Detecting isomorphisms in the homotopy category

We show that the homotopy category of unpointed spaces admits no set of objects jointly reflecting isomorphisms by giving an explicit counterexample involving large symmetric groups. We also show that, in contrast, the spheres jointly reflect equivalences in the homotopy 2-category of spaces. The non-existence of such a set in the homotopy category was originally claimed by Heller, but his argument relied on the statement that for every set of spaces, long enough transfinite sequential diagrams admit weak colimits which are privileged with respect to the given set. Using the theory of graphs of groups, we show that this statement is false, by proving that for every ordinal with uncountable cofinality, there is a diagram indexed by that ordinal which admits no weak colimit that is privileged with respect to the spheres.


Introduction
Let Hot denote the homotopy category of spaces, and let Hot * ,c denote the homotopy category of pointed, connected spaces.Whitehead's theorem says that in Hot * ,c , the set of spheres jointly reflects isomorphisms.One is naturally led to wonder whether there is a set of spaces in Hot which jointly reflects isomorphisms.
In [1], Brown proved that a functor Hot op * ,c → Set is representable if and only if it is halfexact, in the sense that it sends coproducts and weak pushouts in Hot * ,c to products and weak pullbacks in Set.In [4], Heller proved an abstract representability theorem: if C is a category with coproducts and weak pushouts and C contains a "bounded" set G of objects that jointly reflects isomorphisms (see Definition 1.1 below), then a functor C op → Set is representable if and only if it is half-exact.In the same paper, Heller gave an example of a half-exact functor Hot op → Set which is not representable.He then claimed without proof [4,Prop. 1.2] that every set of spaces in Hot is bounded, and concluded [4,Cor. 2.3] that no set of spaces jointly reflects isomorphisms in Hot.
We show that it is not true that every set of spaces is bounded, reopening the question of whether there is a set of spaces that jointly reflects isomorphisms in Hot.We thus also give an independent proof that no set of spaces jointly reflects isomorphisms.
We now give the definitions needed in order to precisely state our results.Definition 1.1.Let C be any category and let G ⊆ C be a set of objects.
(1) We say that G jointly reflects isomorphisms if a morphism f : X → Y in C is an isomorphism whenever C(S, f ) : C(S, X) → C(S, Y ) is a bijection for every S ∈ G. G-privileged weak colimit.(5) We say that G is left cardinally bounded, or just bounded, if it is β-bounded for each sufficiently large regular cardinal β.
We use the word "set" to mean what is sometimes called a "small set," i.e., an object of the category Set.All of our ordinals and cardinals are "small."We regard a cardinal as an ordinal which is least in its cardinality class.The cofinality of an ordinal α is the smallest ordinal that is the order type of a cofinal subset of α.A cardinal is regular if it is equal to its cofinality.
As mentioned above, Hot denotes the homotopy category of spaces, by which we mean the localization of the category of spaces at the weak homotopy equivalences, or equivalently, the category whose objects are CW-complexes and whose morphisms are homotopy classes of continuous maps.It is well-known that every small diagram in Hot has a weak colimit, and that weak colimits are not unique.
We can now state our main results more precisely.First we give the result that shows that [4, Prop.1.2] is false.
Theorem 3.1.The set G = {S n | n ≥ 0} of spheres in Hot is not κ-bounded for any ordinal κ of uncountable cofinality.That is, for each such κ, there exists a diagram D : κ → Hot that admits no G-privileged weak colimit.
Note that Theorem 3.1 applies to all uncountable regular cardinals, showing that the set of spheres is not left cardinally bounded.By adding one more space to the set, we can remove the uncountability assumption: Corollary 3.2.Let T denote a countably infinite, discrete space.Then the set {S n | n ≥ 0} ∪ {T } is not κ-bounded in Hot for any limit ordinal κ.
