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1answer
73 views

Morphisms in a symmetric monoidal closed category.

Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon - \otimes I \to 1_{\mathcal ...
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64 views

Cartesian monoidal functors

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 ...
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1answer
91 views

Is the inverse to a monoidal equivalence also monoidal?

Let ${\cal C,D}$ be two categories, and let $$ F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C}, $$ be an equivalence of categories. Let us now further assume that ${\cal C}$ can ...
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2answers
182 views

The abstract definition of commutative monoids

In trying to begin to understand the idea of a $k$-tuply monoidal $n$-category, I'm already a bit stuck on the idea (Baez, nLab) that a commutative monoid can be defined as a monoid object in the ...
5
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1answer
96 views

Coherence Conditions and Strict Monoidal Equivalences

Consider the following: two monoidal categories $({\cal C},\otimes)$, and $({\cal D},\odot)$, and a functor $F:{\cal C} \to {\cal D}$, that gives an equivalence (of ordinary categories) between ${\cal ...
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0answers
86 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
2
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1answer
126 views

Endomorphisms in a symmetric monoidal category

Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is ...
3
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1answer
236 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 ...
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1answer
357 views

Coherence for symmetric monoidal categories

Let $\mathcal{C}$ be a monoidal category. By Mac Lane's coherence theorem for monoidal categories, there are strong monoidal functors $F : \mathcal{C} \to \mathcal{C}_s$ and $G : \mathcal{C}_s \to ...
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122 views

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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0answers
114 views

Is push-forward of coherent sheaves a tensor functor?

Given a finite map between two Noetherian schemes $f: X \rightarrow Y$, is $f_*: \operatorname{Coh}{(X)} \rightarrow \operatorname{Coh}{(Y)}$ a tensor functor? If this is not true in general, is it ...
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1answer
94 views

Is every functor monoidal between monoidal categories where monoidal product is interpreted as sum?

Namely, for every functor $F$, is $(F, (A_0, A_1)\mapsto[F(\iota_0(A_0, A_1)), F(\iota_1(A_0, A_1))])$ monoidal? (For reference, $\iota_i(A_0, A_1):A_i\to A_0+A_1$ is an injection of the categorical ...
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votes
1answer
207 views

Reference request: Deligne's reconstruction theorem

I've heard this result referenced a few times on MO now. It is supposed to be a theorem of Deligne that gives some natural conditions under which an (abelian?) tensor category $C$ is the category of ...
3
votes
1answer
144 views

May left and right unitors be equal in a monoidal category?

Monoidal category. $\lambda_I = \rho_I : I\otimes I\to I$? If this equality can not be proved, in what categories it is false?
4
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2answers
152 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 ...
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1answer
108 views

Characterisation of duals in rigid monoidal categories

In "On Fusion Categories" by Etingof, Nikshych and Ostrik Proposition 2.1 there is used the following characterisation of a right dual in a rigid monoidal category (let us restrict to strict monoidal ...
5
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1answer
367 views

Drinfeld Center

Let $\mathscr{C}$ be a strict monoidal category. I will denote the product of $\mathscr{C}$ by $\otimes$. The Drinfeld center $\mathscr{Z(C)}$ of $\mathscr{C}$ is the category with object $(X,\phi)$ ...
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2answers
216 views

$(-1)\otimes (-1) \cong I$

Is there a monoidal category $\mathcal C$ whose unit object is $I$ (i.e. $I\otimes A\cong A\cong A\otimes I$ for all $A\in \text{Ob}_\mathcal C$), with an object "$-1$" such that $$ ...