Free extensions of double categories (original) (raw)
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2012
This paper introduces the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories; it explores its use in defining a double category of fractions, and shows that the sub-2-category of groupoidal weakly globular double categories forms an algebraic model of homotopy 2-types.
A Double Categorical Model of Weak 2-CATEGORIES
2013
We introduce the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories. We show that this model is suitably equivalent to bicategories and give an explicit description of the functors involved in this biequivalence. As an application we show that groupoidal weakly globular double categories model homotopy 2-types.
Cartesian Double Categories with an Emphasis on Characterizing Spans
arXiv: Category Theory, 2018
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us to set the whole theory inside the welcoming 2-category of double categories, and to overcome technical problems that were caused by working with left adjoints inside a general bicategory. Cartesian double categories that are also fibrant are of particular interest to us. After describing some important properties of Cartesian and fibrant double categories, we give a characterization of the double category of Spans as a Cartesian double category. Lastly, we talk about profunctors and give a potential framework for their characterization as Cartesian double categories.
Categories enriched on two sides
Journal of Pure and Applied Algebra, 2002
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Model structures on the category of small double categories
Algebraic & Geometric Topology, 2008
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 categorificationnerve 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 and discuss properties of free double categories, quotient double categories, colimits of double categories, several nerves, and horizontal categorification.
Double Categories, 2CATEGORIES, Thin Structures and Connections
1999
The main result is that two possible structures which may be imposed on an edge symmetric double category, namely a connection pair and a thin structure, are equivalent. A full proof is also given of the theorem of Spencer, that the category of small 2-categories is equivalent to the category of edge symmetric double categories with thin structure.
Complete theories in 2-categories
Cahiers de topologie et géométrie différentielle catégoriques, tome 29, n o 1 (1988), p. 9-57 http://www.numdam.org/item?id=CTGDC\_1988\_\_29\_1\_9\_0 © Andrée C. Ehresmann et les auteurs, 1988, tous droits réservés. L'accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 9 COMPLETE THEORIES IN 2-CATEGORIES by Renato BETTI and Marco GRANDIS CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATEGORI QUES Vol. XXIX-1 (1988) RÉSUMÉ. On étudie les théories a valeurs dans une 2-cat6gorie concrète A sous forme de 2-foncteurs "des mod6les" T:
Journal of Pure and Applied Algebra, 1994
The concept of regular category Cl] has several 2-dimensional analogues depending upon which special arrows are chosen to mimic monies. Here, the choice of the conservative arrows, leads to our notion of faithfully conservative bicategory X in which two-sided discrete fibrations become the arrows of a bicategory 9 = DFib(X). While the horncategories 9(B, A) have finite limits, it is important to have conditions under which these finite "local" limits are preserved by composition (on either side) with arrows of 9. In other words, when are all fibrations in LX? flat? Novel axioms on X are provided for this, and we call a bicategory S' modulated when sP' 1s such a X. Thus, we have constructed a proarrow equipment ( )* : A? + ~2 (in the sense of [28]) with A = Y C c00p. Moreover, A is locally finitely cocomplete and certain collages exist .
Monoidal 2-Categories: A Review
arXiv: Category Theory, 2020
We review the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of weak 3-category(or tricategory).