Related papers: Centers and homotopy centers in enriched monoidal …
Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal…
Polynomials in a category have been studied as a generalization of the traditional notion in mathematics. Their construction has recently been extended to higher groupoids, as formalized in homotopy type theory, by Finster, Mimram, Lucas…
We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a…
We develop the theory of tricategorical limits and colimits, and show that they can be modelled up to biequivalence via certain homotopically well-behaved limits and colimits enriched over the monoidal model category $\mathbf{Gray}$ of…
Certain aspects of Street's formal theory of monads in 2-categories are extended to multimonoidal monads in symmetric strict monoidal 2-categories. Namely, any symmetric strict monoidal 2-category $\mathcal M$ admits a symmetric strict…
We give a rather general construction of double categories and so, under further conditions, double groupoids, from a structure we call a `double module'. We also give a homotopical construction of a double groupoid from a triad consisting…
We adapt the notion of an algebraic theory to work in the setting of quasicategories developed recently by Joyal and Lurie. We develop the general theory at some length. We study one extended example in detail: the theory of commutative…
We discuss generalised duality theory for monoidal categories and its applications to the categories of exact endofunctors, graded vector spaces, and topological vector spaces.
In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories $E$, and use these to study model structures on categories of chain…
We study Quillen model categories equipped with a monoidal skew closed structure that descends to a genuine monoidal closed structure on the homotopy category. Our examples are 2-categorical and include permutative categories and…
This new version includes a connection of the main construction to the Gottlieb group, which was absent in the previous versions. However, the first version included material about Lie algebras which will become available soon as a separate…
The Hochschild cohomology of a differential graded algebra, or a differential graded category, admits a natural map to the graded center of its homology category: the characteristic homomorphism. We interpret it as an edge homomorphism in a…
We develop a theory of enriched categories over a (higher) category M equipped with a class W of morphisms called homotopy equivalences. We call them Segal M_W -categories. Our motivation was to generalize the notion of "up-to-homotopy…
This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories…
Polynomial functors are a categorical generalization of the usual notion of polynomial, which has found many applications in higher categories and type theory: those are generated by polynomials consisting a set of monomials built from sets…
In this survey paper we give account of several approaches to the strictification and non-strictification of monoidal categories, which are constructions that turn a monoidal category into a (non-)strict one monoidally equivalent to the…
In this paper, we introduce the notion of Grothendieck enriched categories for categories enriched over a sufficiently nice Grothendieck monoidal category $\mathcal{V}$, generalizing the classical notion of Grothendieck categories. Then we…
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.
When a category is equipped with a 2-cell structure it becomes a sesquicategory but not necessarily a 2-category. It is widely accepted that the latter property is equivalent to the middle interchange law. However, little attention has been…
We introduce Hochschild (co-)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the…