, T) be a cofibration test category and suppose that T is closed under pushouts along cofibrations.Then any cofibrant object in Ind(C) is a filtered colimit of objects in T.
Proof.The "fat small object argument" of [MRV14] shows that if C in Ind(C) is cofibrant, then it is a retract of a colimit colim i∈I c i indexed by a directed poset I that has a least element ⊥, such that c ⊥ is the initial object ∅ and such that c ⊥ → c i is a (finite) composition of pushouts of generating cofibrations for any i.(This follows from Theorem 4.11 of [MRV14] together with the fact that all objects in T are compact.)In particular, since T is closed under pushouts along cofibrations, it follows that c i ∈ T for every i ∈ I. Since ind-categories are idempotent complete, it follows that any retract of such a colimit is an object of Ind(T) as well.In particular, any cofibrant object of Ind(C) lies in Ind(T).
We leave it to the reader to dualize Lemma 7.11 to the context of fibration test categories.
Proposition 7.12.Let φ : (C 1 , T 1 ) → (C 2 , T 2 ) be a morphism of cofibration test categories (or fibration test categories) and suppose that T 1 is closed under pushouts along cofibrations (or closed under pullbacks along fibrations, respectively).If the restriction φ : T 1 → T 2 is homotopically essentially surjective and homotopically fully faithful, then the induced Quillen adjunction of Proposition 7.5 is a Quillen equivalence.
Proof.We prove the statement for ind-categories.As in the proof of Proposition 7.8, it suffices to show that the unit C → φ * φ !C is a weak equivalence for every cofibrant C in Ind(C 1 ).By Lemma 7.11 any cofibrant object is a filtered colimit of objects of T 1 , so by Remark 7.4 it suffices to show that t → φ * φ !t is a weak equivalence for every t ∈ T 1 .This follows exactly as in the proof of Proposition 7.8.
Recall from Section 2.2 that if E is a complete category and if C ⊆ E is a small full subcategory closed under finite limits, then the functor U : Pro(C) → E that sends a proobject to its limit in E has a left adjoint (•) Pro , the pro-C completion functor.Dually, if E is cocomplete and C is closed under finite colimits, then the canonical functor U : Ind(C) → E has a right adjoint (•) Ind .In the situation where E is a simplicial model category and C is a (co)fibration test category, these adjunctions are almost by definition Quillen pairs.Note that in the case of pro-categories, this is the Quillen pair mentioned in item (3) of Theorem 1.1.Proposition 7.13.Let E be a simplicial model category in which every object is fibrant and C ⊆ E a full subcategory closed under finite colimits and finite tensors with the inherited structure of a cofibration test category (in the sense of Example 3.6).Then U : Ind(C) E : (•) Ind is a simplicial Quillen adjunction.Dually, if every object in E is cofibrant and C ⊆ E is a full subcategory closed under finite limits and finite cotensors, given the inherited structure of a fibration test category (as in Example 5.3), then Proof.The first adjunction arises by applying Lemma 7.1 to the inclusion C → E. We need to show that the left adjoint U preserves the generating (trivial) cofibrations.Note that U agrees with the inclusion C → E when restricted to C ⊆ Ind(C).Since the generating (trivial) cofibrations are defined as the (trivial) cofibrations in C ⊆ E between cofibrant objects, they are preserved by U.
The case for pro-C completion follows dually.
Example 7.14.The proposition above shows that the profinite completion functors for sSet KQ and Grpd are left Quillen.These Quillen adjunctions fit into a commutative diagram where N is the nerve adjunction from Example 7.7.There is a similar diagram of Quillen adjunctions for the (profinite) Joyal model structure and the model category of (profinite) categories.

Bousfield localizations
Suppose we are given a cofibration test category (C, T) and that we wish to shrink the full subcategory of test objects T to a smaller one T ⊆ T. If T is closed under finite pushout-products, then (C, T ) is a cofibration test category by Example 3.7, hence we obtain two model structures Ind(C, T) and Ind(C, T ) on the category Ind(C).Since the (trivial) cofibrations of (C, T ) are those of (C, T) between objects of T , the sets of generating (trivial) cofibrations of Ind(C, T ) are contained in those of Ind(C, T).In particular, the identity functor is right Quillen when viewed as a functor Ind(C, T) → Ind(C, T ).Since there are fewer weak equivalences in Ind(C, T) than in Ind(C, T ), this right Quillen functor is close to being a right Bousfield localization.Recall that a right Bousfield localization of a model category is a model structure on the same category with the same class of fibrations, but with a larger class of weak equivalences.The model category Ind(C, T ) is not necessarily a right Bousfield localization of Ind(C, T) since it has fewer generating trivial cofibrations, and hence it might have more fibrations than Ind(C, T).However, it is a general fact about model categories that in such a situation, there exists a model structure on Ind(C) with the weak equivalences of Ind(C, T ) and the fibrations of Ind(C, T).
Lemma 8.1.Let E α and E β be cofibrantly generated model structures on the same category E and suppose that sets of generating cofibrations I α and I β and sets of generating trivial cofibration J α and J β respectively, are given.If I α ⊆ I β and J α ⊆ J β , and if E α has more weak equivalences than E β , then there exists a cofibrantly generated model structure on E with the weak equivalences of E α and the fibrations of E β .
