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Model structures on the category of small double categories

Thomas M Fiore, Simona Paoli and Dorette Pronk

Algebraic & Geometric Topology 8 (2008) 1855–1959
Bibliography
1 J Adámek, J Rosický, Locally presentable and accessible categories, London Math. Society Lecture Note Ser. 189, Cambridge University Press (1994) MR1294136
2 A Bastiani, C Ehresmann, Multiple functors. I. Limits relative to double categories, Cah. Top. Géom. Différ. Catég. 15 (1974) 215 MR0379626
3 M A Bednarczyk, A M Borzyszkowski, W Pawlowski, Generalized congruences—epimorphisms in ${\mathcal{C}at}$, Theory Appl. Categ. 5 (1999) 266 MR1725510
4 J Bénabou, Introduction to bicategories, from: "Reports of the Midwest Category Seminar", Springer (1967) 1 MR0220789
5 J E Bergner, A survey of $(\infty,1)$–categories, to appear in “Proceedings of the IMA Workshop ‘n–Categories: Foundations and Applications’ June 2004, University of Minnesota” arXiv:math/0610239
6 J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 MR2276611
7 J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397 MR2321038
8 R Blackwell, G M Kelly, A J Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1 MR1007911
9 R Boerger, Kongruenzrelationen auf Kategorien, dissertation, Westfälische Wilhelms-Universität Münster (1977)
10 F Borceux, Handbook of categorical algebra. 1, Encyclopedia of Math. and its Applications 50, Cambridge University Press (1994) MR1291599
11 R Brown, P J Higgins, The equivalence of $\infty $–groupoids and crossed complexes, Cah. Top. Géom. Différ. 22 (1981) 371 MR639048
12 R Brown, P J Higgins, The equivalence of $\omega $–groupoids and cubical $T$-complexes, Cah. Top. Géom. Différ. 22 (1981) 349 MR639047
13 R Brown, P J Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233 MR617135
14 R Brown, P J Higgins, Tensor products and homotopies for $\omega$–groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987) 1 MR906402
15 R Brown, I Içen, Towards a $2$–dimensional notion of holonomy, Adv. Math. 178 (2003) 141 MR1994346
16 R Brown, K C H Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra 80 (1992) 237 MR1170713
17 R Brown, G H Mosa, Double categories, $2$–categories, thin structures and connections, Theory Appl. Categ. 5 (1999) 163 MR1694653
18 R Brown, C B Spencer, Double groupoids and crossed modules, Cah. Top. Géom. Différ. Catég. 17 (1976) 343 MR0440553
19 M Bunge, R Paré, Stacks and equivalence of indexed categories, Cah. Top. Géom. Différ. 20 (1979) 373 MR558105
20 R J M Dawson, R Paré, Characterizing tileorders, Order 10 (1993) 111 MR1253709
21 R J M Dawson, R Paré, General associativity and general composition for double categories, Cah. Top. Géom. Différ. Catég. 34 (1993) 57 MR1213297
22 R J M Dawson, R Paré, What is a free double category like?, J. Pure Appl. Algebra 168 (2002) 19 MR1879928
23 R J M Dawson, R Paré, D A Pronk, The structure of spans, preprint
24 R J M Dawson, R Paré, D A Pronk, Free extensions of double categories, Cah. Topol. Géom. Différ. Catég. 45 (2004) 35 MR2040662
25 R J M Dawson, R Paré, D A Pronk, Paths in double categories, Theory Appl. Categ. 16 (2006) 460 MR2259260
26 E J Dubuc, Kan extensions in enriched category theory, Lecture Notes in Math. 145, Springer (1970) MR0280560
27 A Ehresmann, C Ehresmann, Multiple functors. II. The monoidal closed category of multiple categories, Cah. Top. Géom. Différ. 19 (1978) 295 MR546074
28 A Ehresmann, C Ehresmann, Multiple functors. III. The Cartesian closed category $\mathrm{Cat}_{n}$, Cah. Top. Géom. Différ. 19 (1978) 387 MR515164
29 A Ehresmann, C Ehresmann, Multiple functors. IV. Monoidal closed structures on $\mathrm{Cat}_{n}$, Cah. Top. Géom. Différ. 20 (1979) 59 MR544529
30 C Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. $(3)$ 80 (1963) 349 MR0197529
31 C Ehresmann, Catégories et structures, Dunod (1965) MR0213410
32 T Everaert, R W Kieboom, T Van der Linden, Model structures for homotopy of internal categories, Theory Appl. Categ. 15 (2005/06) 66 MR2210576
33 T M Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Mem. Amer. Math. Soc. 182 (2006) MR2229946
34 T M Fiore, Pseudo algebras and pseudo double categories, J. Homotopy Relat. Struct. 2 (2007) 119 MR2369164
35 T M Fiore, S Paoli, A Thomason model structure on the category of small $n$–fold categories arXiv:0808.4108
36 R Fritsch, D M Latch, Homotopy inverses for nerve, Bull. Amer. Math. Soc. $($N.S.$)$ 1 (1979) 258 MR513754
37 P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Math. und ihrer Grenzgebiete 35, Springer (1967) MR0210125
38 R Garner, Double clubs, Cah. Topol. Géom. Différ. Catég. 47 (2006) 261 MR2303000
39 P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag (1999) MR1711612
