Model structures on the category of small double categories

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. 18D05, 18G55; 55P99, 55U10

[1]  Jean-Louis Loday,et al.  Spaces with finitely many non-trivial homotopy groups , 1982 .

[2]  P. J. Higgins,et al.  On the algebra of cubes , 1981 .

[3]  Richard Garner.  Double Clubs , 2006 .

[4]  Cahiers DE Topologie,et al.  Cat as a closed model category , 1980 .

[5]  Charles Ehresmann,et al.  Multiple functors. II. The monoidal closed category of multiple categories , 1978 .

[6]  M. Grandis.  KAN EXTENSIONS IN DOUBLE CATEGORIES (ON WEAK DOUBLE CATEGORIES, PART III) , 2008 .

[7]  John Greenlees,et al.  SIMPLICIAL HOMOTOPY THEORY (Progress in Mathematics 174) , 2001 .

[8]  Ronald Brown,et al.  Double groupoids and crossed modules , 1976 .

[9]  Ronald Brown,et al.  Determination of a double Lie groupoid by its core diagram , 1992 .

[10]  Bertrand Toën.  Vers une axiomatisation de la théorie des catégories supérieures. , 2005 .

[11]  Sally Popkorn,et al.  A Handbook of Categorical Algebra , 2009 .

[12]  A. Power,et al.  A 2-categorical pasting theorem , 1990 .

[13]  André Joyal,et al.  Strong stacks and classifying spaces , 1991 .

[14]  Charles Ehresmann,et al.  Multiple functors. IV. Monoidal closed structures on $Cat_n$ , 1979 .

[15]  Homotopy inverses for nerve , 1979 .

[16]  Free extensions of double categories , 2004 .

[17]  Robert J. MacG. Dawson,et al.  General associativity and general composition for double categories , 1993 .

[18]  Ross Street,et al.  The algebra of oriented simplexes , 1987 .

[19]  J. Morton.  A Double Bicategory of Cobordisms With Corners , 2006 .

[20]  R. J. Macg,et al.  PATHS IN DOUBLE CATEGORIES , 2006 .

[21]  A Full and Faithful Nerve for 2-Categories , 2004, Appl. Categorical Struct..

[22]  Thomas M. Fiore.  Pseudo Limits, Biadjoints, and Pseudo Algebras: Categorical Foundations of Conformal Field Theory , 2004, math/0408298.

[23]  S. Lack,et al.  2-nerves for bicategories , 2006, math/0607271.

[24]  P. J. Higgins,et al.  The equivalence of ∞-groupoids and crossed complexes , 2007 .

[25]  Ronald T. Brown,et al.  DOUBLE CATEGORIES, 2-CATEGORIES, THIN STRUCTURES AND CONNECTIONS , 1999 .

[26]  J. R. Isbell,et al.  Some Remarks Concerning Categories and Subspaces , 1957, Canadian Journal of Mathematics.

[27]  J. Adámek,et al.  Locally presentable and accessible categories , 1994 .

[28]  J. P. May,et al.  Parametrized homotopy theory , 2006 .

[29]  Simona Paoli.  Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids , 2006, math/0607330.

[30]  Towards a 2-dimensional notion of holonomy , 2000, math/0009082.

[31]  Michael Johnson,et al.  The combinatorics of n-categorical pasting☆ , 1989 .

[32]  Marco Grandis,et al.  Limits in double categories , 1999 .

[33]  S. Lack,et al.  A Quillen model structure for Gray-categories , 2002, 1001.2366.

[34]  Arne Strøm,et al.  The homotopy category is a homotopy category , 1972 .

[35]  J. Benabou.  Introduction to bicategories , 1967 .

[36]  J. Benabou,et al.  SIMPLICIAL MATRICES AND THE NERVES OF WEAK n-CATEGORIES I : NERVES OF BICATEGORIES , 2002 .

[37]  Paul G. Goerss,et al.  Simplicial Homotopy Theory , 2009, Modern Birkhäuser Classics.

[38]  Julia E. Bergner.  A model category structure on the category of simplicial categories , 2004 .

