Model structures on the category of small double categories

Fiore, Thomas M¹ ; Paoli, Simona² ; Pronk, Dorette³

¹Department of Mathematics, University of Chicago, 5734 South University, Chicago, IL 60637, USA, and Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
²Department of Mathematics, Macquarie University, NSW 2109, Australia
³Chase Building, Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia, Canada B3H 3J5, Canada

Algebraic and Geometric Topology, Tome 8 (2008) no. 4, pp. 1855-1959

Résumé

In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2–monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions for and discuss properties of free double categories, quotient double categories, colimits of double categories, horizontal nerve and horizontal categorification.

Détail

DOI : 10.2140/agt.2008.8.1855

Keywords: categorification, colimit, double category, fundamental category, fundamental double category, horizontal categorification, internal category, model structure, transfer of model structure, $2$–category, $2$–monad

Bibliographie

[1] J Adámek, J Rosický, Locally presentable and accessible categories, London Math. Society Lecture Note Ser. 189, Cambridge University Press (1994)

[2] A Bastiani, C Ehresmann, Multiple functors. I. Limits relative to double categories, Cah. Top. Géom. Différ. Catég. 15 (1974) 215

[3] M A Bednarczyk, A M Borzyszkowski, W Pawlowski, Generalized congruences—epimorphisms in ${\mathcal{C}at}$, Theory Appl. Categ. 5 (1999) 266

[4] J Bénabou, Introduction to bicategories, from: "Reports of the Midwest Category Seminar", Springer (1967) 1

[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”

[6] J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043

[7] J E Bergner, Three models for the homotopy theory of homotopy theories, Topology 46 (2007) 397

[8] R Blackwell, G M Kelly, A J Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989) 1

[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)

[11] R Brown, P J Higgins, The equivalence of $\infty $–groupoids and crossed complexes, Cah. Top. Géom. Différ. 22 (1981) 371

[12] R Brown, P J Higgins, The equivalence of $\omega $–groupoids and cubical $T$-complexes, Cah. Top. Géom. Différ. 22 (1981) 349

[13] R Brown, P J Higgins, On the algebra of cubes, J. Pure Appl. Algebra 21 (1981) 233

[14] R Brown, P J Higgins, Tensor products and homotopies for $\omega$–groupoids and crossed complexes, J. Pure Appl. Algebra 47 (1987) 1

[15] R Brown, I Içen, Towards a $2$–dimensional notion of holonomy, Adv. Math. 178 (2003) 141

[16] R Brown, K C H Mackenzie, Determination of a double Lie groupoid by its core diagram, J. Pure Appl. Algebra 80 (1992) 237

[17] R Brown, G H Mosa, Double categories, $2$–categories, thin structures and connections, Theory Appl. Categ. 5 (1999) 163

[18] R Brown, C B Spencer, Double groupoids and crossed modules, Cah. Top. Géom. Différ. Catég. 17 (1976) 343

[19] M Bunge, R Paré, Stacks and equivalence of indexed categories, Cah. Top. Géom. Différ. 20 (1979) 373

[20] R J M Dawson, R Paré, Characterizing tileorders, Order 10 (1993) 111

[21] R J M Dawson, R Paré, General associativity and general composition for double categories, Cah. Top. Géom. Différ. Catég. 34 (1993) 57

[22] R J M Dawson, R Paré, What is a free double category like?, J. Pure Appl. Algebra 168 (2002) 19

[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

[25] R J M Dawson, R Paré, D A Pronk, Paths in double categories, Theory Appl. Categ. 16 (2006) 460

[26] E J Dubuc, Kan extensions in enriched category theory, Lecture Notes in Math. 145, Springer (1970)

[27] A Ehresmann, C Ehresmann, Multiple functors. II. The monoidal closed category of multiple categories, Cah. Top. Géom. Différ. 19 (1978) 295

[28] A Ehresmann, C Ehresmann, Multiple functors. III. The Cartesian closed category $\mathrm{Cat}_{n}$, Cah. Top. Géom. Différ. 19 (1978) 387

[29] A Ehresmann, C Ehresmann, Multiple functors. IV. Monoidal closed structures on $\mathrm{Cat}_{n}$, Cah. Top. Géom. Différ. 20 (1979) 59

[30] C Ehresmann, Catégories structurées, Ann. Sci. École Norm. Sup. $(3)$ 80 (1963) 349

[31] C Ehresmann, Catégories et structures, Dunod (1965)

[32] T Everaert, R W Kieboom, T Van Der Linden, Model structures for homotopy of internal categories, Theory Appl. Categ. 15 (2005/06) 66

[33] T M Fiore, Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory, Mem. Amer. Math. Soc. 182 (2006)

[34] T M Fiore, Pseudo algebras and pseudo double categories, J. Homotopy Relat. Struct. 2 (2007) 119