40 M Grandis, Higher cospans and weak cubical categories (cospans in algebraic topology. I), Theory Appl. Categ. 18 (2007) 321 MR2326435
41 M Grandis, Cubical cospans and higher cobordisms (cospans in algebraic topology. III), J. Homotopy Relat. Struct. 3 (2008) 273
42 M Grandis, R Paré, Limits in double categories, Cah. Top. Géom. Différ. Catég. 40 (1999) 162 MR1716779
43 M Grandis, R Paré, Adjoint for double categories. Addenda to: “Limits in double categories” [Cah. Topol. Géom. Différ. Catég. 40 (1999) 162–220; MR1716779], Cah. Topol. Géom. Différ. Catég. 45 (2004) 193 MR2090335
44 M Grandis, R Paré, Lax Kan extensions for double categories (on weak double categories. IV), Cah. Topol. Géom. Différ. Catég. 48 (2007) 163 MR2351267
45 M Grandis, R Paré, Kan extensions in double categories (on weak double categories. III), Theory Appl. Categ. 20 (2008) 152 MR2395245
46 P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003) MR1944041
47 M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999) MR1650134
48 J R Isbell, Some remarks concerning categories and subspaces, Canad. J. Math. 9 (1957) 563 MR0094405
49 M Johnson, The combinatorics of $n$–categorical pasting, J. Pure Appl. Algebra 62 (1989) 211 MR1026875
50 A Joyal, R Street, Pullbacks equivalent to pseudopullbacks, Cah. Top. Géom. Différ. Catég. 34 (1993) 153 MR1223657
51 A Joyal, M Tierney, Strong stacks and classifying spaces, from: "Category theory (Como, 1990)", Lecture Notes in Math. 1488, Springer (1991) 213 MR1173014
52 A Joyal, M Tierney, Quasi-categories vs Segal spaces, from: "Categories in algebra, geometry and mathematical physics", Contemp. Math. 431, Amer. Math. Soc. (2007) 277 MR2342834
53 G M Kelly, Elementary observations on $2$–categorical limits, Bull. Austral. Math. Soc. 39 (1989) 301 MR998024
54 G M Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005) MR2177301
55 T Kerler, V V Lyubashenko, Non-semisimple topological quantum field theories for $3$–manifolds with corners, Lecture Notes in Math. 1765, Springer (2001) MR1862634
56 J Kock, Note on commutativity in double semigroups and two-fold monoidal categories, J. Homotopy Relat. Struct. 2 (2007) 217 MR2369167
57 S Lack, A $2$–categories companion, to appear in “Proceedings of the IMA Workshop ‘n–Categories: Foundations and Applications’ June 2004, University of Minnesota” arXiv:math/0702535
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, Homotopy-theoretic aspects of $2$–monads, J. Homotopy Relat. Struct. 2 (2007) 229 MR2369168
61 S Lack, S Paoli, An operadic approach to internal structures, Appl. Categ. Structures 13 (2005) 205 MR2167790
62 S Lack, S Paoli, $2$–nerves for bicategories, $K$-Theory 38 (2008) 153 MR2366560
63 J L Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179 MR651845
64 S Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998) MR1712872
65 S Mac Lane, I Moerdijk, Sheaves in geometry and logic, Universitext, Springer (1994) MR1300636
66 J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surveys and Monogr. 132, Amer. Math. Soc. (2006) MR2271789
67 J Mersch, Structures quotients, Bull. Soc. Roy. Sci. Liege 33 (1964) 45 MR0178040
68 J Mersch, Le problème du quotient dans les catégories, Mém. Soc. Roy. Sci. Liège Coll. in-8 $(5)$ 11 (1965) 103 MR0213414
69 J Morton, A double bicategory of cobordisms with corners arXiv:0611930
70 S Paoli, Internal categorical structures in homotopical algebra, to appear in “Proceedings of the IMA Workshop ‘n–Categories: Foundations and Applications’ June 2004, University of Minnesota”
71 S Paoli, Semistrict models of connected $3$–types and Tamsamani's weak $3$–groupoids, J. Pure Appl. Algebra 211 (2007) 801 MR2344230
72 A J Power, A $2$–categorical pasting theorem, J. Algebra 129 (1990) 439 MR1040947
73 A J Power, An $n$–categorical pasting theorem, from: "Category theory (Como, 1990)", Lecture Notes in Math. 1488, Springer (1991) 326 MR1173022
74 C Rezk, A model category for categories (2000)
75 C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 MR1804411
76 M Shulman, Comparing composites of left and right derived functors arXiv:0706.2868
77 M Shulman, Framed bicategories and monoidal fibrations arXiv:0706.1286
78 R Street, Categorical structures, from: "Handbook of algebra, Vol. 1", North-Holland (1996) 529 MR1421811
79 A Strøm, The homotopy category is a homotopy category, Arch. Math. $($Basel$)$ 23 (1972) 435 MR0321082
80 R W Thomason, Cat as a closed model category, Cah. Top. Géom. Différ. 21 (1980) 305 MR591388
81 B Toën, Vers une axiomatisation de la théorie des catégories supérieures, $K$–Theory 34 (2005) 233 MR2182378
82 V Trnková, Sum of categories with amalgamated subcategory, Comment. Math. Univ. Carolinae 6 (1965) 449 MR0190208
83 K Worytkiewicz, K Hess, P E Parent, A Tonks, A model structure à la Thomason on $2$–$\mathbf{Cat}$, J. Pure Appl. Algebra 208 (2007) 205 MR2269840