Proof.It easily follows by checking the hypotheses of Theorem 11.3.1 of [Hir03] that the sets I α ∪ J β and J β determine a cofibrantly generated model structure on E in which the weak equivalences agree with those of E α .This model structure has the desired properties.As an example, we check item (4b) of Theorem 11.3.1 of [Hir03], and leave the other hypotheses to the reader.This comes down to showing that if E → F has the right lifting property with respect to J β and is a weak equivalence in E α , then it must have the right lifting property with respect to I α ∪ J β .It suffices to show that E → F has the right lifting property with respect to I α .Since E → F has the right lifting property with respect to J β , it has so with respect to J α ⊆ J β , hence it is a fibration in E α .Since it is also a weak equivalence in E α , it follows that it has the right lifting property with respect to I α and hence with respect to If E is a simplicial model category with a given full subcategory T ⊆ E , then R T E denotes (if it exists) the right Bousfield localization of E in which a map E → E is a weak equivalence if and only if Map(t, E) → Map(t, E ) is a weak equivalence for every t ∈ T. We call such a map a T-colocal weak equivalence.Dually, L T E denotes (if it exists) the left Bousfield localization of E in which E → E is a weak equivalence if and only if Map(E , t) → Map(E, t) is a weak equivalence for every t ∈ T. Such a map is called a T-local weak equivalence.Proposition 8.2.Let (C, T) be a cofibration test category and let T ⊆ T be a full subcategory.Then the right Bousfield localization R T Ind(C) exists and is cofibrantly generated.
Dually, if (C, T) is a fibration test category and T ⊆ T a full subcategory, then the left Bousfield localization L T Pro(C) exists and is fibrantly generated.
Proof.We first prove the proposition in the special case that T is closed under finite pushout-products, and then deduce the general case from this.In this special case, (C, T ) is a cofibration test category as in Example 3.7, so we obtain a cofibrantly generated model category on Ind(C, T ) in which the weak equivalences are the T -colocal ones.We also have the model structure on Ind(C) corresponding to the cofibration test category (C, T), which by construction has more generating (trivial) cofibrations than Ind(C, T ).By applying Lemma 8.1, we obtain the desired right Bousfield localization R T Ind(C).

Example: complete Segal profinite spaces vs profinite quasicategories
Recall that in Example 5.7, we defined the profinite Joyal model structure.In this section, we will define another candidate for the homotopy theory of profinite ∞-categories, namely a profinite version of Rezk's model category of complete Segal spaces.We then show that there are two Quillen equivalences between the model category of complete Segal profinite spaces and the profinite Joyal model structure.After establishing these Quillen equivalences, we characterize in both these model categories the weak equivalences between the fibrant objects as the essentially surjective and fully faithful maps, where fully faithfulness is defined in terms of the Quick model structure.It is worth mentioning that in Remark A.11, we moreover give a precise description of the underlying ∞-category of these model categories.
Let us start with a short review of the theory of complete Segal spaces, originally defined by Rezk in [Rez01].Consider the category bisSet = sSet ∆ op of bisimplicial sets, or simplicial spaces, equipped with the Reedy model structure (with respect to the Kan-Quillen model structure on sSet).We denote this model category by bisSet R .Objects of bisSet have two simplicial parameters.We denote the "inner" one by n, m, . . .and refer to it as the space parameter, and we denote the "outer" one (corresponding to the ∆ op in sSet ∆ op ) by s, t, r, . ... For any pair of simplicial sets X and Y, one can define the external product X × Y by (X × Y) t,n = X t × Y n .Note that the external product ∆ t × ∆ n is the functor ∆ op × ∆ op → Set represented by ([t], [n]).In particular, the internal hom of bisSet can be defined by (Y X ) t,n = Hom((∆ t × ∆ n ) × X, Y).This internal hom allows one to regard bisSet as a simplicial category in multiple ways; the two simplicial enrichments that we will use are given by Map 1 (X, Y) := (Y X ) •,0 and Map 2 (X, Y) := (Y X ) 0,• .
The category bisSet is tensored and cotensored with respect to both of these enrichments. As gives the model category bisSet CSS for complete Segal spaces.Here J is the nerve of the groupoid with two objects and exactly one isomorphism between any ordered pair of objects.It is part of a cosimplicial object J • in sSet, J t being the nerve of the groupoid with t + 1 objects and exactly one isomorphism between any ordered pair of objects.
All three of the model structures bisSet R , bisSet SS and bisSet CSS are sSet KQ -enriched model structures with respect to the enrichment Map 2 mentioned above.
The model category bisSet CSS is Quillen equivalent to sSet J .In fact, there are Quillen pairs in both directions, whose right Quillen functors are the evaluation at the inner coordinate n = 0 ev 0 : bisSet → sSet; (ev 0 X) t = X t,0 and the singular complex functor with respect to J • Sing J : sSet → bisSet; Sing J (X) t,n = Map(J n , X) t = Hom(∆ t × J n , X).
These Quillen equivalences are described in detail in [JT07].One can prove, using the Quillen equivalence ev 0 together with the fact that bisSet CSS is a cartesian closed model category, that bisSet CSS is a sSet J -enriched model category with respect to the simplicial enrichment Map 1 mentioned above.Both of the above right Quillen functors are simplicial functors that preserve cotensors with respect to this simplicial enrichment.This is explained in detail in the proof of Proposition E.2.2 of [RV22].Now let L (2) be the category of doubly lean bisimplicial sets, i.e. those bisimplicial sets X for which X t,n is finite for each t and n, and such that X ∼ = cosk t,n (X) for some t and n.Here cosk t,n : bisSet → bisSet is the functor that restricts X ∈ bisSet to a functor ∆ op ≤t × ∆ op ≤n → Set and then right Kan extends along ∆ op ≤t × ∆ op ≤n → ∆ op × ∆ op .This agrees with the notion of doubly lean as defined at the end of Section 2.2, and it follows from (the dual of) Theorem 2.3 that the inclusion L (2) → bis Set extends to an equivalence Pro(L (2) ) bis Set.
