Model structures on the category of small double categories

Algebraic and Geometric Topology

Volumn 8, Issue 4, 2008, Pages 1855-1959

  Fiore, Thomas M ^a,b   Paoli, Simona ^c   Pronk, Dorette ^d  

^a UNIVERSITY OF CHICAGO   (United States)

^b UNIVERSITAT AUTÒNOMA DE BARCELONA   (Spain)

^c MACQUARIE UNIVERSITY   (Australia)

^d DALHOUSIE UNIVERSITY   (Canada)

Author keywords

2 category; 2 monad; Categorification; Colimit; Double category; Fundamental category; Fundamental double category; Horizontal categorification; Internal category; Model structure; Transfer of model structure

Indexed keywords

EID: 73049085471     PISSN: 14722747     EISSN: 14722739     Source Type: Journal    
DOI: 10.2140/agt.2008.8.1855     Document Type: Article

References (83)

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. G éom. Différ. Cat ég. 15 (1974) 215-292 MR0379626
3. M A Bednarczyk, A M Borzyszkowski, W Pawlowski, Generalized congruences-epimorphisms in Cat, Theory Appl. Categ. 5 (1999) 266 -280 MR1725510
4. J Bénabou, Introduction to bicategories, from: "Reports of the Midwest Category Seminar", Springer (1967) 1-77 MR0220789
5. J E Bergner, A survey of (∞, 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) 2043-2058 MR2276611
7. J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397-436 MR2321038
8. R Blackwell, G M Kelly, A J Power, Twodimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1-41 MR1007911
9. R Boerger, Kongruenzrelationen auf Kategorien, dissertation, Westfälische WilhelmsUniversität Münster (1977)
10. F Borceux, Handbook of categorical algebra. 1, Encyclopedia of Math. and its Applications 50, Cambridge University Press (1994) MR1291599 Basic category theory
11. R Brown, P J Higgins, The equivalence of ∞ -groupoids and crossed complexes, Cah. Top. G éom. Différ. 22 (1981) 371-386 MR639048
12. R Brown, P J Higgins, The equivalence of ω -groupoids and cubical Tcomplexes, Cah. Top. G éom. Différ. 22 (1981) 349-370 MR639047
13. R Brown, P J Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233-260 MR617135
14. R Brown, P J Higgins, Tensor products and homotopies for ω-groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987) 1-33 MR906402
15. R Brown, I Içen, Towards a 2-dimensional notion of holonomy, Adv. Math. 178 (2003) 141-175 MR1994346
16. R Brown, K C H Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra 80 (1992) 237-272 MR1170713
17. R Brown, G H Mosa, Double categories, 2-categories, thin structures and connections, Theory Appl. Categ. 5 (1999) 163-175 MR1694653
18. R Brown, C B Spencer, Double groupoids and crossed modules, Cah. Top. Géom. Différ. Cat ég. 17 (1976) 343-362 MR0440553
19. M Bunge, R Paré, Stacks and equivalence of indexed categories, Cah. Top. Géom. Différ. 20 (1979) 373-399 MR558105
20. R J M Dawson, R Paré, Characterizing tileorders, Order 10 (1993) 111-128 MR1253709
21. R J M Dawson, R Paré, General associativity and general composition for double categories, Cah. Top. Géom. Différ. Cat ég. 34 (1993) 57-79 MR1213297
22. R J M Dawson, R Paré, What is a free double category like?, J. Pure Appl. Algebra 168 (2002) 19-34 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. Cat ég. 45 (2004) 35-80 MR2040662
25. R J M Dawson, R Paré, D A Pronk, Paths in double categories, Theory Appl. Categ. 16 (2006) 460-521 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. G éom. Différ. 19 (1978) 295-333 MR546074
28. A Ehresmann, C Ehresmann, Multiple functors. III. The Cartesian closed category Cat n, Cah. Top. G éom. Différ. 19 (1978) 387-443 MR515164
29. A Ehresmann, C Ehresmann, Multiple functors. IV. Monoidal closed structures on Catn, Cah. Top. G éom. Différ. 20 (1979) 59-104 MR544529
30. C Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. (3) 80 (1963) 349-426 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) 66-94 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) 119-170 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) 258-262 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. G éom. Différ. Catég. 47 (2006) 261-317 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) 321-347 MR2326435
