Related papers: Topological *-autonomous categories, revisited
Our aim is to give a fairly complete account on the construction of compatible model structures on exact categories and symmetric monoidal exact categories, in some cases generalizing previously known results. We describe the close…
We study monoidal 2-categories and bicategories in terms of categorical extensions and the cohomological data they determine in appropriate cohomology theories with coefficients in Picard groupoids. In particular, we analyze the hierarchy…
We prove that the homotopy theory of parsummable categories (as defined by Schwede) with respect to the underlying equivalences of categories is equivalent to the usual homotopy theory of symmetric monoidal categories. In particular, this…
We introduce a notion of globular multicategory with homomorphism types. These structures arise when organizing collections of "higher category-like" objects such as type theories with identity types. We show how these globular…
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
We develop a `universal' support theory for derived categories of constructible (analytic or \'etale) sheaves, holonomic D-modules, mixed Hodge modules and others. As applications we classify such objects up to the tensor triangulated…
We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples…
We establish a bi-equivalence between the bi-category of topoi with enough points and a localisation of a bi-subcategory of topological groupoids
We show that the categories of compact Lie groups and complex reductive groups (not necessarily connected) are homotopy equivalent topological categories. In other words, the corresponding categories enriched in the homotopy category of…
Some aspects of basic category theory are developed in a finitely complete category $\C$, endowed with two factorization systems which determine the same discrete objects and are linked by a simple reciprocal stability law. Resting on this…
We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.
In view of the fact that many of the most familiar examples of category bases are equivalent to some topology, it is natural to ask whether category bases are always topological in nature. The answer is in the negative. In this paper, we…
It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a topologically enriched locally contractible small category is Quillen equivalent to the standard Quillen model…
To any dg-category $T$ (over some base ring $k$), we define a $D^{-}$-stack $\mathcal{M}_{T}$ in the sense of \cite{hagII}, classifying certain $T^{op}$-dg-modules. When $T$ is saturated, $\mathcal{M}_{T}$ classifies compact objects in the…
A cocycle category H(X,Y) is defined for objects X and Y in a model category, and it is shown that the set of morphisms [X,Y] is isomorphic to the set of path components of H(X,Y) provided the ambient model category is right proper and…
This rough note describes some attempts to define a notion of enriched topology (and the associated theory of enriched stacks) on a category enriched over a symmetric monoidal model category, and poses some related questions.
This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the…
For a collection of subcategories satisfying a fixed set of conditions, for example thick subcategories of a triangulated category, we define a topological space called classifying space of subcategories. We show that this space classifies…
The main source of inspiration for the present paper is the work of R. Rosebrugh and R.J. Wood on constructive complete distributive lattices where the authors employ elegantly the concepts of adjunction and module in their study of ordered…
We extend the theory of Sweeder's measuring comonoids to the framework of duoidal categories: categories equipped with two compatible monoidal structures. We use one of the tensor products to endow the category of monoids for the other with…