The proof of Theorem 3.1 is somewhat involved and forms the bulk of the paper.We first show that it is sufficient to find a counterexample in the homotopy category HoGpd of groupoids.Then, given κ as in the statement, we consider the diagram D : κ → HoGpd sending α to the free group on 2 + α generators.We make use of the theory of graphs of groups [7] and the associated fundamental groupoid [5] in order to construct a sufficiently pathological cocone D → Z which we use to show that D admits no G ′ -privileged weak colimit, where G ′ = {BZ}.This involves a detailed understanding of the morphisms in Z and how they are expressed as words in the given generators.It follows that the diagram κ → Hot sending α to the wedge of α circles has no G-privileged weak colimit, where G is as in the statement of Theorem 3.1.
Heller's argument for his claim [4, Prop.1.2] that any set G of objects in Hot is bounded was to take the cocone W to be the homotopy colimit, i.e., a generalized telescope.Since such homotopy colimits are G-privileged, our result above implies that they are not, in general, even weak colimits in Hot.This is in contrast to the situation for telescopes of sequences indexed by ω, and for other homotopy colimits of diagrams indexed by freelygenerated categories.In the introduction to [2], Franke suggests using a Bousfield-Kan spectral sequence to show that Heller's claim is false, by comparing weak colimits to homotopy colimits, but we were unable to find an example in which we could prove that a certain differential was non-zero.
In the homotopy category of pointed, connected spaces, the set of spheres jointly reflects isomorphisms-this is the classical form of Whitehead's theorem.However, we conjecture that the set of spheres is not bounded in Hot * ,c .If this is true, it means that Heller's abstract representability theorem, as stated, does not imply Brown's representability theorem.That said, Heller's argument only requires a set of objects that jointly reflects isomorphisms and is β-bounded for some regular cardinal β.Thus, since the set of spheres is ℵ 0 -bounded, the proof of Heller's theorem goes through in Hot * ,c .
Next we state the result that shows that the statement of [4, Cor.2.3] is nevertheless correct.
Theorem 2.1.The category Hot contains no set G of spaces that jointly reflects isomorphisms.That is, there exists no set G of spaces such that, if f : X → Y is a map of spaces and f * : Hot(S, X) → Hot(S, Y ) is a bijection for every S ∈ G, then f is an isomorphism in Hot.
This second result is easier to prove, and so we prove it first, in Section 2. Our method is a generalization of [6, Proposition 4.1], which gives a "phantom homotopy equivalence," that is, a map in Hot which while not an isomorphism is seen as one by all finite complexes.Our proof also shows that there is no set of connected spaces that jointly reflects isomorphisms in the homotopy category of connected spaces.Moreover, Theorem 2.1 implies similar results in other settings.For example, since Hot is a reflective subcategory of the homotopy category of (∞, 1)-categories, it follows that there is no set of (∞, 1)-categories that jointly reflects isomorphisms in that category.
Since the (∞, 1)-category S of spaces certainly contains a set of objects jointly reflecting equivalences-namely the set whose only element is the one-point space-while its 1-categorical truncation Hot does not, one might ask which behavior the n-categorical truncations of S exhibit for larger values of n.In fact, we show in Theorem 4.3 that in the 2-category Hot of spaces, morphisms, and homotopy classes of homotopies between them, the set of spheres does jointly reflect equivalences, which is the natural generalization of joint reflection of isomorphisms to 2-category theory.Intuitively, the reason for the divergent behavior of Hot and Hot is that the 2-morphisms of Hot retain the information about based homotopies that is lost in Hot.

