Related papers: Constructing categorical idempotents

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.

We construct the Heisenberg counterpart of a Clifford categorification. It is a modification of Khovanov's Heisenberg categorification. We express generators of the Heisenberg category as a complex of generators of the Clifford category.…

Let $\mathcal{A}$ be an essentially small abelian category. We prove that if $\mathcal{A}$ admits a generator $M$ with ${\rm End}_{\mathcal{A}}(M)$ right artinian, then $\mathcal{A}$ admits a projective generator. If $\mathcal{A}$ is…

Using the syzygy method, established in our earlier paper, we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such…

Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of…

We study tensor structures on (Rep G)-module categories defined by actions of a compact quantum group G on unital C*-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of…

I classify projective modules over idempotent semirings that are free on a monoid. The analysis extends to the case of the semiring of convex, piecewise-affine functions on a polyhedron, for which projective modules correspond to convex…

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…

We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. The extensive and general theory underlying this…

Over a finite-dimensonal algbera $A$, simple $A$-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings, and they have also appeared in…

For a complete and cocomplete category $\mathcal{C}$ with a well-behaved class of `projectives' $\bar{\mathcal{P}}$, we construct a model structure on the category $s\mathcal{C}$ of simplicial objects in $\mathcal{C}$ where the weak…

We provide the expected constructions of weakly $\omega$-categorified models (in the sense of Bressie) of the theory of groups and quandles which arise by replacing the homotopies used to give equivalence relations in the theory of…

We discuss what it means for a symmetric monoidal category to be a module over a commutative semiring category. Each of the categories of (1) cartesian monoidal categories, (2) semiadditive categories, and (3) connective spectra can be…

In previous work, we showed that there are appropriate model category structures on the category of simplicial categories and on the category of Segal precategories, and that they are Quillen equivalent to one another and to Rezk's complete…

We construct a compact closed category out of any symmetric monoidal category by freely adding adjoints to its objects. The morphisms of the completion are defined as string diagrams annotated by objects and morphisms from the original…

We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…

The purpose of this paper is to consider some basic constructions in the category of compact quantum groups --for example de case of extensions, of Drinfeld twists, of matched pairs, of extensions, of linked pairs and of cocycle Singer…

We construct free monoids in a monoidal category with finite limits and countable colimits, in which tensoring on either side preserves reflexive coequalizers and colimits of countable chains.

We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in…

We define a higher-order generalisation of the CPM construction based on arbitrary finite abelian group symmetries of symmetric monoidal categories. We show that our new construction is functorial, and that its closure under iteration can…