Each of the three model structures bisSet R , bisSet SS and bisSet CSS gives rise to the structure of a fibration test category on L (2) by the general scheme of Example 5.3.We will mainly be interested in the Reedy and the complete Segal model structures, so denote the corresponding fibration test categories by L (2) CSS will be called the Reedy model structure for profinite spaces and model structure for complete Segal profinite spaces, and denoted bis Set R and bis Set CSS , respectively.A fibrant object in bis Set CSS will be called a complete Segal profinite space.
Since we can view bisSet CSS as a simplicial model category in two ways, the full subcategory L (2) CSS inherits two different structures of a fibration test category, namely one with respect to the enrichment Map 1 and one with respect to Map 2 .The (trivial) fibrations of both fibration test category structures agree, so they will induce the same model structures on Pro(L (2) ) ∼ = bis Set.This shows that we can view bis Set CSS as a sSet J -enriched model category through the enrichment Map 1 , and as a sSet KQ -enriched model category through Map 2 . 1 In what follows, we will consider the simplicial enrichment Map 1 , since this one is compatible with the right Quillen functors ev 0 and Sing J discussed above.By Proposition 7.13, the profinite completion functor bisSet → bis Set is a left Quillen functor, whose right adjoint is given by the functor U : bis Set → bisSet that sends a bisimplicial profinite set to its underlying bisimplicial set.Levelwise, this is the functor that sends a profinite set to its underlying set.

Since L
(2) CSS has fewer test objects than L (2) R , we see that bis Set CSS has more weak equivalences than bis Set R .By Proposition 6.8, the cofibrations are the monomorphisms in both model structures, hence bis Set CSS is a left Bousfield localization of bis Set R .In particular, the construction of the model structure bis Set CSS given in Example 8.4 agrees with the one given here.
The right Quillen functors ev 0 and Sing J mentioned above restrict to morphisms of fibration test categories between L J and L (2) CSS , where L J is the category of lean simplicial sets (with the fibration test category structure from Example 5.7).This amounts to showing that ev 0 maps doubly lean bisimplicial sets to lean simplicial sets, and that Sing J maps lean simplicial sets to doubly lean bisimplicial sets.In the case of ev 0 , this follows directly from the definition, while the case of Sing J requires some work.
Proof.Let X be a lean simplicial set and suppose that X is n-coskeletal.It suffices to show that Sing J (X) •,m and Sing J (X) t,• are both n-coskeletal and degreewise finite simplicial sets for any t, m ∈ N. Since J m is a degreewise finite simplicial set for every m, we see that Sing J (X) •,m = Map(J m , X) is an n-coskeletal degreewise finite simplicial set for every m.This automatically shows that Sing J (X) t,• is a degreewise finite simplicial set as well.It therefore remains to show that, for every n-coskeletal simplicial set X and every t, the simplicial set Sing J (X) t,• ∼ = Hom(J • × ∆ t , X) ∼ = Hom(J • , X ∆ t ) is n-coskeletal.Since any cotensor X Y of an n-coskeletal simplicial set X is again n-coskeletal, it suffices to prove the case t = 0. To this end, let ∂J k+1 denote the simplicial subset or equivalently, the left Kan extension of J • : ∆ → sSet along the Yoneda embedding ∆ → sSet, evaluated at ∂∆ k+1 ∈ sSet.The inclusion ∂J k+1 → J k+1 restricts to an isomorphism sk n ∂J k+1 → sk n J k+1 for any k ≥ n.Combining this with the canonical isomorphism Hom(∂∆ k+1 , Hom(J • , X)) ∼ = Hom(∂J k+1 , X), it follows that Hom(J • , X) is n-coskeletal.
Denote the profinite Joyal model structure by s Set J .We can apply Proposition 7.12 to ev 0 : CSS to show that the induced functors between s Set J and bis Set CSS are right Quillen equivalences.We will denote these functors by ev 0 and Sing J as well.
Proposition 9.3.The functors ev 0 : bis Set CSS → s Set J and Sing J : s Set J → bis Set CSS are right Quillen equivalences.
Proof.Since there is a natural isomorphism ev 0 Sing J (X) ∼ = X, it suffices to show that ev 0 : bis Set CSS → s Set J is a right Quillen equivalence.The same then follows for Sing J by the 2 out of 3 property.Since ev 0 : bisSet CSS → sSet J is a (simplicial) right Quillen equivalence, its restriction ev 0 : L (2) CSS → L J is a morphism of fibration test categories that is homotopically fully faithful when restricted to test objects.Furthermore, it is homotopically essentially surjective since X ∼ = ev 0 (Sing J X) for any lean quasi-category X.By Proposition 7.12, we conclude that induced functor ev 0 : bis Set CSS → s Set J (and hence Sing J : s Set J → bis Set CSS ) is a right Quillen equivalence.
One can prove "profinite versions" of many of the properties that complete Segal spaces enjoy.The general strategy for proving such a profinite version of a given property is to reduce it to its classical counterpart.We will illustrate this by showing that the weak equivalences between complete Segal profinite spaces coincide with (a profinite version of) the Dwyer-Kan equivalences.This is done by exploiting two facts: that bis Set CSS is a left Bousfield localization of the Reedy model structure bis Set R (with respect to s Set Q ), and that the weak equivalences between fibrant objects in s Set Q can be detected underlying in sSet KQ .Denote the functor that sends a simplicial profinite set to its underlying simplicial set by U : s Set → sSet.Note that this functor is right Quillen as functor from Quick's model structure to the Kan-Quillen model structure, and that its left adjoint is the profinite completion functor.
