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The purpose of this note is to show that, if $mathcal{V}$ is a closed monoidal category, the following three notions are equivalent. (1) Category with $mathcal{V}$-structure and cylinder. (2) Tensored $mathcal{V}$-category. (3) Tensor-closed $mathcal{V}$-module. As an application we will show that, if $mathcal{V}$ is closed and symmetric, then given a category $mathcal{S}$ there is an one-to-one correspondence between the set of $mathcal{V}$-structures with cylinder and path on $mathcal{S}$ introduced by Quillen and the set of closed $mathcal{V}$-module structures on $mathcal{S}$ introduced by Hovey.
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We show that und
Restriction categories were established to handle maps that are partially defined with respect to composition. Tensor topology realises that monoidal categories have an intrinsic notion of space, and deals with objects and maps that are partially def
We introduce a notion of $n$-commutativity ($0le nle infty$) for cosimplicial monoids in a symmetric monoidal category ${bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${bf V,}$ while $n=infty$ corresponds to commutative cosimplicial
We give a proof of a formula for the trace of self-braidings (in an arbitrary channel) in UMTCs which first appeared in the context of rational conformal field theories (CFTs). The trace is another invariant for UMTCs which depends only on modular da
We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally