Related papers: Double categorical equivalences

In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use…

We propose definitions of regular and exact (virtual) double categories, proving a number of results which parallel many basic results in the theory of regular and exact categories. We show that any regular virtual double category admits a…

We investigate a canonical way of defining bisimilarity of systems when their semantics is given by a coreflection, typically in a category of transition systems. We use the fact, from Joyal et al., that coreflections preserve open…

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided…

We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of $\infty$-operads. This model is directly analogous to a recent construction on the category of…

This is a continuation, completion, and generalization of our previous joint work with B. Chorny. We supply model structures and Quillen equivalences underlying Goodwillie's constructions on the homotopy level for functors between…

For a cofibrantly generated Quillen model category, we show that the cofibrant replacement functor constructed using the small object argument admits a cotriple structure. If all acyclic cofibrations are monomorphisms, the fibrant…

We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent…

We consider the problem of constructing the free bifibration generated by a functor of categories $p : D \to C$. This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Par\'e, and…

We characterize virtual double categories of enriched categories, functors, and profunctors by introducing a new notion of double-categorical colimits. Our characterization is strict in the sense that it is up to equivalence between virtual…

In this paper, we construct a model structure for $(\infty,1)$-categories on the category of simplicial spaces, whose fibrant objects are the Segal spaces. In particular, we show that it is Quillen equivalent to the models of…

In the present article, we describe constructions of model structures on general bicomplete categories. We are motivated by the following question: given a category $\mathcal{C}$ with a subcategory $w\mathcal{C}$ closed under retracts, when…

We study the accessibility properties of trivial cofibrations and weak equivalences in a combinatorial model category and prove an estimate for the accessibility rank of weak equivalences. In particular, we show that the class of weak…

Using full images of accessible functors, we prove some results about combinatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible.

This paper introduces the construction of a weakly globular double category of fractions for a category and studies its universal properties. It shows that this double category is locally small and considers a couple of concrete examples.

In this paper we study the question of how to transfer homotopic structure from the category sD of simplicial objects in a fixed category D to D. To this end we use a sort of homotopy colimit s : sD --> D, which we call simple functor. For…

We prove the (2,1)-categorical analogue of the small object argument and give a (2,1)-model structure on the category of small coherent categories, coherent functors and natural isomorphisms. It is induced by a higher dimensional example of…

We introduce a functor $\mathcal V\colon \mathrm{DblCat}_{h,nps}\to \mathrm{2Cat}_{h,nps}$ extracting from a double category a $2$-category whose objects and morphisms are the vertical morphisms and squares. We give a characterisation of…

Most categorical models for dependent types have traditionally been heavily set based: contexts form a category, and for each we have a set of types in said context -- and for each type a set of terms of said type. This is the case for…

Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide.…