Proposition 9.4.A map X → Y between fibrant objects in s Set Q is a weak equivalence if and only if UX → UY is a weak equivalence in sSet KQ .
Proof.This follows from Theorem E.3.1.6 of [Lur], which states that the functor between the underlying ∞-categories of s Set Q and sSet KQ induced by U (which is called "Mat" by Lurie) is conservative.Another way to deduce this proposition is to show that the weak equivalences between fibrant objects in s Set Q are the π * -isomorphisms (as in the proof of Proposition 3.9 of [BHR19]) and that the the underlying group/set Uπ n (X, x) of the profinite group/set π n (X, x) agrees with π n (UX, x) for any fibrant X ∈ s Set Q and any x ∈ X 0 .
Since bis Set CSS is a left Bousfield localization of the model category bis Set R , which coincides with the Reedy model structure on bis Set with respect to s Set Q by Proposition 6.9, we see that a map between complete Segal profinite spaces is a weak equivalence if and only if it is levelwise a weak equivalence in s Set Q .In particular, we obtain the following result: Proposition 9.5.A map X → Y between complete Segal profinite spaces is a weak equivalence if and only if for every t, the map X t,• → Y t,• is a weak equivalence in s Set Q .In particular, X → Y is a weak equivalence between complete Segal profinite spaces if and only if UX → UY is a weak equivalence between complete Segal spaces.
For a complete Segal profinite space X and two objects x, y ∈ X 0,0 , i.e. two maps ∆ 0 × ∆ 0 → X, we can mimic the classical definition of the mapping space map X (x, y) by defining map X (x, y) as the pullback • is a fibration in s Set Q , and hence map X (x, y) is a fibrant object in s Set Q .Since U : bis Set CSS → bisSet CSS preserves limits, we see that U(map X (x, y)) ∼ = map UX (x, y) for any complete Segal profinite space X.If f : X → Y is a map between complete Segal profinite spaces, then for any x, y ∈ X 0,0 , we obtain a map map X (x, y) → map Y ( f x, f y) from the universal property of the pullback.We call a map between complete Segal profinite spaces f : X → Y fully faithful if, for any x, y ∈ X 0,0 , the map map X (x, y) → map Y ( f x, f y) is a weak equivalence in s Set Q .It follows from Proposition 9.4 that X → Y is fully faithful if and only if UX → UY is a fully faithful map of complete Segal spaces.One can also mimic the classical definitions of a homotopy and of homotopy equivalences in a complete Segal space, and use this to define what it means for a map of complete Segal profinite spaces to be essentially surjective.An equivalent, but easier, way is to say that X → Y is essentially surjective if and only if the induced map π 0 X 0,• → π 0 Y 0,• is an epimorphism of profinite sets.Since Uπ 0 Z ∼ = π 0 UZ for any fibrant object Z in s Set Q , and since epimorphisms of profinite sets are detected underlying, we see that a map of complete Segal profinite spaces X → Y is essentially surjective if and only if UX → UY is so.One can lift Proposition 9.5 and Theorem 9.7 to analogous results about weak equivalences between profinite quasi-categories using the Quillen equivalences ev 0 and Sing J between s Set J and bis Set CSS .Proposition 9.8.A map X → Y between profinite quasi-categories is a weak equivalence in s Set J if and only if UX → UY is a weak equivalence in sSet J .
Proof.Let f : X → Y be a map between profinite quasi-categories.If f is a weak equivalence in s Set J , then U f : UX → UY is a weak equivalence of quasi-categories since U : s Set J → sSet J is right Quillen.Conversely, suppose U f is a weak equivalence of quasi-categories.Then Sing J U f : Sing J (UX) → Sing J (UY) is a weak equivalence between complete Segal spaces.Note that Sing J •U U • Sing J since both functors preserve cofiltered limits and since they agree on lean simplicial sets.By Proposition 9.5, we see that Sing J X → Sing J Y is a weak equivalence between complete Segal profinite spaces.Since ev 0 is right Quillen, we see that the original map ev 0 Sing J X ∼ = X → Y ∼ = ev 0 Sing J Y is a weak equivalence in s Set J .
For a profinite quasi-category X and two 0-simplices x, y ∈ X 0 (i.e.maps ∆ 0 → X), we define map X (x, y) as the pullback map Since the right-hand vertical map is obtained by cotensoring with the cofibration ∂∆ 1 → ∆ 1 , it must be a fibration in s Set J .In particular, map X (x, y) is fibrant in s Set J .One can show that, analogous the classical case, map X (x, y) is actually fibrant in s Set Q .However, the proof of this is technical and not necessary for what follows, so it is not included.A map f : X → Y of profinite quasi-categories induces a morphism map X (x, y) → map Y ( f x, f y) for any x, y ∈ X 0 by the universal property of the pullback.We say that f is fully faithful if map X (x, y) → map Y ( f x, f y) is a weak equivalence in s Set J for any x, y ∈ X 0 .2For a 1-simplex α ∈ X 1 with d 1 α = x and d 0 α = y, i.e a 0-simplex in map X (x, y), we say that α is a homotopy equivalence if ∆ 1 α − → X extends to a map J 1 → X.Here J 1 is viewed as a simplicial profinite set through the inclusion sFinSet → s Set.We say that a map of profinite quasi-categories f : X → Y is essentially surjective if for any y ∈ Y 0 , there exists an x ∈ X 0 and an α ∈ map Y ( f x, y) such that α is a homotopy equivalence.