Acknowledgments:
The first author would like to thank George Raptis for suggesting an argument that the spheres should generate Hot, simpler than that originally given for the tori.Both authors thank the referee for many valuable comments that helped to improve the paper, including the citation to [6] that now does the bulk of the work in Section 4.

Hot admits no set that jointly reflects isomorphisms
We make the following definitions.For an ordinal α, write Σ α for the group of all bijections of the set α, ignoring order.When β < α, there is a natural inclusion Σ β ֒→ Σ α , and we define Σ c α to be the union of the images of Σ β for all β < α.We typically consider Σ c α when α is a cardinal, considered as the smallest ordinal with that cardinality, and we call the elements of Σ c α essentially constant permutations.Theorem 2.1.The category Hot contains no set G of spaces that jointly reflects isomorphisms.(See Definition 1.1.) Proof.Let G be a set of spaces and let α be a regular cardinal larger than the cardinality of π 1 (S, s 0 ) for each S ∈ G and each s 0 ∈ S. We must construct a map f : X → Y which is not a homotopy equivalence but which induces bijections on homotopy classes of maps from spaces in G.
Our example will be Bs : (Here and in what follows, if γ is a successor ordinal, we write γ ′ for its predecessor.)We must check that sσ ∈ Σ c α .First, it is essentially constant: if β < α and σ fixes each γ ≥ β, then for γ > β we have (sσ)(γ) = γ, if γ is a limit ordinal, and (sσ we see that s is a homomorphism: s(στ ) and (sσ)(sτ ) both fix all limit ordinals, while for successors we have as desired.Note that setting τ = σ −1 , respectively σ = τ −1 , we confirm that sσ is indeed a bijection.
Let H be a group with classifying space BH and let X be a connected space.If Gp denotes the category of groups, recall that Hot(X, BH) is isomorphic to Gp(π 1 (X), H) modulo conjugation by elements of H. (See, for example, [8,Corollary V.4.4].)In particular, we have a natural isomorphism Hot(X, BH) ∼ = Hot(Bπ 1 (X), BH).It also follows that for groups G and H, Hot(BG, BH) is isomorphic to Gp(G, H) modulo conjugation by elements of H, and that an element of Hot(BG, BH) is a homotopy equivalence if and only if it is represented by an isomorphism.
Note that s is not surjective, since sσ always preserves limit ordinals.Therefore, Bs : BΣ c α → BΣ c α is not a homotopy equivalence.However, we will show that it induces an isomorphism on G. First observe that it suffices to prove this for connected components of spaces in G.It follows that it is enough to prove this for spaces of the form BG, where G is a group of cardinality less than α.
Any map BG → BΣ c α arises from a homomorphism ϕ : G → Σ c α , well-defined up to conjugation.Since α is regular, there is a limit ordinal β < α so that ϕ(g) ∈ Σ β for every g ∈ G.We claim that s • ϕ is conjugate to ϕ by an element τ ∈ Σ c α defined as follows: It is straightforward to check that τ is a permutation, and it clearly fixes ordinals greater than or equal to β + β, which is less than α.For g ∈ G, let σ = ϕ(g).Then, noting that τ −1 (γ) = γ + 1 for any γ < β, we have We have used that if γ ≥ β, then σ(γ) = γ, and the consequence that if γ < β, then σ(γ) < β.
In summary, we have shown that Bs induces the identity on Hot(S, BΣ c α ) for every S ∈ G, proving the claim.Remark 2.2.Since the map Bs : BΣ c α → BΣ c α used in the proof has connected domain and codomain, it follows that there is no set of connected spaces that jointly reflects isomorphisms in the homotopy category of connected spaces.
We explain the origin of the maps s and τ .Morally, s is conjugation by the successor operation on ordinals, with limit ordinals handled specially.The map τ implements this by "making room" for the relevant limit ordinals in a range outside of the support of a particular permutation σ.In fact, if we denote the map τ above by τ β , then s itself is conjugation by τ α in Σ c γ for a regular cardinal γ > α.Remark 2.3.The referee pointed out an alternate proof of Theorem 2.1, which makes use of the techniques employed in [4, Lemma 2.2], namely the use of HNN-extensions.It also involves a map between classifying spaces, but is less explicit.In addition, the referee and N. Kuhn pointed out that the case when α = ω was proved in [6, Proposition 4.1], using an approach very similar to the approach given here.

The lack of privileged weak colimits
In this section, we give an example showing that Heller's privileged weak colimits do not generally exist.
Theorem 3.1.The set G = {S n | n ≥ 0} of spheres in Hot is not κ-bounded for any ordinal κ of uncountable cofinality, e.g., for any uncountable regular cardinal.That is, for each such κ, there exists a diagram D : κ → Hot that admits no G-privileged weak colimit.
In particular, D admits no G-privileged weak colimit for any set G containing the spheres.Note that the set of spheres is ℵ 0 -bounded, so we learn that boundedness for one ordinal does not imply it for ordinals with larger cofinality.Corollary 3.2.Let T denote a countably infinite, discrete space.Then the set {S n | n ≥ 0} ∪ {T } is not κ-bounded in Hot for any limit ordinal κ.
Proof.Since κ is a limit ordinal, it has infinite cofinality.If κ has uncountable cofinality, then Theorem 3.1 applies.If κ has countable cofinality, then {T } is not κ-bounded.
In Section 3.1, we reduce the problem to finding a counterexample in the homotopy category of groupoids.In Section 3.2, we recall the theory of graphs of groups, and prove some general results about the word problem in the fundamental groupoid of a graph of groups.Finally, in Section 3.3, we give a counterexample in the homotopy category of groupoids and complete the proof of Theorem 3.1.