[35] T M Fiore, S Paoli, A Thomason model structure on the category of small $n$–fold categories

[36] R Fritsch, D M Latch, Homotopy inverses for nerve, Bull. Amer. Math. Soc. $($N.S.$)$ 1 (1979) 258

[37] P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Math. und ihrer Grenzgebiete 35, Springer (1967)

[38] R Garner, Double clubs, Cah. Topol. Géom. Différ. Catég. 47 (2006) 261

[39] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Math. 174, Birkhäuser Verlag (1999)

[40] M Grandis, Higher cospans and weak cubical categories (cospans in algebraic topology. I), Theory Appl. Categ. 18 (2007) 321

[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

[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

[44] M Grandis, R Paré, Lax Kan extensions for double categories (on weak double categories. IV), Cah. Topol. Géom. Différ. Catég. 48 (2007) 163

[45] M Grandis, R Paré, Kan extensions in double categories (on weak double categories. III), Theory Appl. Categ. 20 (2008) 152

[46] P S Hirschhorn, Model categories and their localizations, Math. Surveys and Monogr. 99, Amer. Math. Soc. (2003)

[47] M Hovey, Model categories, Math. Surveys and Monogr. 63, Amer. Math. Soc. (1999)

[48] J R Isbell, Some remarks concerning categories and subspaces, Canad. J. Math. 9 (1957) 563

[49] M Johnson, The combinatorics of $n$–categorical pasting, J. Pure Appl. Algebra 62 (1989) 211

[50] A Joyal, R Street, Pullbacks equivalent to pseudopullbacks, Cah. Top. Géom. Différ. Catég. 34 (1993) 153

[51] A Joyal, M Tierney, Strong stacks and classifying spaces, from: "Category theory (Como, 1990)", Lecture Notes in Math. 1488, Springer (1991) 213

[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

[53] G M Kelly, Elementary observations on $2$–categorical limits, Bull. Austral. Math. Soc. 39 (1989) 301

[54] G M Kelly, Basic concepts of enriched category theory, Repr. Theory Appl. Categ. (2005)

[55] T Kerler, V V Lyubashenko, Non-semisimple topological quantum field theories for $3$–manifolds with corners, Lecture Notes in Math. 1765, Springer (2001)

[56] J Kock, Note on commutativity in double semigroups and two-fold monoidal categories, J. Homotopy Relat. Struct. 2 (2007) 217

[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”

[58] S Lack, A Quillen model structure for $2$–categories, $K$–Theory 26 (2002) 171

[59] S Lack, A Quillen model structure for bicategories, $K$-Theory 33 (2004) 185

[60] S Lack, Homotopy-theoretic aspects of $2$–monads, J. Homotopy Relat. Struct. 2 (2007) 229

[61] S Lack, S Paoli, An operadic approach to internal structures, Appl. Categ. Structures 13 (2005) 205

[62] S Lack, S Paoli, $2$–nerves for bicategories, $K$-Theory 38 (2008) 153

[63] J L Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179

[64] S Mac Lane, Categories for the working mathematician, Graduate Texts in Math. 5, Springer (1998)

[65] S Mac Lane, I Moerdijk, Sheaves in geometry and logic, Universitext, Springer (1994)

[66] J P May, J Sigurdsson, Parametrized homotopy theory, Math. Surveys and Monogr. 132, Amer. Math. Soc. (2006)

[67] J Mersch, Structures quotients, Bull. Soc. Roy. Sci. Liege 33 (1964) 45

[68] J Mersch, Le problème du quotient dans les catégories, Mém. Soc. Roy. Sci. Liège Coll. in-8 $(5)$ 11 (1965) 103

[69] J Morton, A double bicategory of cobordisms with corners

[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

[72] A J Power, A $2$–categorical pasting theorem, J. Algebra 129 (1990) 439

[73] A J Power, An $n$–categorical pasting theorem, from: "Category theory (Como, 1990)", Lecture Notes in Math. 1488, Springer (1991) 326

[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

[76] M Shulman, Comparing composites of left and right derived functors

[77] M Shulman, Framed bicategories and monoidal fibrations

[78] R Street, Categorical structures, from: "Handbook of algebra, Vol. 1", North-Holland (1996) 529

[79] A Strøm, The homotopy category is a homotopy category, Arch. Math. $($Basel$)$ 23 (1972) 435

[80] R W Thomason, Cat as a closed model category, Cah. Top. Géom. Différ. 21 (1980) 305

[81] B Toën, Vers une axiomatisation de la théorie des catégories supérieures, $K$–Theory 34 (2005) 233

[82] V Trnková, Sum of categories with amalgamated subcategory, Comment. Math. Univ. Carolinae 6 (1965) 449

[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