Since U : s Set → sSet preserves pullbacks, we see that U map X (x, y) ∼ = map UX (x, y).By Proposition 9.8, a map X → Y of profinite quasi-categories is fully faithful if and only if UX → UY is so.Since Hom(J 1 , X) ∼ = Hom(J 1 , UX) for any X ∈ s Set, we also see that X → Y is essentially surjective if and only if UX → UY is so.Definition 9.9.A map between profinite quasi-categories is called a Dwyer-Kan equivalence or DK-equivalence if it is essentially surjective and fully faithful.
Theorem 9.10.A map between profinite quasi-categories is a Dwyer-Kan equivalence if and only if it is a weak equivalence in s Set J .
Proof.A map X → Y of profinite quasi-categories is a DK-equivalence if and only if UX → UY is.Since the weak equivalences between fibrant objects in sSet J are exactly the DK-equivalences, we conclude from Proposition 9.8 that a map of profinite quasicategories X → Y is a DK-equivalence if and only if it is a weak equivalence.

A Comparison to the ∞-categorical approach
The goal of this appendix is to compare the model structures on Ind(C) and Pro(C) constructed in this paper to the ∞-categorical approach to ind-and pro-categories.Since the cases of ind-and pro-categories are dual, we only treat the case of ind-categories and dualize the main result at the end of this appendix.
Given a cofibration test category C, the underlying ∞-category of the completed model structure on Ind(C) will be denoted by Ind(C) ∞ .Recall that this ∞-category is defined as the homotopy-coherent nerve of the full simplicial subcategory spanned by the fibrant-cofibrant objects.We will show that if (C, T) is a cofibration test category with a suitable assumption on T, then the ∞-category Ind(C) ∞ is equivalent to Ind(N(T)).Here N(T) is the homotopy-coherent nerve of the simplicial category T, and Ind denotes the ∞-categorical version of the ind-completion as defined in Definition 5.3.5.1 of [Lur09].
Warning A.1.There is a subtlety here that we should point out: if (C, T) is a cofibration test category with respect to the Joyal model structure on sSet, meaning that items (2) and (3) of Definition 3.3 hold with respect to the trivial cofibrations and weak equivalences of sSet J , then the "mapping spaces" of T are quasi-categories but not necessarily Kan complexes.Recall that any quasi-category X contains a maximal Kan complex, which we will denote by k(X).Since this functor k preserves cartesian products, any category enriched in quasi-categories can be replaced by a category enriched in Kan complexes by applying the functor k to the simplicial hom.If (C, T) is a cofibration test category with respect to sSet J , then we will abusively write N(T) for the simplicial set obtained by first applying the functor k to all the mapping spaces in T, and then applying the homotopycoherent nerve.Similarly, by the underlying infinity category Ind(C) ∞ of Ind(C), we mean the quasi-category obtained by taking the full subcategory on fibrant-cofibrant objects, applying k to all mapping spaces, and then taking the homotopy-coherent nerve.
Since Ind(C) ∞ is the underlying ∞-category of a combinatorial model category, we see that it is complete and cocomplete.Furthermore, since T is a full subcategory of the fibrant-cofibrant objects in Ind(C), we see that the inclusion T → Ind(C) induces a fully faithful inclusion N(T) → Ind(C) ∞ .By Proposition 5.3.5.10 of [Lur09], this inclusion extends canonically to a filtered colimit preserving functor : Ind(N(T)) → Ind(C) ∞ .In order for this functor to be an equivalence, any object in Ind(C) needs to be equivalent to a filtered homotopy colimit of objects in T. This means that T should be "large enough" for this to hold.It turns out that this is the case if T is closed under pushouts along cofibrations (in the sense of Definition 7.10).
Theorem A.2. Let (C, T) be a cofibration test category and suppose that T is closed under pushouts along cofibrations.Then the canonical functor F : Ind(N(T)) → Ind(C) ∞ is an equivalence of quasi-categories.
Remark A.3.Note that in many of the examples discussed in this paper, the category T of test objects is closed under pushouts along cofibrations.For example, this is the case if (C, T) has inherited the structure of a cofibration test category from some model category E in the sense of Example 3.6.
Remark A.4.If (C, T) is a cofibration test category, then one can always "enlarge" the full subcategory T together with the sets of (trivial) cofibrations to obtain a cofibration test category (C, T ) such that T is closed under pushouts along cofibrations, and for which the completed model structures Ind(C, T) and Ind(C, T ) coincide.To see this, note that we can define T to consists of all objects in C that are cofibrant in Ind(C, T), and that we can define the (trivial) cofibrations of (C, T ) to be the trivial cofibrations of Ind(C, T) between objects of T ; that is, we endow C with the structure of a cofibration test category inherited from Ind(C, T) (see Example 3.6).It is then clear that the model structures Ind(C, T) and Ind(C, T ) coincide, and that T is closed under pushouts along cofibrations.In particular, we see by Theorem A.2 that the underlying ∞-category of Ind(C, T) can be described as the ind-category of the small ∞-category N(T ), which contains N(T) as a full subcategory.
Before proving this theorem, we will prove the following rectification result.Lemma A.5.Let (C, T) be a cofibration test category such that T is closed under pushouts along cofibrations, and let I be a poset with the property that I <i is finite for every i.The following lemma is needed for the proof.
Lemma A.6.Let (C, T) be a cofibration test category and let {Y i } i∈I be a diagram in T indexed by a finite poset such that for any i ∈ I, the map is a finite composition of pushouts of cofibrations of (C, T).Then, for any k ∈ I, the map is a finite composition of pushouts of cofibrations.In particular, if T is closed under pushouts along cofibrations, then colim i Y i is an object of T.