3.1.
Reducing from spaces to groupoids.To prove Theorem 3.1 we will work primarily in the homotopy category HoGpd of groupoids, that is, the category of groupoids and isomorphism classes of functors.It is well known that the geometric realization of groupoids induces a reflective embedding B : HoGpd → Hot whose left adjoint is the fundamental groupoid functor Π 1 and whose image consists of the 1-types, i.e., the spaces X with π n (X, x) = 0 for all x ∈ X and n > 1.All this follows from the adjunction between π 1 and the classifying space functor B that was used in the proof of Theorem 2.1.Proof.We prove the contrapositive.Let λ : B • D → X be a G-privileged weak colimit, with X ∈ Hot.Then, since left adjoints preserve weak colimits, Π 1 (λ) : D → Π 1 X is a weak colimit.We will show that it is G ′ -privileged.
First, since λ is G-privileged, every map a : S n → X factors through a 1-type BD(j) for some j.Thus, when n > 1, a is freely homotopic to a constant, which implies that π n (X, x) is trivial for all x ∈ X.We conclude that X is a 1-type itself, so that X ≃ B (Π 1 X).
Since B is fully faithful, we see that Π 1 (λ) : One can show that the composite isomorphism is induced by Π 1 (λ).
Thus it suffices to exhibit appropriately pathological diagrams in HoGpd, and then to upgrade them to Hot.We aim to give a diagram in HoGpd admitting no weak colimit privileged with respect to the set G ′ = {BZ}.Here BZ denotes the groupoid freely generated by an automorphism, i.e., the groupoid with one object * whose endomorphism group is the integers.Of course, B(BZ) is homotopy equivalent to S 1 , so G in Lemma 3.3 can be taken to be the set of spheres.
Remark 3.4.Note that, for any groupoid G, a functor f : BZ → G corresponds to an object f ( * ) of G and an automorphism f * : f ( * ) → f ( * ).Furthermore, two such functors f, g : BZ → G are naturally isomorphic if and only if the automorphisms f * and g * are conjugate in G.In particular, a functor f : BZ → G factors through h : H → G in HoGpd if and only if f * is conjugate to an automorphism in the image of h.

3.2.
Graphs of groups.To construct our example, we recall the notion of a graph of groups, and prove Corollaries 3.7 and 3.8, and Lemma 3.9 that will be used in the next section.
Definition 3.5.A graph of groups Γ is given by: • A graph, i.e., a set X of vertices, a set Y of oriented edges, functions s, t : Y ⇒ X, and an involution (−) : Y → Y interchanging s and t. • Groups G x and G y for x ∈ X and y ∈ Y equipped with monomorphisms µ y : G y → G s(y) such that G y = G ȳ.For simplicity, we assume that the groups G x are disjoint.For more on graphs of groups, see [7, Section I.5] and [3, Section 1.B].
Higgins [5] defined the fundamental groupoid Π 1 Γ of a graph of groups.The groupoid Π 1 Γ is the groupoid on objects X with generating morphisms the elements of the groups G x , endowed with x as domain and codomain, together with the elements of Y viewed as morphisms y : s(y) → t(y).These generators are subject to the relations holding in the groups G x , as well as new relations µ ȳ(a) = yµ y (a)ȳ, for every y and every a ∈ G y .Note in particular that ȳ = y −1 , and we shall use both notations.It may aid the intuition to consider Π 1 Γ as the fundamental groupoid of the space built from X BG x with cylinders BG y × I glued in for each set {y, ȳ} of elements of Y related by the involution.
By definition, the groupoid Π 1 Γ is a quotient of the groupoid K with object set X and with morphisms freely generated by ( G x ) Y , subject to the relations holding in the groups G x .A morphism x 0 → x n in K is given by a word (a n , y n , . . ., y 1 , a 0 ), with y i ∈ Y , s(y 1 ) = x 0 , t(y n ) = x n , and s(y i+1 ) = t(y i ) =: x i for 1 ≤ i < n, while a i ∈ G x i for 0 ≤ i ≤ n.
The natural realization functor K → Π 1 Γ will be denoted by |(a n , y n , . . ., y 1 , a Higgins proves that every morphism of Π 1 Γ is uniquely the image under | • | of a so-called "normal" word.We will not recall this concept, as we need only Higgins' corollary regarding the less rigid irreducible words. A morphism (a n , y n , . . ., y 1 , a 0 ) in K is called reducible if n > 1 and for some i, y i−1 = ȳi and a i−1 ∈ µ y i (G y i ).Otherwise, the morphism is said to be irreducible.Note that a reducible word can be shortened by the move (. . ., a i , y i , µ y i (â i−1 ), ȳi , a i−2 , . ..) → (. . ., a i µ ȳi (â i−1 )a i−2 , . ..) to a word with the same realization.Therefore, every element of Π 1 Γ is the realization of an irreducible word.We will use a key result of [5].Proposition 3.6 ([5, Corollary 5]).Let w be an irreducible word in K.If |w| is an identity morphism in Π 1 Γ, then w = (e), where e is an identity element of some G x .
Define the length ℓ(w) of the word w = (a n , y n , . . ., y 1 , a 0 ) to be n.We deduce the following: Corollary 3.7.Let Γ be a graph of groups and consider a word w in the groupoid K.If ℓ(w) > 0 and |w| is equal to the realization of a zero-length word, then w is reducible.
Proof.Suppose that w = (a n , y n , . . ., y 1 , a 0 ) for n > 0 and that |w| = |(a)| for some a in some G x .Let w ′ = (a n , y n , . . ., y 1 , a 0 a −1 ).Then |w ′ | is an identity morphism in Π 1 Γ, so by Proposition 3.6, w ′ is reducible.Since reduction occurs at interior points, w must be reducible as well.
Corollary 3.8.Given a graph of groups Γ and a vertex x, the vertex group G x embeds in the automorphism group of x in the fundamental groupoid Π 1 Γ.
Because of this, we regard elements of the vertex groups as elements of the fundamental groupoid without explicitly naming the inclusion map.
Proof.The map sends a ∈ G x to the realization of the word (a).Since the word (a) is irreducible, if the realization is an identity in Π 1 Γ, Proposition 3.6 tells us that a is the identity element of G x .Therefore, this map is injective.
We next record some facts about free groups.Lemma 3.9.Let A ⊆ B be nonabelian free groups, with A free on generators {a i } and B free on {a i } ∪ {b j }.

