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تسجيل مستخدم جديدTensor Topology
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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 under mild conditions subunits endow any monoidal category with a kind of topological intuition: there are well-behaved notions of restriction, localisation, and support, even though the subunits in general only form a semilattice. We develop universal constructions completing any monoidal category to one whose subunits universally form a lattice, preframe, or frame.
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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 $m
athcal{V}$-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.
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
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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
ta, and contains the expression of the Frobenius-Schur indicator as a special case. Furthermore, we discuss some applications of the trace formula to the realizability problem of modular data and to the classification of UMTCs.
We consider algebras in a modular tensor category C. If the trace pairing of an algebra A in C is non-degenerate we associate to A a commutative algebra Z(A), called the full centre, in a doubled version of the category C. We prove that two simple al
gebras with non-degenerate trace pairing are Morita-equivalent if and only if their full centres are isomorphic as algebras. This result has an interesting interpretation in two-dimensional rational conformal field theory; it implies that there cannot be several incompatible sets of boundary conditions for a given bulk theory.
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
monoids. If ${bf V}$ has a monoidal model structure we show (under some mild technical conditions) that the total object of an $n$-cosimplicial monoid has a natural $E_{n+1}$-algebra structure. Our main applications are to the deformation theory of tensor categories and tensor functors. We show that the deformation complex of a tensor functor is a total complex of a $1$-commutative cosimplicial monoid and, hence, has an $E_2$-algebra structure similar to the $E_2$-structure on Hochschild complex of an associative algebra provided by Delignes conjecture. We further demonstrate that the deformation complex of a tensor category is the total complex of a $2$-commutative cosimplicial monoid and, therefore, is naturally an $E_3$-algebra. We make these structures very explicit through a language of Delannoy paths and their noncommutative liftings. We investigate how these structures manifest themselves in concrete examples.
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