Related papers: An enriched small object argument over a cofibrant…

We introduce a notion of a filtered model structure and use this notion to produce various model structures on pro-categories. This framework generalizes several known examples. We give several examples, including a homotopy theory for…

Weak (Hopf) bialgebras are described as (Hopf) bimonoids in appropriate duoidal (also known as 2-monoidal) categories. This interpretation is used to define a category wba of weak bialgebras over a given field. As an application, the "free…

We prove that a weak factorization system on a locally presentable category is accessible if and only if it is small generated in the sense of R. Garner. Moreover, we discuss an analogy of Smith's theorem for accessible model categories.

We introduce a notion of "weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of…

We provide a more economical refined version of Evrard's categorical cocylinder factorization of a functor [Ev1,2]. We show that any functor between small categories can be factored into a homotopy equivalence followed by a (co)fibred…

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…

In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure…

In this paper, we study properties of maps between fibrant objects in model categories. We give a characterization of weak equivalences between fibrant object. If every object of a model category is fibrant, then we give a simple…

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal…

There is a well-known correspondence between coherent theories (and their interpretations) and coherent categories (resp. functors), hence the (2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent functors and all…

Argumentation is the process of constructing arguments about propositions, and the assignment of statements of confidence to those propositions based on the nature and relative strength of their supporting arguments. The process is modelled…

We define a new model structure on the category of small categories, which is intimately related to the notion of coverings and fundamental groups of small categories. Fibrant objects in the model structure coincide with groupoids, and the…

A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is…

In this paper, we discuss certain circumstances in which the category of tame functors inherits an abelian category structure with minimal resolutions and a model category structure with minimal cofibrant replacements. We also present a…

Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an…

We develop a correspondence between presentations of compactly generated triangulated categories as localizations of derived categories of ring spectra and proxy-small objects, and explore some consequences. In addition, we give a…

We prove that, assuming Vop\venka's principle, every small projectivity class in a locally presentable category is accessible.

We introduce Hopf categories enriched over braided monoidal categories. The notion is linked to several recently developed notions in Hopf algebra theory, such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We…

We construct a machine which takes as input a locally small symmetric closed complete multicategory $\mathsf V$. And its output is again a locally small symmetric closed complete multicategory $\mathsf V\text-\mathcal{C}at$, the…