# Questions tagged [symmetric-monoidal-categories]

The symmetric-monoidal-categories tag has no usage guidance.

95
questions

2
votes

0
answers

126
views

### Deligne/Milne Tannakian Categories, prop. 1.17

In Tannakian Categories (page 14, prop. 1.17) we are given the following:
Let $(\mathcal{C},\otimes)$ be a rigid abelian tensor category. If $U$ is a subobject of $\mathbf{1}$, then $\mathbf{1} = U \...

2
votes

1
answer

93
views

### Subobjects in rigid abelian tensor categories

$\require{AMScd}$
In Tannakian Categories (prop. 1.17) we are given the following:
Let $(\mathcal{C},\otimes)$ be a rigid abelian tensor category. If $U$ is a subobject of $\mathbf{1}$, then $\mathbf{...

2
votes

0
answers

62
views

### Adjoint to "strict twocategory of strict twofunctors"

Let C be the category of strict twofunctors, featuring the addition of a Grothendieck universe. Strict twocategories are categories enriched over the category of categories.
C has an internal hom ...

1
vote

1
answer

149
views

### Morphism of tensor functors in rigid tensor categories

This is a cross-post from MSE.
$\require{AMScd}$
The following proposition (1.13) is given in Tannakian Categories (loosely paraphrased with some change in notation).
Assume that $(F,c),(G,d):\...

2
votes

0
answers

97
views

### Trace morphism in Deligne/Milne's "Tannakian categories"

I originally posted this on MSE, but only got a comment linking an article (Bontea and Nikshych's "Pointed braided tensor categories"). So I'll repost the question in full here:
Is there a ...

6
votes

0
answers

99
views

### Is the symmetry compatibility condition in Fox's theorem necessary?

Let $(\mathscr V, \otimes, 1, \sigma)$ be a symmetric strict monoidal category whose unit is terminal. Suppose that every object $A$ is equipped with the structure of a cocommutative comonoid $1 \...

1
vote

0
answers

239
views

### Invertible objects in tensor categories (with P. Deligne/J.S. Milne's definition)

$\newcommand\id{\mathrm{id}}$I've asked this question on MSE, but the only response I've gotten so far is a comment, which I failed to understand. I'll add another question I have, related to this.
...

1
vote

1
answer

158
views

### Are the minimal nondegenerate extensions universal?

We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...

1
vote

0
answers

129
views

### Is a symmetric monoidal category ("tensor-category" in P. Deligne & J.S. Milne's vocabulary) neccessarily locally small?

Let $(\mathcal{C},\otimes,\mathbf{1},\phi,\psi)$ (I will denote this by just $(\mathcal{C},\otimes)$) be a tensor-category (in P. Deligne & J.S. Milne's vocabulary, see https://www.jmilne.org/math/...

3
votes

2
answers

356
views

### $R$-Module objects in cartesian closed categories

I am looking for a reference for the following statement.
Theorem. Let
$C$ be a regular, well-powered, countably complete cartesian closed category,
$R$ be a (commutative) ring object in $C$,
$R\...

5
votes

0
answers

82
views

### Tensor product of modules in model vs. infinity categories

Let $C$ be a combinatorial symmetric monoidal model category and let $A$ be a associative algebra object in $C$, that is cofibrant as an object in $C$. In Higher Algebra 4.3.3.17, Lurie proves an ...

1
vote

1
answer

228
views

### Symmetric-monoidal-associative smash product up to homotopy

I am thinking about sequential spectra. I am trying to figure out if the smash product here is symmetric monoidal associative up to homotopy. See definition 3.16 in the above.
Recall that a sequential ...

5
votes

1
answer

372
views

### A result on symmetric closed monoidal categories

In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...

3
votes

2
answers

246
views

### Is the free algebra functor over an $\infty$-operad symmetric monoidal?

Suppose $F: \mathcal{O}^\otimes \to \mathcal{P}^\otimes$ is a map of $\infty$-operads, and $\mathcal{C}$ is a symmetric monoidal $\infty$-category that admits small colimits, such that the tensor ...

7
votes

1
answer

157
views

### The change-of-monoid adjunction between categories of modules induced by a morphism of monoids

Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...

