Related papers: On the construction of functorial factorizations f…
In this note, we discuss several aspects of the functoriality of universal abelian factorizations associated to representations of quivers into abelian categories. After recalling the general construction of universal abelian…
We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several…
Model structures for many different kinds of functor calculus can be obtained by applying a theorem of Bousfield to a suitable category of functors. In this paper, we give a general criterion for when model categories obtained via this…
We present an efficient and user-friendly method for constructing any cofibrantly generated model structure on the category of double categories whose trivial fibrations are the "canonical" ones: the double functors which are surjective on…
The homotopy category of a model structure on a weakly idempotent complete additive category is proved to be equivalent to the additive quotient of the category of cofibrant-fibrant objects with respect to the subcategory of…
If M is a model category and Z is an object of M, then there are model category structures on the category of objects of M over Z and the category of objects of M under Z under which a map is a cofibration, fibration, or weak equivalence if…
Generalizing Eisenbud's matrix factorizations, we define factorization categories. Following work of Positselski, we define their associated derived categories. We construct specific resolutions of factorizations built from a choice of…
We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…
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.
Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized…
There is an ``algebraisation'' of the notion of weak factorisation system (w.f.s.) known as a natural weak factorisation system. In it, the two classes of maps of a w.f.s. are replaced by two categories of maps-with-structure, where the…
We develop a categorical framework for reasoning about abstract properties of differentiation, based on the theory of fibrations. Our work encompasses the first-order fragments of several existing categorical structures for differentiation,…
We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…
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…
In this paper we develop the theory of topological categories over a base category, that is, a theory of topological functors. Our notion of topological functor is similar to (but not the same) the existing notions in the literature (see…
In this paper the concept of compatible weak factorization systems in general categories is introduced as a counterpart of compatible complete cotorsion pairs in abelian categories. We describe a method to construct model structures on…
The Grothendieck construction is a classical correspondence between diagrams of categories and coCartesian fibrations over the indexing category. In this paper we consider the analogous correspondence in the setting of model categories. As…
We define a natural 2-categorical structure on the base category of a large class of Grothendieck fibrations. Given any model category $\mathbf{C}$, we apply this construction to a fibration whose fibers are the homotopy categories of the…
We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform)…
Derived D-Geometry is considered as a convenient language for a coordinate-free investigation of nonlinear partial differential equations up to symmetries. One of the first issues one meets in the functor of points approach to derived…