Proof.This follows from the dual of [BS16a, Proposition 2.17].For the convenience of the reader, we spell out their argument in our setting.Throughout this proof, we call a map in C good if it is a finite composition of pushouts of cofibrations.Note that any pushout of a good map is again a good map.A subposet S ⊆ I is called a sieve if for any i ∈ S and any j ≤ i in I, one has j ∈ S. Write Y S = colim j∈S Y j for any sieve S and Y <i for Y I <i = colim j<i Y j for any i ∈ I.
We will prove inductively that for two sieves S ⊆ T, the map Y S → Y T is good.This certainly holds if |T| = 0, so suppose this holds for |T| < n and let sieves S ⊆ T with |T| = n be given.If S = T then there is nothing to prove, so suppose that S T and choose some maximal i ∈ T \ S. We then obtain a diagram where the square is a pushout.The map Y <i → Y i is good by assumption while Y S → Y T\{i} is good by the induction hypothesis, so we conclude that Y S → Y T is good.This completes the induction and the lemma now follows by considering the sieves S = I ≤k and T = I.
Proof of Lemma A.5.To distinguish colimits in quasi-categories from homotopy colimits and ordinary colimits in simplicial categories, we will call them ∞-colimits.By a homotopy colimit of a diagram Z : J → T, we mean a cocone Z j → W that induces an equivalence Map(W, t) ∼ −→ holim j∈J Map(Z j , t) for every t ∈ T. The following proof is for the case that (C, T) is a cofibration test category with respect to the Kan-Quillen model structure on sSet.The same proof works if (C, T) is a cofibration test category with respect to sSet J ; however, one has to replace Map(−, −) with the maximal Kan complex k(Map(−, −)) contained in it, and one has to replace ∆ 1 by the simplicial set H (as defined in Lemma 2.1) in the construction of the mapping cylinder below.We will construct the diagram Y : I → T and the equivalence N(Y) X inductively.Let i ∈ I be given and suppose that Y| I <i : I <i → T and N(Y| I <i ) X| N(I <i ) have been constructed and have the desired properties.We need to construct Y| I ≤i : I ≤i → T and an equivalence N(Y I ≤i ) X| I ≤i extending these.Denote colim j<i Y j by Y <i .If I <i is empty, then Y <i is the initial object of C and hence an object of T by definition.If I <i is not empty, then it follows from the assumptions on Y| I <i and Lemma A.6 that Y <i is an object of T. Also note that the assumptions on Y| I <i and the fact that Ind(C) is a simplicial model structure ensure that, for any t ∈ T, the diagram j → Map(Y j , t) is fibrant in the injective model structure on sSet (I <i ) op .In particular, we see that in T, where the second map is a weak equivalence.The first map can be written as a composition of the following two pushouts of cofibrations: Define Y i = Y <i ⊗ ∆ 1 ∪ Y <i ⊗{1} X i .This defines a diagram Y| I ≤i : I ≤i → T. The above factorization of Y <i → X i shows that we obtain a natural equivalence N(Y| I ≤i ) X| N(I ≤i ) extending the equivalence N(Y| I <i ) X| N(I <i ) .
We are now ready to prove Theorem A.2.
Proof of Theorem A.2.The terms "colimit", "homotopy colimit" and "∞-colimit" are used in the same way as in the proof of Lemma A.5.We will denote mapping spaces in a simplicial category by "Map", while mapping spaces in a quasi-category are denoted by "map"; that is, with a lowercase m.We will prove that the functor F : Ind(N(T)) → Ind(C) ∞ is fully faithful and essentially surjective.To see that F is fully faithful, we need to show that map Ind(N(T)) (X, Y) → map Ind(C) ∞ (F(X), F(Y)) is a weak equivalence for any X, Y ∈ Ind(N(T)).Since F preserves filtered ∞-colimits, it suffices to show this for X ∈ N(T).Write Y = colim i Y i as a filtered ∞-colimit of a diagram Y : I → N(T) (which we also denote by Y).By Proposition 5.3.1.18 of [Lur09] and Lemma E.1.6.4 of [Lur], we may assume without loss of generality that I is the nerve of a directed poset, which we also denote by I, with the property that I <i is finite for any i ∈ I.By Lemma A.5, we may replace Y by a strict diagram Z : I → T. Since a diagram as described in Lemma A.5 is cofibrant in the projective model structure on Ind(C) I , we see that the ind-object Z = {Z i } i∈I is the homotopy colimit of the diagram i → Z i .By Theorem 4.2.4.1 of [Lur09], the object Z is an ∞-colimit of the diagram Y : N(I) → Ind(C) ∞ , It is clear that the 0-simplices of Fill(D) correspond to n-simplices in X that fill D. A 1-simplex in Fill(D) between two such n-simplices f , g in X is exactly an (n + 1)-simplex h : ∆ n+1 → X such that d n h = f , d n+1 h = g and d i h = d i s m f for any i < n.Given such an (n + 1)-simplex h, the sequence (s 0 f , s 1 f , . . ., s n−1 f , h) defines a homotopy ∆ n × ∆ 1 → X between f and g relative to ∂∆ n . 3In particular, by minimality of X, the existence of such an (n + 1)-simplex h implies that f = g, and hence the number of elements in π 0 (Fill(D)) equals the number of fillers of D : ∂∆ n → X.
