# Questions tagged [symmetric-monoidal-categories]

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### Reference for free symmetric monoidal categories with duals on symmetric monoidal categories

The inclusion $U$ of the (2,1)-category of symmetric monoidal categories with duals into the (2,1)-category of symmetric monoidal categories admits a left 2-adjoint functor $L$ for formal reasons. In ...
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### Tannaka without Yoneda?

I am studying enriched categories, and as I wrote in my previous question How is the morphism of composition in the enriched category of modules constructed?, this is very difficult because there are ...
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### How is the morphism of composition in the enriched category of modules constructed?

I asked this a week ago at MSE, but without success. I am studying enriched categories and I have a feeling that I am doing something wrong because all the way each step, each elementary proposition, ...
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### Symmetric monoidal structure(s) on the $\infty$-category of dg-categories

Let $k$ be a commutative ring with $1$, and let $\mathsf{dgCat}_k$ be the category of $k$-linear dg-categories, as defined in [1, Section 2]. We may equip $\mathsf{dgCat}_k$ with the Morita model ...
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### Frobenius algebras and traces of modules

$\newcommand{\Hom}{\mathscr{Hom}}$ Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable; Let $A$ be a commutative algebra in $C$, ...
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### Formal completion of a quotient stack

$\newcommand{\Rep}{\operatorname{Rep}}$ $\newcommand{\mo}{\operatorname{-mod}}$ $\renewcommand{\hat}{\widehat}$ I apologize in advance if this is a naive question but my background in algebraic ...
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### Mac Lane's proof of coherence for symmetric monoidal categories

This question only concerns the final part of the proof, so I assume that the symmetric monoidal category is a strict monoidal category $\mathsf{C}$ with the braiding $s$. Let $X_1,...,X_n$ be ...
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### Deligne's internal characterisation of Tannakian categories - glueing of algebras

I am currently reading the proof of Deligne's internal characterization of Tannakian categories (Thm. 7.1) in Deligne: Catégories tannakiennes. My question is similar to this one. Given a ...
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### A question about the Tannaka-Krein reconstruction of finite groups

In Chapter 15 Section 15.2.1 of Quantum Groups and Noncommutative Geometry, 2nd edition, the authors raised a question: can we reconstruct a finite group $G$ from its category of finite dimensional ...
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### Krein's theorem in the Tannaka-Krein duality

In the Tannaka-Krein duality theory, the Krein theorem describes the conditions under which a given category $\varPi$ is a category of finite-dimensional representations of some compact group $G$: ...
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### "Fundamental theorem for Hopf modules"

I am studying Hopf algebras in categories, and I hope, somebody could help me with the following. Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (...
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### Symmetric monoidal category with trivial switch morphisms

Is there a specific terminology for a symmetric monoidal category in which for any object $x$ the switch map $x\otimes x\to x\otimes x$ is the identity ? (Or alternatively the action of the symmetric ...
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### Tannaka duality for closed monoidal categories

I asked this some time ago at mathstackexchange, and people there explained to me the mathematical part of what I was asking, but the question about references remains open. In my impression, people ...
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### Symmetric monoidal structure on algebras

I stuck at a relatively simple thing of formalization in infinity setting. I use here the formalism of quasi categories, i.e. simplicial sets with inner horn fillings. Suppose $O^{\otimes}$ is an ...
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### Inequivalent compact closed symmetric monoidal structures on the same category

I am looking for interesting examples of categories admitting multiple monoidally inequivalent closed (or compact closed) symmetric monoidal structures. We know how to construct disconnected toy ...
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### Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?

Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent (by a zig-zag of symmetric monoidal Quillen equivalences) to a symmetric monoidal combinatorial ...
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### Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s ...
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### Derived tensor products and Tor of commutative monoids

Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: \text{Hom}(M\otimes N,L)...
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### If a $\otimes$-idempotent object has a dual, must it be self-dual?

Let $C$ be a symmetric monoidal category. Recall that a dual for $X \in C$ is an object $X^\vee$ and maps $\eta: I \to X \otimes X^\vee$ and $\varepsilon: X^\vee \otimes X \to I$ (where $I$ is the ...
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### Monoidal tensor product which preserves directed limits

Given a symmetric monoidal category $Q$, is there a construction of a (preferably full and faithful strong) monoidal embedding of $Q$ into some symmetric monoidal closed category $M$ which has all ...
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### What are the advantages of various "models" for the motivic stable homotopy category

People use several distinct models for the motivic stable homotopy category (so, there are some choices for the underlying category and a collection of available model structures). I would like to ask ...
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### Representation categories and homology

Let $G$ be a finite group. Let $\mathcal{C}=Rep-G$ be the rigid $\mathbb{C}$-linear symmetric monoidal category of finite dimensional complex representations of $G$. Can we recover some homological ...
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Has the monad induced by the free commutative monoid functor already been studied anywhere? Does it have any particular properties (other than not being cartesian)? I would prefer a reference on ...
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### Left adjoint for categories of commutative monoids?

The $n$Lab writes (prop. 2.2 in https://ncatlab.org/nlab/show/category+of+monoids) : Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the ...
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### How nontrivial can "central extensions of ribbon fusion categories" be?

In a sense, this is a follow up on this question, but one PhD programme later. Let $\mathcal{C}$ be ribbon fusion. By $\mathcal{C}'$, we denote the symmetric centre, i.e. the full subcategory of ...
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### Temporal semantics for string diagrams

Suppose I have a string diagram $D$ which involves a set of strings $S$ and atomic processes $A$. Formally, we should think of this as a canonically chosen map in the free symmetric monoidal category (...
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### Branching behavior in string diagrams/monoidal categories?

I am currently working through Peter Selinger's paper "Towards a Quantum Programming Language", and trying to connect it with what I already know about monoidal categories and string diagrams. ...
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### Chevalley property for Tannakian categories

Are all Tannakian categories have the Chevalley property? I know that the categories $\mathrm{Rep}_k(G)$ have the Chevalley property for affine algebraic groups $G$ over a field $k$, but I don't know ...
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### Deligne tensor product of semisimple tensor categories

Let $T_1, T_2$ be two semisimple tensor categories over a field $k$ (i.e. symmetric rigid monoidal abelian $k$-linear). Then is the Deligne product, $T_1\otimes T_2$ also a $k$-tensor category? ...