the ordinary tensor product of modules is defined using bilinear maps (bimorphisms), that are linear in eachcomponent. keeping this in mind, linton and banaschewski with nelson defined and studied the tensor product in an equational category and in a general (concrete) category k, respectively, using bimorphisms, that is, defined via the hom-functor on k. also, the so-called sesquilinear, or on...

The Grothendieck construction is a process to form a single category from a diagram of small categories. In this paper, we extend the definition of the Grothendieck construction to diagrams of small categories enriched over a symmetric monoidal category satisfying certain conditions. Symmetric monoidal categories satisfying the conditions in this paper include the category of k-modules over a c...

Let k be a commutative ring and let C be the operad of differential graded k-modules obtained as the singular k-chains of the linear isometries operad [4, §V.9]. We show that the category of C-algebras is a proper closed model category. We use the amenable description of the coproduct in this category [4, V.3.4] to analyze the coproduct of and develop a homotopy theory for algebras over an arbi...

We show that there are no gaps in the lengths of the indecomposable objects in an abelian k-linear category over a field k provided all simples are absolutely simple. To derive this natural result we prove that any distributive minimal representation-infinite k-category is isomorphic to the linearization of the associated ray category which is shown to have an interval-finite universal cover wi...

We show that every essentially small finitely semisimple k-linear additive spherical category in which k = End(1) is a field, is equivalent to its dual over the long canonical forgetful functor. This includes the special case of modular categories. In order to prove this result, we show that the universal coend of the spherical category with respect to the long forgetful functor is self-dual as...

noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $math...

In this paper we take up again the deformation theory for K-linear pseudofunctors initiated in [4] with a two-fold purpose. Firstly, we introduce the notion of weak 2-cosemisimplicial object in a 2-category and show that the deformation complex X•(F) introduced in [4] can be obtained from one such object in the 2-category CatK of small K-linear categories. In doing this, we describe a family of...

For a symmetric monoidal-closed category X and any object K, the category of K-Chu spaces is small-topological over X and small cotopological over X . Its full subcategory of M-extensive K-Chu spaces is topological over X when X is Mcomplete, for any morphism class M. Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addi...