3.3.
A counterexample in the homotopy category of groupoids.We now apply the generalities above to the problem of weak colimits in HoGpd.
We fix for the rest of the paper an ordinal κ of uncountable cofinality, and introduce the main characters in our counterexample.Note that Theorem 3.1 will follow if we replace κ = [0, κ) by the interval [2, κ), since the two categories are isomorphic.We use the latter because it allows us to use simple indexing while ensuring that all of the vertex groups below are non-abelian.We have a cocone A : D → Z with A α : D(α) → Z the natural inclusion of the vertex group.To see that these maps do constitute a cocone, we note that y β α is the unique component of a natural isomorphism We do not need this fact, but it may provide motivation to the reader to know that Z is the "standard" weak colimit of the diagram D, defined as the homotopy coequalizer of the natural diagram Critically, we do not have the relations y β α y γ β = y γ α in Z which would allow us to lift A into a cocone in the 2-category of groupoids.We now intend to show that D admits no privileged weak colimit by, roughly, showing that this failure is unavoidable: no choice of isomorphisms A β ∼ = A α • D β α can give A such a lift.Write Z Y for the subgroupoid of Z generated by the edges of the graph.Any morphism of Z Y can be uniquely written as a reduced word in the generators y β α .We say that such a morphism passes through a vertex α if this unique word involves a generator with source or target α.The identity id α is said to pass through α and no other vertex.Lemma 3.11.Let u : β → α in Z and let 2 ≤ γ ≤ min(α, β).Then u is in Z Y and does not pass through any vertex less than γ if and only if u is the unique component of a natural isomorphism Suppose that u is in Z Y and does not pass through any vertex less than γ.It suffices to show that y β α conjugates D γ β into D γ α when γ ≤ β ≤ α.In this case, µ y β α is an identity map, and so the claim follows from the defining relations of Z: . For the converse, let u be the realization of an irreducible word w = (a n , y n , . . ., y 1 , a 0 ).We proceed by induction on n.If n = 0, then α = β and u = |(a 0 )| ∈ G β .The assumption that D γ β (a) = uD γ β (a)u −1 shows that u centralizes a nonabelian subgroup of a free group.By Lemma 3.9 (1), we see that u is trivial as desired.And clearly u does not pass through a vertex less than γ; indeed, it passes through only β, and β ≥ γ.
For the inductive step, assume n > 0. Then s(y 1 ) = β and t(y n ) = α.Let t(y 1 ) = δ, and note that δ = β.In terms of w, the assumption on u is that the word for every a ∈ G γ .Thus, by Corollary 3.7, w ′ is reducible.Since by assumption w is irreducible, any reduction must occur at the central entry.So, letting ε := min(β, δ), we must have for some non-identity element â in G min(γ,ε) .So by Lemma 3.9 (2), we see that β for every a ∈ G γ , which means that γ ≤ ε, since the inclusions of vertex groups are strict.
Let Z X denote the subgroupoid of Z containing those morphisms in the image of G x for some x.By Corollary 3.8, Z X is isomorphic to the disjoint union of the groups G x .Lemma 3.12.Consider a morphism z : α → α in Z.If there are morphisms u : α → β and v : α → γ in Z such that uzu −1 is in Z X and vzv −1 is in Z Y , then z = id α .
Proof.Let y = vzv −1 .Note that the inclusion Z Y → Z has a retraction r : Z → Z Y defined by sending the generators of each vertex group to identity elements.Since uv −1 yvu −1 is in Z X , we have that r(uv −1 yvu −1 ) = r(uv −1 ) y r(uv −1 ) −1 is an identity, and so y is an identity.Since y = vzv −1 is an identity, we have that z is an identity as well.
Proof.Assume that this is not the case.Let δ 0 = 2 and δ 1 = 3. Inductively, for each n ∈ ω let δ n be an ordinal exceeding every vertex that u δ n−1 passes through.This is possible because κ is a limit ordinal.
For each n, u can be written uniquely as a reduced word in the free groupoid Z Y .Let y n be a letter in this word which is of the form y β α with β < δ n ≤ α.Such a letter must exist since u δ n−1 δn starts at a vertex less than δ n and ends at δ n .Note that y n cannot occur in the reduced form of any u δ k−1 δ k with k = n.For k < n, this holds by definition of δ n , and for k > n, this holds by our assumption that each u β α only passes through γ with γ ≥ β.In particular, the y n 's are distinct.
Using that κ has uncountable cofinality, choose δ ω < κ to be an ordinal exceeding every δ n .Consider the decompositions In the expression u δ 1 δω u δ 0 δ 1 , a y 1 occurs in the reduced form of the right-hand factor, and does not occur in the left-hand factor, so the reduced form of u δ 0 δω must contain a y 1 .Similarly, the second decomposition involves a y 2 , which can't be cancelled from either side, so the reduced form of u δ 0 δω must contain a y 2 .Continuing, we see that the reduced form of u δ 0 δω must contain countably many distinct letters, a contradiction.
Recall that κ is an arbitrary ordinal of uncountable cofinality.
For each γ < β < α, denote by w αβγ ∈ W the unique component of the composite natural transformation On the other hand, by assumption on F , w αβγ is conjugate to a morphism in the image of some F θ : D(θ) → W , say to F θ (w ′ αβγ ).Composing with f , we see that u αβγ is conjugate to f (F θ (w ′ αβγ )).Finally, using kγ , we see u αβγ is conjugate to A θ (w ′ αβγ ), in particular, to an element of Z X .Since we saw above that u αβγ is in Z Y , Lemma 3.12 shows that u αβγ = id γ .
Finally, Lemma 3.13 implies that at least one u β α passes through a vertex less than β, contradicting what we saw above.
Proof of Theorem 3.1.By Proposition 3.14 and Remark 3.4, the diagram D admits no weak colimit privileged with respect to the set G ′ = {BZ}.Thus by Lemma 3.3, B • D admits no weak colimit in Hot which is privileged with respect to the set of spheres.

