Algebraic & Geometric Topology Volume 10, issue 4 (2010)

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A Thomason model structure on the category of small $n$–fold categories

Thomas M Fiore and Simona Paoli

Algebraic & Geometric Topology 10 (2010) 1933–2008

DOI: 10.2140/agt.2010.10.1933

Bibliography

1	J Adámek, J Rosický, Locally presentable and accessible categories, London Math. Society Lecture Note Series 189, Cambridge Univ. Press (1994) MR1294136
2	A Bastiani, C Ehresmann, Multiple functors. I. Limits relative to double categories, Cahiers Topologie Géom. Différentielle 15 (1974) 215 MR0379626
3	C Berger, A cellular nerve for higher categories, Adv. Math. 169 (2002) 118 MR1916373
4	J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 MR2276611
5	J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397 MR2321038
6	J E Bergner, A survey of $(\infty,1)$–categories, from: "Towards higher categories" (editors J Baez, J P May), IMA Vol. Math. Appl. 152, Springer (2010) 69 MR2664620
7	R Brown, P J Higgins, The equivalence of $\infty $–groupoids and crossed complexes, Cahiers Topologie Géom. Différentielle 22 (1981) 371 MR639048
8	R Brown, P J Higgins, The equivalence of $\omega $–groupoids and cubical $T$–complexes, Cahiers Topologie Géom. Différentielle 22 (1981) 349 MR639047
9	R Brown, P J Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233 MR617135
10	R Brown, P J Higgins, Tensor products and homotopies for $\omega$–groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987) 1 MR906402
11	R Brown, G H Mosa, Double categories, $2$–categories, thin structures and connections, Theory Appl. Categ. 5 (1999) 163 MR1694653
12	D C Cisinski, La classe des morphismes de Dwyer n'est pas stable par retractes, Cahiers Topologie Géom. Différentielle Catég. 40 (1999) 227 MR1716777
13	D C Cisinski, Les préfaisceaux comme modèles des types d'homotopie, Astérisque (2006) MR2294028
14	R J M Dawson, R Paré, What is a free double category like?, J. Pure Appl. Algebra 168 (2002) 19 MR1879928
15	R J M Dawson, R Paré, D A Pronk, Paths in double categories, Theory Appl. Categ. 16 (2006) 460 MR2259260
16	J W Duskin, Simplicial matrices and the nerves of weak $n$–categories. II. Bicategory morphisms and simplicial maps, Preprint (2001)
17	J W Duskin, Simplicial matrices and the nerves of weak $n$–categories. I. Nerves of bicategories, Theory Appl. Categ. 9 (2001/02) 198 MR1897816
18	A Ehresmann, C Ehresmann, Multiple functors. II. The monoidal closed category of multiple categories, Cahiers Topologie Géom. Différentielle 19 (1978) 295 MR546074
19	A Ehresmann, C Ehresmann, Multiple functors. III. The Cartesian closed category $\mathrm{Cat}_{n}$, Cahiers Topologie Géom. Différentielle 19 (1978) 387 MR515164
20	A Ehresmann, C Ehresmann, Multiple functors. IV. Monoidal closed structures on $\mathrm{Cat}_{n}$, Cahiers Topologie Géom. Différentielle 20 (1979) 59 MR544529
21	C Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. $(3)$ 80 (1963) 349 MR0197529
22	C Ehresmann, Catégories et structures, Dunod (1965) MR0213410
23	T M Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Mem. Amer. Math. Soc. 182 (2006) MR2229946
24	T M Fiore, Pseudo algebras and pseudo double categories, J. Homotopy Relat. Struct. 2 (2007) 119 MR2369164
25	T M Fiore, S Paoli, D Pronk, Model structures on the category of small double categories, Algebr. Geom. Topol. 8 (2008) 1855 MR2449004
26	R Fritsch, D M Latch, Homotopy inverses for nerve, Bull. Amer. Math. Soc. $($N.S.$)$ 1 (1979) 258 MR513754
