Related papers: On functors between categories of modules over tru…

The balanced tensor product M (x)_A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M x N. The balanced tensor product M [x]_C N of two module categories over a monoidal…

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…

In this paper, we describe a general theory of modules over an algebra over an operad. We also study functors between categories of modules. Specializing to the operad E_d of little d-dimensional disks, we show that each (d-1)-manifold…

In [arXiv:1509.02937], the notion of a module tensor category was introduced as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow \mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal category…

Let $\mathcal{S}$ be a small category admitting binary products. We show that the whole theory of monoidal $\mathcal{S}$-fibered categories, which is customarily formulated in terms of the usual internal tensor product, can be rephrased…

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…

A finite tensor category is called pointed if all its simple objects are invertible. We find necessary and sufficient conditions for two pointed semisimple categories to be dual to each other with respect to a module category. Whenever the…

We show that every action operad gives rise to a notion of monoidal category via the categorical version of the Borel construction, embedding action operads into the category of 2-monads on $\mathbf{Cat}$. We characterize those 2-monads in…

It is shown that the idempotent completion of the additive hull of the tensor product of the residue category of the category of paths of a locally finite quiver modulo an admissible ideal and a dualizing category is dualizing. Furthermore,…

For a finite free and projective EI category, we prove that Gorenstein-projective modules over its category algebra are closed under the tensor product if and only if each morphism in the given category is a monomorphism.

In this paper we study the relative tensor product of module categories over braided fusion categories using, in part, the notion of the relative center of a module category. In particular we investigate the canonical tensor category…

In the theory of operads we consider functors of generalized symmetric powers defined by sums of coinvariant modules under actions of symmetric groups. One observes classically that the construction of symmetric functors provides an…

The monoidal version of classical Morita theory is a theory of bialgebroids. To make this explicit we construct a bicategory the objects of which are the bialgebroids and in which equivalence of objects means that the corresponding module…

We aim to study Morita theory for tensor triangulated categories. For two finite tensor categories having no projective simple objects, we prove that their stable equivalence induced by an exact $\Bbbk$-linear monoidal functor can be lifted…

This article offers an intuitive introduction to monoidal categories through the lens of painting, presenting abstract mathematical concepts with visual and tactile analogies. Aimed at curious undergraduates and non-specialists, it seeks to…

We present a unified framework for categorical systems theory which packages a collection of open systems, their interactions, and their maps into a symmetric monoidal loose right module of systems over a symmetric monoidal double category…

It is known that finite crossed modules provide premodular tensor categories. These categories are in fact modularizable. We construct the modularization and show that it is equivalent to the module category of a finite Drinfeld double.

This article is devoted to the investigation of the deformation (twisting) of monoidal structures, such as the associativity constraint of the monoidal category and the monoidal structure of monoidal functor. The sets of twistings have a…

We exhibit sufficient conditions for a monoidal monad T on a monoidal category C to induce a monoidal structure on the Eilenberg--Moore category C^T that represents bimorphisms. The category of actions in C^T is then shown to be monadic…

We introduce a theory of modules over a representation of a small category taking values in entwining structures over a semiperfect coalgebra. This takes forward the aim of developing categories of entwined modules to the same extent as…