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# Questions tagged [symmetric-monoidal-categories]

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### Properties of semisimple monoidal category

In my work, I have constructed a semisimple category which has two monoidal structures: the usual direct sum; and a new "tensor product". This "tensor product" have several nice ...
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### Symmetric monoidal structures on the functor taking presheaves

Let $\mathrm{Cat}_\infty$ be the $\infty$-category of small $\infty$-categories, $\mathrm{Pr}^\mathrm{L}$ be the $\infty$-category of presentable $\infty$-categories, and $\mathcal P(\mathcal C)$ be ...
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### "Closed $\mathscr{V}$-modules are uniquely (co)tensored $\mathscr{V}$-categories": shouldn't we assume they are also "mixed monoidal"?

$\newcommand{\M}{\mathcal{M}}\newcommand{\V}{\mathscr{V}}\newcommand{\hom}{\mathsf{hom}}$Throughout this post, $\V$ refers to some "cosmos", where I borrow the word "cosmos" from ...
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### Day convolution for bimonoidal categories

Semiring categories, also called rig categories or bimonoidal categories, are pseudomonoids in the symmetric monoidal bicategory $(\mathsf{SymMonCats},\otimes_{\mathbb{F}},\mathbb{F})$¹. These are a ...
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### Symmetric monoidal structure on categorical nerves

There are several notions of nerves, including nerves of categories, $2$-categories, and simplicial categories. These define functors \begin{align*} \mathrm{N} &\colon \mathrm{Cats}_{(2,1)} ...
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### What is an invertible operad?

Let $\mathcal V$ be a nice symmetric monoidal ($\infty$-)category, and consider the ($\infty$-)category $Op(\mathcal V)$ of $\mathcal V$-enriched (symmetric) operads, symmetric monoidal under the ...
207 views

### Functors that preserve monoids

In the comments section of this question there was a question that I don't know if it has been asked on the site. It is well-known and easily proved that lax monoidal functors preserve monoids. So the ...
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### Functors between module categories that comes from restriction

Suppose you have two $k$ algebras $A, B$ (say also finitely generated if this helps) and a functor $F: A-mod \to B-mod$ such that $| F(M) |= |M|$. Here $|U|$ denotes the underlying $k$ vector space. ...
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### Braided monoidal category, example

Let $H$ be a cocommutative hopf algebra. Let $M$ be the category of $H$-bimodules. Does the category $M$ form a braided monoidal category with tensor product $\otimes_{H}$ ?
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### Tensor product of unit and co-unit in a closed compact category

Consider a compact closed category, i.e., a symmetric monoidal category with a unit $\eta$ and co-unit $\epsilon$. It seems natural to demand that the tensor product of two units (for different ...
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### R-matrices and symmetric fusion categories

Given a $\mathbb{C}$-linear braided fusion category $\mathcal{C}$ containing a fusion rule of the form e.g. \begin{equation}X\otimes Y\cong A\oplus B \oplus C\end{equation} (where $A,B, C, X$ and $Y$ ...
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### Automated rewriting of string diagrams in symmetric monoidal categories

Many algebraic structures, such as Frobenius algebras, or quasi-triangular Hopf algebras, can be formulated in an arbitrary symmetric monoidal category. They are given by a collection of morphisms, ...
1 vote
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### Oplax monoidal functors of $\infty$-categories

In Higher Algebra, a notion of lax symmetric monoidal functors (in what follows, I'll remove the adjective "symmetric", but I'm mainly interested in the symmetric situation) is defined : if you have ...
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### Does Cantor Bernstein hold in a Closed Symmetric Monoidal Category?

In a closed symmetric monoidal category with $[I,X] \cong X$ for all $X$ is it true that having monomorphisms $m :A \rightarrow B$ and $m: B \rightarrow A$ is enough to imply $A \cong B$ ? I tried to ...
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### Very basic question about category theory

I have a symmetric monoidal category, and go to another one by replacing objects with isomorphism classes of objects. What's that called in category theory language? To give an example, consider the ...
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