27	R Fritsch, D M Latch, Homotopy inverses for nerve, Math. Z. 177 (1981) 147 MR612870
28	P Gabriel, F Ulmer, Lokal präsentierbare Kategorien, Lecture Notes in Math. 221, Springer (1971) MR0327863
29	P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Math. und ihrer Grenzgebiete 35, Springer (1967) MR0210125
30	P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag (1999) MR1711612
31	M Golasiński, Homotopies of small categories, Fund. Math. 114 (1981) 209 MR644406
32	M Golasiński, Closed models on the procategory of small categories and simplicial schemes, Russian Math. Surveys 39 (1984) 275 MR764018
33	M Golasiński, Closed models on the procategory of small categories and simplicial schemes, Uspekhi Mat. Nauk 39 (1984) 239 MR764018
34	M Grandis, Higher cospans and weak cubical categories (cospans in algebraic topology. I), Theory Appl. Categ. 18 (2007) 321 MR2326435
35	M Grandis, Cubical cospans and higher cobordisms (cospans in algebraic topology. III), J. Homotopy Relat. Struct. 3 (2008) 273 MR2426182
36	M Grandis, R Paré, Limits in double categories, Cahiers Topologie Géom. Différentielle Catég. 40 (1999) 162 MR1716779
37	M Grandis, R Paré, Adjoint for double categories. Addenda to: “Limits in double categories” [Cah. Topol. Géom. Différ. Catég. \bf40 (1999), no. 3, 162–220; MR1716779], Cah. Topol. Géom. Différ. Catég. 45 (2004) 193 MR2090335
38	M Grandis, R Paré, Lax Kan extensions for double categories (on weak double categories. IV), Cah. Topol. Géom. Différ. Catég. 48 (2007) 163 MR2351267
39	M Grandis, R Paré, Kan extensions in double categories (on weak double categories. III), Theory Appl. Categ. 20 (2008) 152 MR2395245
40	M Heggie, Homotopy cofibrations in $\mathrm{CAT}$, Cahiers Topologie Géom. Différentielle Catég. 33 (1992) 291 MR1197427
41	M Heggie, The left derived tensor product of $\mathrm{CAT}$–valued diagrams, Cahiers Topologie Géom. Différentielle Catég. 33 (1992) 33 MR1163426
42	M Heggie, Homotopy colimits in presheaf categories, Cahiers Topologie Géom. Différentielle Catég. 34 (1993) 13 MR1213295
43	P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003) MR1944041
44	M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999) MR1650134
45	M L del Hoyo, On the subdivision of small categories, Topology Appl. 155 (2008) 1189 MR2421828
46	L Illusie, Complexe cotangent et déformations. II, Lecture Notes in Math. 283, Springer (1972) MR0491681
47	J F Jardine, Cubical homotopy theory: a beginning, Preprint (2002)
48	J F Jardine, Categorical homotopy theory, Homology, Homotopy Appl. 8 (2006) 71 MR2205215
49	A Joyal, Theory of quasi-categories, Vol. I, Preprint
50	A Joyal, Theory of quasi-categories, Vol. II, Preprint
51	A Joyal, The theory of quasi-categories and its applications, Quadern 45, Vol. II (2008)
52	A Joyal, M Tierney, Elements of simplicial homotopy theory, in progress, Chapters 1–4 available as Quadern 47, Centre de Recerca Mat. Barcelona (2008)
53	A Joyal, M Tierney, Strong stacks and classifying spaces, from: "Category theory (Como, 1990)" (editors A Carboni, M C Pedicchio, G Rosolini), Lecture Notes in Math. 1488, Springer (1991) 213 MR1173014
54	A Joyal, M Tierney, Quasi-categories vs Segal spaces, from: "Categories in algebra, geometry and mathematical physics" (editors A Davydov, M Batanin, M Johnson, S Lack, A Neeman), Contemp. Math. 431, Amer. Math. Soc. (2007) 277 MR2342834
55	D M Kan, On c. s. s. complexes, Amer. J. Math. 79 (1957) 449 MR0090047