[39]  G. M. Kelly.  Elementary observations on 2-categorical limits , 1989, Bulletin of the Australian Mathematical Society.

[40]  G. M. Kelly,et al.  BASIC CONCEPTS OF ENRICHED CATEGORY THEORY , 2022, Elements of ∞-Category Theory.

[41]  Stephen Lack,et al.  An Operadic Approach to Internal Structures , 2005, Appl. Categorical Struct..

[42]  P. J. Higgins,et al.  The equivalence of $\omega $-groupoids and cubical $T$-complexes , 1981 .

[43]  T. Kerler,et al.  Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners , 2001 .

[44]  Eduardo J. Dubuc,et al.  Kan Extensions in Enriched Category Theory , 1970 .

[45]  Robert Paré,et al.  Stacks and equivalence of indexed categories , 1979 .

[46]  Charles Rezk,et al.  A model for the homotopy theory of homotopy theory , 1998, math/9811037.

[47]  M. Bullejos,et al.  On the Geometry of 2-Categories and their Classifying Spaces , 2003 .

[48]  M. Grandis.  Cubical cospans and higher cobordisms (Cospans in algebraic topology, III) , 2008, 0806.2359.

[49]  Michael Shulman,et al.  Comparing composites of left and right derived functors , 2007, 0706.2868.

[50]  Lax kan extensions for double categories * (On weak double categories, Part IV) , 2007 .

[51]  P. J. Higgins,et al.  The equivalence of $\infty$-groupoids and crossed complexes , 1981 .

[52]  Michael Barr,et al.  GENERALIZED CONGRUENCES — EPIMORPHISMS IN Cat , 1999 .

[53]  R. Dawson,et al.  What is a free double category like , 2002 .

[54]  S. Lane,et al.  Sheaves In Geometry And Logic , 1992 .

[55]  Multiple functors. III. The cartesian closed category $Cat_n$ , 1978 .

[56]  A. Karimi,et al.  Master‟s thesis , 2011 .

[57]  S. Lane.  Categories for the Working Mathematician , 1971 .

[58]  K. Hess,et al.  A model structure à la Thomason on 2 - CAT , 2004 .

[59]  Philip S. Hirschhorn.  Model categories and their localizations , 2003 .

[60]  T. Linden,et al.  Model structures for homotopy of internal categories , 2004 .

[61]  F. Linton,et al.  Categories et Structures. , 1968 .

[62]  V. Trnková.  Sum of categories with amalgamated subcategory , 1965 .

[63]  Charles Ehresmann,et al.  Multiple functors. I. Limits relative to double categories , 1974 .

[64]  Myles Tierney,et al.  Quasi-categories vs Segal spaces , 2006 .

[65]  Michael Shulman,et al.  Framed bicategories and monoidal fibrations , 2007, 0706.1286.

[66]  Ross Street,et al.  Pullbacks equivalent to pseudopullbacks , 1993 .

[67]  Julia E. Bergner,et al.  THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES , 2005, math/0504334.

[68]  Stephen Lack,et al.  Homotopy-theoretic aspects of 2-monads , 2006 .

[69]  Thomas M. Fiore,et al.  A thomason model structure on the category of small n-fold categories , 2008, 0808.4108.

[70]  PSEUDO ALGEBRAS AND PSEUDO DOUBLE CATEGORIES , 2006, math/0608760.

[71]  NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWO-FOLD MONOIDAL CATEGORIES , 2006, math/0608452.

[72]  P. J. Higgins,et al.  Tensor products and homotopies for ω-groupoids and crossed complexes , 1987 .

[73]  G. M. Kelly,et al.  Two-dimensional monad theory , 1989 .

[74]  Marco Grandis,et al.  Adjoint for double categories , 2004 .

[75]  Peter Gabriel,et al.  Calculus of Fractions and Homotopy Theory , 1967 .

[76]  Marco Grandis.  HIGHER COSPANS AND WEAK CUBICAL CATEGORIES (COSPANS IN ALGEBRAIC TOPOLOGY, I) , 2007 .

[77]  I. Moerdijk,et al.  A Shapiro lemma for diagrams of spaces with applications to equivariant topology , 1995 .