3
votes

1
answer

205
views

### Is the symmetric monoidal product on the $\infty$-category of $R$-modules unique?

In Higher Algebra 4.2.8.19, Lurie shows that the symmetric monoidal structure on spectra is uniquely defined (on the $\infty$-category level) by the following properties:
The sphere spectrum is the ...

8
votes

1
answer

421
views

### Why are enriched (co)ends defined like that?

I'm mainly following references such as Kelly, Loregian and the nLab, and it seems customary there to generalize (co)ends to the enriched context (over a symmetric monoidal category $\mathcal{V}$) by ...

2
votes

0
answers

89
views

### Nerve functor for symmetric monoidal category

The nerve $N(\mathsf{C})$ of a category $\mathsf{C}$ can be seen as a geometric realization of it (via n-simplices). This defines a functor $N: \mathsf{Cat} \rightarrow \mathsf{SSet}$ called nerve ...

5
votes

0
answers

169
views

### 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 ...

8
votes

0
answers

179
views

### 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 ...

2
votes

0
answers

97
views

### "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 ...

3
votes

0
answers

154
views

### When a monoidal closed category is cartesian closed

Let $C$ be a monoidal closed category with tensor $\otimes$ and internal hom $[-, -]$.
Suppose that
$C$ acts by adjoint monads, i.e. $- \otimes X$ is a comonad and $[X, -]$ is a monad, and each $F : ...

4
votes

1
answer

214
views

### $\ast$-autonomous categories with non-invertible dualizing object?

1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:
A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a ...

12
votes

2
answers

398
views

### 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 ...

6
votes

0
answers

856
views

### 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 ...

5
votes

2
answers

580
views

### 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, ...

2
votes

0
answers

231
views

### Taking the homology of a chain complex, seen as a symmetric monoidal functor

I've found, in quite some places online (e.g. it's the last example in the wikipedia page about monoidal functors), a statement similar to this:
Homology can be seen as a symmetric, monoidal functor $...

4
votes

1
answer

300
views

### 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 ...

2
votes

0
answers

111
views

### 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)} ...

2
votes

0
answers

176
views

### 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 ...

3
votes

1
answer

288
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 ...

6
votes

1
answer

477
views

### 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.
...

5
votes

1
answer

467
views

### 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 $\otimes_{H}$ ?

3
votes

1
answer

376
views

### Is every $\otimes$-invertible object "coherently sym-central"?

Let $\mathcal C$ be a symmetric monoidal $\infty$-category, and let $L \in \mathcal C$ be a $\otimes$-invertible object. Then the braiding $L \otimes L \to L \otimes L$ is simply multiplication by $\...

9
votes

0
answers

412
views

### 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 ...

3
votes

1
answer

313
views

### 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 $C$, ...

7
votes

1
answer

475
views

### 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 ...

7
votes

2
answers

416
views

### References about "monoidal fibrations" in $\infty$-category theory

$\newcommand{\cat}{\mathsf} \newcommand{\fun}{\mathrm{Fun}} \newcommand{\calg}{\mathrm{CAlg}}$
Let $\cat C^\otimes,\cat D^\otimes, \cat E^\otimes$ be symmetric monoidal $\infty$-categories, and $p^\...

6
votes

1
answer

163
views

### 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 ...

9
votes

1
answer

326
views

### 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 $Y$ ...

4
votes

1
answer

162
views

### 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

0
answers

471
views

### 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 ...

5
votes

0
answers

142
views

### 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 ...

2
votes

0
answers

221
views

### 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 ...

3
votes

2
answers

338
views

### The symmetric monoidal closed structure on the category of $\mathcal{F}$-cocomplete categories and $\mathcal{F}$-cocontinuous functors

In 6.5 of the book by Kelly,
Basic concepts of enriched category theory. Reprints in Theory and Applications of Categories, No. 10, 2005.
the author claims that the $2$-category $\mathsf{Cat}_{\...

7
votes

2
answers

744
views

### 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 ...

2
votes

0
answers

219
views

### 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 ...

9
votes

2
answers

516
views

### 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 ...

4
votes

1
answer

461
views

### 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 $G$:
...

6
votes

0
answers

489
views

### "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 (...