Since X E/ → X ∂E/ is a left fibration and map(∂E, z) is a Kan complex, the restriction map(E, z) → map(∂E, z) is a Kan fibration and hence Fill(D) is the homotopy fiber of map(E, z) map(∂E, z).In particular, if map(E, z) and map(∂E, z) have finite homotopy groups, then Fill(D) does as well, and hence D has finitely many fillers.If we let y denote the top vertex of the (n − 1)-simplex E, then map(E, z) map(y, z), which has finite homotopy groups by assumption.To see that map(∂E, z) has finite homotopy groups, note that map(∂E, z) = lim x∈nd(∂∆ n ) op map(E| x , z) where nd(∂∆ n ) denotes the poset of non-degenerate simplices of ∂∆ n−1 .This follows from the fact that the join of simplicial sets preserves connected colimits.We see that for any x ∈ nd(∂∆ n ), the Kan complex map(E| x , z) is equivalent to map(y, z), where y denotes the top vertex of E| x .In particular, it has finite homotopy groups.Note that the diagram x → map(E| x , z) is injectively fibrant since the diagram {x} x∈nd(∂∆ n ) is cofibrant in the projective model structure on sSet nd(∂∆ n ) .In particular, map(∂E, z) is a finite homotopy limit of spaces with finite homotopy groups, so it has finite homotopy groups as well.We conclude that there are finitely many n-simplices filling D : ∂∆ n → X.
Remark A.11.It follows as in the proofs of Lemmas A.9 and A.10 that a quasi-category is equivalent to a lean quasi-category if and only it has finitely many objects up to equivalence and all its mapping spaces have finite homotopy groups that vanish above a certain dimension; let us call such quasi-categories π-finite.Applying Theorem A.7 to the fibration test category L J of Example 5.7 shows that the underlying ∞-category of the profinite Joyal model structure s Set J (and hence also of bis Set CSS ) is equivalent to Pro(Cat ∞,π ), where Cat ∞,π denotes the ∞-category of π-finite ∞-categories.
and (2) of Definition 3.3.By adjunction, we conclude that for any C → D that has the right lifting property with respect to maps in J, the map Map(t, C) → Map(t, D) is a fibration.If we are given a map s t in I, then Map(s, C) × Map(s,D) Map(t, D) is fibrant because the map to Map(t, D) is the pullback of the fibration Map(s, C) Map(s, D).By a similar argument as above, Map(t, C) → Map(s, C) × Map(s,D) Map(t, D) is a fibration.The same argument with the set of boundary inclusions {∂∆ a weak equivalence, then the maps Map(s, C) → Map(s, D) and Map(t, C) → Map(t, D) are weak equivalences by definition, hence trivial fibrations by the above.As indicated in the diagram Map(t, C) Note that the geometric realization functor | • | : sSet fin → C extends uniquely to a filtered colimit preserving functor | • | : sSet → Ind(C) that has a right adjoint Sing defined by (Sing C) n = Hom(|∆ n |, C) for any C ∈ Ind(C).Lemma 4.1.Let (C, T) be a cofibration test category as above.Then a map C → D in Ind(C) is a weak equivalence if and only if Map( * , C) → Map( * , D) is a weak equivalence, where * is the terminal object.In particular, C → D is a weak equivalence if and only if Sing C → Sing D is a weak equivalence in sSet KQ .Proof.If C → D is a weak equivalence in Ind(C), then Map( * , C) → Map( * , D) is a weak equivalence by definition.Conversely, suppose that Map( * , C) → Map( * , D) is a weak equivalence and let X be a finite simplicial set.It follows by adjunction that Map(|X|, C) → Map(|X|, D) agrees with Map( * , C) X → Map( * , D) X , hence this map is a weak equivalence.
| • | : sSet → Top factors as sSet |•| − → Ind(C) L − → Top.Proposition 4.2.Let (C, T) be a cofibration test category as above.The adjunctions | • | : sSet KQ Ind(C) : Sing and L : Ind(C) Top : (•) Ind are Quillen equivalences.Proof.It is clear from the definition of the (trivial) cofibrations in (C, T) that | • | : sSet KQ → Ind(C) and L : Ind(C) → Top send generating (trivial) cofibrations to (trivial) cofibrations.In particular, they are left Quillen functors.Since the composition of these adjunctions is the well-known Quillen equivalence | • | : sSet KQ Top : Sing, it suffices to show by the 2 out of 3 property for Quillen equivalences that Definition 4.3.The n-th homotopy group π n (C, c 0 ) of an object C ∈ Ind(C) and a basepoint c 0 : * → C is defined as the set of pointed maps |∆ n /∂∆ n | → C modulo pointed homotopy.
Pro(C) L T Pro(C).The weak equivalences of L T Pro(C) are by definition the T-local equivalences.By Theorem 5.2, any object in Pro(C) (and hence in L T Pro(C)) is cofibrant.By Proposition 7.13, we obtain a simplicial Quillen adjunction E Pro(C) and hence a simplicial Quillen adjunction E L T Pro(C).We conclude that the model structure L T Pro(C) satisfies items (1)-(3) of Theorem 1.1.
The model structures on bis Set obtained by applying Theorem 5.2 to the fibration test categories L (2) R and L For any diagram X : N(I) → N(T), there exists a strict diagram Y : I → T such that N(Y) : N(I) → N(T) is naturally equivalent to X.This diagram Y can be constructed such a way that or any i ∈ I, the map colim j<i Y j → Y i is a composition of two pushouts of cofibrations in T.
