Related papers: Cellular categories

This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax…

We provide a complete description of the category of pseudo-categories (including pseudo-functors, natural and pseudo-natural transformations and pseudo modifications). A pseudo-category is a non strict version of an internal category. It…

In his book on model categories, Hovey asked whether the 2-category $\mathbf{Mod}$ of model categories admits a "model 2-category structure" whose weak equivalences are the Quillen equivalences. We show that $\mathbf{Mod}$ does not have…

We give a classification of substructures (= closed subbifunctors) of a given skeletally small extriangulated category by using the category of defects, in a similar way to the author's classification of exact structures of a given additive…

In the sixties, Grothendieck developed the theory of pro-objects over a category. The fundamental property of the category $Pro(C)$ is that there is an embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small…

We show that the classifying category C(T) of a dependent type theory T with axioms for identity types admits a non-trivial weak factorisation system. We provide an explicit characterisation of the elements of both the left class and the…

We prove, without set theoretic assumptions, that every locally presentable category C endowed with a tractable cofibrantly generated class of cofibrations has a unique minimal (or left induced) Quillen model structure. More generally, for…

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy…

We prove that the 2-category Grt of Grothendieck abelian categories with colimit preserving functors and natural transformations is a bicategory of fractions in the sense of Pronk of the 2-category Site of linear sites with continuous…

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak…

Delta lenses are functors equipped with a functorial choice of lifts, generalising the notion of split opfibration. In this paper, we introduce a Grothendieck construction (or category of elements) for delta lenses, thus demonstrating a…

We show that in the category of groups, every singly-generated class which is closed under isomorphisms, direct limits and extensions is also singly-generated under isomorphisms and direct limits, and in particular is co-reflective. We also…

The notion of retrocell in a double category with companions is introduced and its basic properties established. Explicit descriptions in some of the usual double categories are given. Monads in a double category provide an important…

Based on a Whitehead-type characterization of the sectional category we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of…

We establish a Dwyer-Kan equivalence of relative categories of combinatorial model categories, presentable quasicategories, and other models for locally presentable (infinity,1)-categories. This implies that the underlying quasicategories…

We use pointwise Kan extensions to generate new subcategories out of old ones. We investigate the properties of these newly produced categories and give sufficient conditions for their cartesian closedness to hold. Our methods are of…

We study discrete opfibration classifiers in enhanced 2-categories and show how, under suitable hypotheses, such classifiers can be endowed with the structure of a (lax or pseudo-)T-algebra and classify strict discrete opfibrations in…

We introduce the category of bicomodules for a comonad in a Grothendieck category whose underlying functor is right exact and preserves direct sums. We characterize comonads with a separable forgetful functor by means of cohomology groups…

We show that almost split sequences in the category of comodules over a coalgebra with finite-dimensional right-hand term are direct limits of almost split sequences over finite dimensional subcoalgebras. In previous work we showed that…

We introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of…