41. M Grandis, Cubical cospans and higher cobordisms (cospans in algebraic topology. III), J. Homotopy Relat. Struct. 3 (2008) 273-308
42. M Grandis, R Paré, Limits in double categories, Cah. Top. Géom. Différ. Catég. 40 (1999) 162-220 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
44. M Grandis, R Paré, Lax Kan extensions for double categories (on weak double categories. IV), Cah. Topol. G éom. Différ. Cat ég. 48 (2007) 163-199 MR2351267
45. M Grandis, R Paré, Kan extensions in double categories (on weak double categories. III), Theory Appl. Categ. 20 (2008) 152-185 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) 563-577 MR0094405
49. M Johnson, The combinatorics of n -categorical pasting, J. Pure Appl. Algebra 62 (1989) 211-225 MR1026875
50. A Joyal, R Street, Pullbacks equivalent to pseudopullbacks, Cah. Top. G éom. Différ. Cat ég. 34 (1993) 153-156 MR1223657
51. A Joyal, M Tierney, Strong stacks and classifying spaces, from: Category theory (Como, 1990), Lecture Notes in Math. 1488, Springer (1991) 213-236 MR1173014
52. A Joyal, M Tierney, Quasicategories vs Segal spaces, from: Categories in algebra, geometry and mathematical physics, Contemp. Math. 431, Amer. Math. Soc. (2007) 277-326 MR2342834
53. G M Kelly, Elementary observations on 2 -categorical limits, Bull. Austral. Math. Soc. 39 (1989) 301-317 MR998024
54. G M Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005) MR2177301 Reprint of the 1982 original [Cambridge Univ. Press; MR0651714]
55. T Kerler, V V Lyubashenko, Nonsemisimple 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 twofold monoidal categories, J. Homotopy Relat. Struct. 2 (2007) 217-228 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) 171-205 MR1931220
59. S Lack, A Quillen model structure for bicategories, KTheory 33 (2004) 185-197 MR2138540
60. S Lack, Homotopytheoretic aspects of 2-monads, J. Homotopy Relat. Struct. 2 (2007) 229-260 MR2369168
61. S Lack, S Paoli, An operadic approach to internal structures, Appl. Categ. Structures 13 (2005) 205-222 MR2167790
62. S Lack, S Paoli, 2-nerves for bicategories, KTheory 38 (2008) 153-175 MR2366560
63. J L Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179-202 MR651845
64. S MacLane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998) MR1712872
65. S MacLane, I Moerdijk, Sheaves in geometry and logic, Universitext, Springer (1994) MR1300636 A first introduction to topos theory, Corrected reprint of the 1992 edition
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) 45-58 MR0178040
68. J Mersch, Le problème du quotient dans les catégories, Mém. Soc. Roy. Sci. Liège Coll. in 8 (5) 11 (1965) MR0213414
69. J Morton, A double bicategory of cobordisms with corners arXiv: 061193
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) 801-820 MR2344230
72. A J Power, A 2-categorical pasting theorem, J. Algebra 129 (1990) 439-445 MR1040947
73. A J Power, An n-categorical pasting theorem, from: Category theory (Como, 1990), Lecture Notes in Math. 1488, Springer (1991) 326-358 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) 973-1007 MR1804411
76. M Shulman, Comparing composites of left and right derived functors arXiv:0706.2868
77. M Shulman, Framed bicategories and monoidal fibrations arXiv:0706128
78. R Street, Categorical structures, from: Handbook of algebra, Vol. 1", NorthHolland (1996) 529-577 MR1421811
79. A Strøm, The homotopy category is a homotopy category, Arch. Math. (Basel) 23 (1972) 435-441 MR0321082
80. R W Thomason, Cat as a closed model category, Cah. Top. G éom. Différ. 21 (1980) 305-324 MR591388
81. B Toën, Vers une axiomatisation de la théorie des cat égories supérieures, K-Theory 34 (2005) 233-263 MR2182378
82. V Trnková, Sum of categories with amalgamated subcategory, Comment. Math. Univ. Carolinae 6 (1965) 449-474 MR0190208
83. K Worytkiewicz, K Hess, P E Parent, A Tonks, A model structure à la Thomason on 2-Cat, J. Pure Appl. Algebra 208 (2007) 205-236 MR2269840