Map(Y <i , t) ∼ = lim j<i Map(Y j , t) holim j<i Map(Y j , t), so Y <i is a homotopy colimit of the diagram Y| I <i .By Theorem 4.2.4.1 of [Lur09], it follows that it is also the ∞-colimit of the diagram N(Y| I <i ) : N(I <i ) → N(T).In particular, if we define the diagram Y : I ≤i → T by Y j = Y j for all j < i and Y i = Y <i , then the natural equivalence N(Y| I <i )X| N(I <i ) extends to a natural map N(Y ) → X| N(I ≤i ) .The map Y <i = Y i → X i factors through the mapping cylinder

Remark 3.4. Note
that property (4) implies that Map(t, C) is fibrant for every t ∈ T and C ∈ Ind(C).Namely, writing C as a filtered colimit colim i c i with c i ∈ C for every i, we see that Map(t, C) = colim i Map(t, c i ).Hence it suffices to show that Map(t, c) is fibrant for every object c in C.This is equivalent to c having the right lifting property with respect to certain maps of the form t ⊗ Λ n k → t ⊗ ∆ n , which is indeed the case by items (1), (2) and (4).The definition of a cofibration test category depends on whether we work with the Kan-Quillen model structure sSet KQ or the Joyal model structure sSet J .However, since sSet KQ is a left Bousfield localization of sSet J , any cofibration test category with respect to sSet KQ is also a cofibration test category with respect to sSet J .To see this, suppose that (C, T) is a cofibration test category with respect to sSet KQ .It is clear that items (1), (

Example 3.8. Let Top be
a convenient category of topological spaces, such as k-spaces or compactly generated (weak) Hausdorff spaces.The Quillen model structure on Top is a simplicial model structure, in which tensors are given by C ⊗ X = C × |X| for any C ∈ Top and X ∈

sSet. Let C ⊆ Top be any small full subcategory of Top that is closed under
finite colimits and finite tensors, and moreover contains the space |X| for any finite simplicial set X. Define T ⊆ C to be the full subcategory consisting of the objects |X| for any finite simplicial set X, and define a map |X| → |Y| in T to be a (trivial) cofibration if it is the geometric realization of a (trivial) cofibration X Y in the Kan-Quillen model structure on sSet.Using that there are natural isomorphisms |Y| ⊗ D) is a trivial fibration by the 2 out of 3 property.
Lemma 3.13.Let C be a cofibration test category and let s ∼ t be a trivial cofibration in C. Then any pushout of s ∼ t in Ind(C) is a weak equivalence in the sense of Theorem 3.9.Proof.The following proof works if C is a cofibration test category with respect to the Kan-Quillen model structure on sSet.The same proof works in the case that C is a cofibration test category with respect to sSet J if one replaces every instance of ∆ 1 by H, where H is as in the proof of Lemma 2.1.We will first show that i : s ∼ t is a deformation retract.By item (4) of Definition 3.3, there exists a lift in s s t,= ∼ i r | • | : sSet KQ Ind(C) : Sing is a Quillen equivalence.By Lemma 4.1, a map C → D in Ind(C) is a weak equivalence if and only if Sing C → Sing D is so.In particular, this adjunction is a Quillen equivalence if and only if the unit X → Sing |X| is a weak equivalence for any simplicial set X.If X is a finite simplicial set, then X → Sing |X| agrees by definition with the unit of the adjunction | • | : sSet KQ Top : Sing, which is always a weak equivalence.Since weak equivalences are stable under filtered colimits in sSet KQ , it follows that the unit X → Sing |X| of | • | : sSet KQ Ind(C) : Sing is a weak equivalence for any simplicial set X.
(most notably Lemma 7.4.7 and Lemma 7.4.10),using that Quick's and Morel's model structures on s Set are simplicial model structures.([BHH17]).A map X → Y of simplicial profinite sets is a weak equivalence in Quick's model structure if and only if Map(Y, K) → Map(X, K) is a weak equivalence for any lean Kan complex K.It is a weak equivalence in Morel's model structure if and only if Map(Y, K) → Map(X, K) is a weak equivalence for any lean Kan complex K whose homotopy groups are finite p-groups.From this proposition and the definition of the completed model structure (see Theorem 5.2), we see that the weak equivalences of Pro(L KQ ) (or Pro(L p )) agree with the weak equivalences in Quick's model structure (or Morel's model structure, respectively) on s Set.The completed model structure on Pro(L KQ ) coincides with Quick's model structure.For any prime number p, the completed model structure on Pro(L p ) coincides with Morel's model structure.
If t ∈ T 2 , then since φ : T 1 → T 2 is homotopically essentially surjective, there is a t ∈ T 1 together with an equivalence φ(t) ∼ −→ t .Since C and D are fibrant in Ind(C 2 ), the map Map(t , C) → Map(t , D) is a weak equivalence if and only if Map(φ(t), C) → Map(φ(t), D) is so.Since φ !extends φ and the adjunction φ !φ * is enriched, we see that * (C)) Map(t, φ * (D)) described in [Rez01, §10 & §12], one can localize the Reedy model structure on bisSet by the Segal maps Sp∆ t × ∆ 0 ∆ t × ∆ 0 ,where Sp∆ t = ∆[0, 1] ∪ . ..∪ ∆[t − 1, t]is the spine of the t-simplex.This gives the model category bisSet SS for Segal spaces.Localizing one step further, by the map [Rez01]ion 9.6.A map between complete Segal profinite spaces is called a Dwyer-Kan equivalence or DK-equivalence if it is essentially surjective and fully faithful.A map between complete Segal profinite spaces is a Dwyer-Kan equivalence if and only if it is a weak equivalence in bis Set CSS .Proof.As explained above Definition 9.6, f : X → Y is essentially surjective and fully faithful if and only if UX → UY is so.By Proposition 7.6 of[Rez01], this is the case if and only if UX → UY is a weak equivalence in bisSet CSS .By Proposition 9.5, this is equivalent to X → Y being a weak equivalence in bis Set CSS .