The spheres reflect equivalences in the 2-category of spaces
We saw in Theorem 2.1 that in the homotopy category of spaces there is no set of objects that jointly reflects isomorphisms.In this section, we show that in the homotopy 2-category of spaces, the spheres do jointly reflect equivalences.We first define the terms we are using.Definition 4.1.By Hot, we mean the 2-category whose objects are spaces of the homotopy type of a CW-complex and whose hom-categories are the fundamental groupoids of mapping spaces, that is Hot(X, Y ) = Π 1 (Y X ).Definition 4.2.A set G of objects in a 2-category K jointly reflects equivalences if, whenever f : X → Y is a morphism in K such that, for every S ∈ G, the induced functor K(S, f ) : K(S, X) → K(S, Y ) is an equivalence of categories, then f itself must be an equivalence in K.
We shall show in Theorem 4.3 that the 2-category Hot admits a set G of objects that jointly reflects equivalences, namely G = {S n | n ≥ 0}.Note that a map f in Hot is an equivalence if and only if it is a homotopy equivalence.This theorem is a corollary of Theorem 1 in [6], which shows that for a map f : X → Y of (arcwise connected) spaces which is surjective on all fundamental groups, bijectivity of f on higher homotopy groups is equivalent to that on free homotopy classes of maps from spheres.
With this, we are prepared to show that the spheres satisfy the analogue of Whitehead's theorem for Hot.Proof.Let f : X → Y be such that Hot(S n , f ) : Hot(S n , X) → Hot(S n , Y ) is an equivalence of groupoids, for every n.Consider an inclusion of * into S 0 ∼ = * ⊔ * .Since this has a retraction, the functor Hot( * , X) → Hot( * , Y ) is a retract of the equivalence Hot(S 0 , X) → Hot(S 0 , Y ) and is therefore also an equivalence.That is, f induces an equivalence Π 1 (X) → Π 1 (Y ) of fundamental groupoids.Thus f induces an isomorphism on π 0 and on every π 1 .
Therefore, we can apply Theorem 1 of [6], so that f will be a homotopy equivalence as soon as it induces a bijection on free homotopy classes of maps from S n .Now, the set of free homotopy classes of maps S n → X is simply the set of connected components in the groupoid Hot(S n , X).Since f induces an equivalence Hot(S n , X) → Hot(S n , Y ), a fortiori it induces an isomorphism on connected components, and the theorem is proven.