56	G M Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005) MR2177301
57	J Kock, Polynomial functors and trees, Internat. Math. Res. Not. (2010)
58	S Lack, A Quillen model structure for $2$–categories, $K$–Theory 26 (2002) 171 MR1931220
59	S Lack, A Quillen model structure for bicategories, $K$–Theory 33 (2004) 185 MR2138540
60	S Lack, S Paoli, $2$–nerves for bicategories, $K$–Theory 38 (2008) 153 MR2366560
61	D M Latch, The uniqueness of homology for the category of small categories, J. Pure Appl. Algebra 9 (1977) 221 MR0460421
62	M J Lee, Homotopy for functors, Proc. Amer. Math. Soc. 36 (1972) 571, 648 MR0334212
63	T Leinster, A survey of definitions of $n$–category, Theory Appl. Categ. 10 (2002) 1 MR1883478
64	T Leinster, Nerves of algebras, Lecture notes from CT04 (2004)
65	T Leinster, M Weber, et al, How I learned to love the nerve construction, The n–Category Café, A group blog on math, physics and philosophy (2008)
66	J L Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179 MR651845
67	J Lurie, Derived algebraic geometry I: Stable $\infty$–categories arXiv:math/0608228
68	J Lurie, Higher topos theory, Annals of Math. Studies 170, Princeton Univ. Press (2009) MR2522659
69	S Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998) MR1712872
70	J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surveys and Monogr. 132, Amer. Math. Soc. (2006) MR2271789
71	I Moerdijk, J A Svensson, A Shapiro lemma for diagrams of spaces with applications to equivariant topology, Compositio Math. 96 (1995) 249 MR1327146
72	J C Morton, Double bicategories and double cospans, J. Homotopy Relat. Struct. 4 (2009) 389 MR2591970
73	S Paoli, Internal categorical structures in homotopical algebra, from: "Towards higher categories" (editors J Baez, J P May), IMA Vol. Math. Appl. 152, Springer (2010) 85 MR2664621
74	R Pellissier, Weak enriched categories – Categories enrichies faibles arXiv:math/0308246
75	D Quillen, Higher algebraic $K$–theory. I, from: "Algebraic $K$–theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972)" (editor H Bass), Lecture Notes in Math. 341, Springer (1973) 85 MR0338129
76	C Rezk, A model category for categories, Preprint (2000)
77	C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 MR1804411
78	M Shulman, Comparing composites of left and right derived functors arXiv:0706.2868
79	M Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008) 650 MR2534210
80	C Simpson, A closed model structure for $n$–categories, internal ${H}om$, $n$–stacks and generalized Seifert–Van Kampen arXiv:9704.5006
81	C Simpson, Homotopy theory of higher categories arXiv:1001.4071
82	R Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283 MR920944
83	Z Tamsamani, Sur des notions de $n$–catégorie et $n$-groupoïde non strictes via des ensembles multi-simpliciaux, $K$–Theory 16 (1999) 51 MR1673923
84	R W Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979) 91 MR510404
85	R W Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle 21 (1980) 305 MR591388
86	B Toën, Vers une axiomatisation de la théorie des catégories supérieures, $K$–Theory 34 (2005) 233 MR2182378
87	F Waldhausen, Algebraic $K$–theory of spaces, from: "Algebraic and geometric topology (New Brunswick, NJ, 1983)" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318 MR802796
88	M Weber, Familial $2$–functors and parametric right adjoints, Theory Appl. Categ. 18 (2007) 665 MR2369114
89	K Worytkiewicz, K Hess, P E Parent, A Tonks, A model structure à la Thomason on \bf$2$–Cat, J. Pure Appl. Algebra 208 (2007) 205 MR2269840