Tagged Questions

In mathematics, a monoidal category (or tensor category) is a category C equipped with a bifunctor ⊗ : C × C → C which is associative up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism. (Def: http://en.m.wikipedia.org/wiki/...

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Grothendieck's yoga of six operations - in relatively basic terms?

I'm reading about the basic interactions between sheaves over topological spaces and arrows in $\mathsf{Top}$, in particular, about the inverse/direct image functors $f^\ast \dashv f_\ast$, the proper ...
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Semirings induced by symmetric monoidal categories with finite coproducts

A symmetric monoidal category with finite coproducts is by definition a symmetric monoidal category $(\mathcal{C},\otimes,1,\dotsc)$ such that the underlying category $\mathcal{C}$ has finite ...
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What is the decategorification of a triangulated category?

The decategorification of an essentially small category $\mathcal C$ is the set $\lvert\mathcal C\rvert$ of isomorphism classes of $\mathcal C$. If $\mathcal C$ carries additional structure, then so ...
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Why can I choose to work in a strict monoidal category without loss of generality?

Let $\mathcal A$ be a monoidal category. We know that $\mathcal A$ is monoidally equivalent to a strict monoidal category $\mathcal A^{\mathrm{str}}$. In many books/papers it is assumed without loss ...
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Rings that cannot be representations rings

Given a monoidal category $\mathcal{C}$ one can define the Green ring (or representation ring) $r(\mathcal{C})$ as the abelian group generated by the isomorphism classes $[V]$ of $\mathcal{C}$ modulo ...
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What natural monoidal structure and braiding exists on the category of modules of the convolution algebra of an action groupoid?

Let $S$ be a set with an action $\triangleright$ of a finite group $G$. The action groupoid $S // G$ has as objects the set $S$, and the morphisms from $s_1$ to $s_2$ are just the $g \in G$ that ...
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Tensor of tensored categories

Given two $V$-categories $C$ and $D$ tensored over a symmetric monoidal category $V$, could I form the "tensor" of $C$ and $D$? More precisely, is there a $V$-category $T(C,D)$ such that $V$-functors ...
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Generalization of analytic functors

A functor $F\colon \bf Sets\to Sets$ is said to be analytic if it results from the left Kan extension of a functor $f\colon \mathbf{Bij}(\mathbb N)\to \bf Sets$ (the "species" of the functors $F$) ...
I'm led to define a promonad in $\bf D$ as a monoid in the category of endo-profunctors of a category $\bf D$, where the product of two profunctors is their composition as profunctors: $$F\odot G := \... 1answer 86 views The algebra of natural transformations of the n-th power tensor functor Let k be a 0 characteristic field, n an positive integer and S_n the n-th symmetric group. Let's work in the symmetric monoidal category of k-vector spaces and linear maps that we denote ... 1answer 341 views What is the categorical perspective on representations of topological groups? One categorical definition of a group G is that it is a category C with a single object X such that every morphism in the set C(X,X) is invertible, i.e. such that C(X,X) is precisely the set-... 1answer 62 views A question regarding monoidal closed categories If a category \mathcal{C} is (symmetric) monoidal closed, is the opposite category \mathcal{C}^{\text{op}} also monoidal closed? It is not clear to me whether by dualising the natural bijection ... 2answers 218 views “Change-of-base” between enriched categories I would like to prove that a monoidal functor$$\Phi\colon \mathbf{V}\to \mathbf{V'}$$induces a functor$$\Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}$$and in particular I ... 1answer 36 views Are units in rigid (autonomous) categories some sort of natural transformation? In a rigid category \mathcal{C}, let us choose left and right duals and left and right (co)units for every object. This gives us, for example, a dualisation functor -^*:\mathcal{C} \to \mathcal{C}^{... 1answer 83 views Uniqueness of Duals in a Monoidal Category Given a monoidal category {\cal C}, and X \in {\cal C}, we define a left dual of X to be an object X^* such that there exist morphisms \epsilon:X^* \otimes X \to I, and \eta:I \to X \otimes ... 1answer 96 views At a closed monoidal category, how can I derive a morphism C^A\times C^B\to C^{A+B}? Let A, B and C be objects of a closed monoidal category which is also bicartesian closed. How can I derive a morphism C^A\times C^B\to C^{A+B}? (-)\times (-) denotes the product, (-)+(-) ... 1answer 147 views Reasons for coherence for bi/monoidal categories Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this ... 2answers 85 views The rigid additive tensor category freely generated by an object I've found myself to be absolutely mystified by something in Deligne and Milne's notes on Tannakian categories. Namely, on p. 16 they are showing that there is a rigid additive tensor category \... 1answer 99 views Coherence result for (braided) monoidal functors Is there any coherence result for (braided) monoidal functors? (like Mac Lane's coherence theorem for monoidal categories) What I have in mind is a theorem like the following: Let F be a (... 1answer 67 views Rooted trees morphisms and categories If I take the monoidal category freely generated by a single object A and a morphism f: A \otimes A \to A, I end up with the monoidal category whose morphisms are forests of binary rooted trees. ... 1answer 77 views Associative Law for a Monoid in a Monoidal Category. Let B be a monoidal category, and c a monoid in B. Powers of c are defined by taking c^{n} to be the \otimes -string of length n of c in which the parentheses are all in front. We ... 1answer 72 views Iterated Coproduct in a Monoidal Category; finding the unit of a monoid. Suppose B is a monoidal category and further that the functors -\bigotimes a:B\rightarrow B and a\bigotimes -:B\rightarrow B preserve coproducts. The we have \theta :\coprod _{b} a\bigotimes b_{... 0answers 47 views When modular tensor categories are equivalent? I would like to know when we say that two modular tensor categories are equivalent. Is it true that two modular tensor categories are equivalent if they are equivalent as monoidal categories? Or do ... 0answers 61 views Tensoring a connective chain complex with a simplicial set Let \mathrm{Ch}_{\geq 0}(R) be the category of chain complexes of R-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ... 0answers 103 views Mac Lane's Coherence Theorem: Why not just use the functors themselves? I have a softball question on Mac Lane's proof. Suppose B=\left ( B, \square , \alpha ,\rho ,\lambda \right ) is a monoidal category. Fix b\in B. Define W, the (monoidal) category of binary ... 0answers 98 views What's the difference between a cartesian monoidal category and a semicartesian monoidal category? According to ncatlab: In a semicartesian monoidal category, any tensor product of objects x \otimes y comes equipped with morphisms$$ p_x : x \otimes y \to x  p_y : x \otimes y \to y$... 0answers 133 views Logic in closed symmetric monoidal categories; reference request. Suppose we want an algebraic theory$T$to be interpretable in any closed symmetric monoidal category$\mathbf{C}.$I am thinking in particular of the case where$\mathbf{C}$is the category of models ... 0answers 55 views Characterization of certain maps in$Hom(A \otimes A^{*}, A \otimes A^{*})$Let$(M, \otimes, I)$be a symmetric monoidal category and let$(A, B, \eta, \epsilon)$be a dual pair in$M$. Consider maps$i_{A}: Hom(A, A) \rightarrow Hom(A \otimes B, A \otimes B)$,$i_{B}: Hom(...
Let $\mathcal{C}$ and $\mathcal{D}$ be categories with finite products, and consider them as monoidal categories in the obvious way. Every functor $\mathcal{C} \to \mathcal{D}$ can be canonically ...