2 ( 2 )
A weak colimit of a diagram D : I → C is a cocone through which every cocone factors, not necessarily uniquely.(3) A cocone W of D : I → C is G-privileged if the canonical map colim α∈I C(S, D(α)) → C(S, W ) is a bijection for every S ∈ G. (4) For an ordinal β, we say that G is β-bounded if every diagram D : β → C has a

Lemma 3 . 3 .
Suppose given a diagram D : J → HoGpd, a set G ′ of groupoids, and a set G of spaces containing BG ′ as well as S n for all n.If D admits no G ′ -privileged weak colimit in HoGpd, then B • D : J → Hot admits no G-privileged weak colimit in Hot.

( 1 ) 2 )
If b ∈ B and for all a ∈ A we have bab −1 = a, then b is the identity.(If b ∈ B satisfies bab −1 ∈ A for some a ∈ A, then either a is the identity or b ∈ A. Proof.Fix b ∈ B. For part (1), if we take a = a i then the assumption that ba i b −1 = a i shows that an irreducible word for b must have last letter a i or a −1 i for every i, which is absurd since there are at least two i's.For part (2), we assume a is nontrivial and b / ∈ A. Factor b as b ′ b ′′ , where b ′′ ∈ A while b ′ is represented by an irreducible word with rightmost letter some b j .Then bab −1 = b ′ a ′ b ′−1 , where a ′ := b ′′ a b ′′−1 is a non-trivial element of A. The conclusion now follows from the observation that no reductions are possible in the concatenation of the irreducible words for b ′ , a ′ and b ′−1 , since concatenating those words gives no letter adjacent to its inverse.

Proposition 3 .
14.There exists a diagram C : [2, κ) → HoGpd valued in the homotopy category of groupoids such that for any weak colimit with cocone F : C → W , there exists an automorphism in W which is not conjugate to any morphism in the image of any legF α : C(α) → W of F .Proof.We claim that the diagram D (see Definition 3.10) is an example of such a C.Towards a contradiction, suppose F : D → W is a weakly colimiting cocone such that every automorphism in W is conjugate to one in the image of some component of F .Write

F
α for functors representing the maps D(α) → W . Since F is a cocone in HoGpd, for each β < α ∈ [2, κ) we may choose a natural isomorphismh β α : F β ∼ = F α • D β α between functors D(β) → W in Gpd.Denote by ĥβ α the unique component of h β α .As usual we shall denote (h β a ) −1 by h α β , and similarly for ĥ, as well as u below.Recall the natural cocone A : D → Z from Definition 3.10 and suppose given a representative f : W → Z of a factorization of the cocone A through F .For each α, pick a natural isomorphism k

Theorem 4 . 3 .
The set G = {S n } of spheres jointly reflects equivalences in the 2-category Hot of spaces.