# Questions tagged [symmetric-monoidal-categories]

The tag has no usage guidance.

95 questions
Filter by
Sorted by
Tagged with
126 views

• 275
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
149 views

• 9,666
1 vote
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. ...
• 275
1 vote
158 views

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 ...
• 1,986
1 vote
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 ...
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,...
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 ...
• 894
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 ...
• 73
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 ...
• 455
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 ...
• 351
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 ...
• 233
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 ...
• 265
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 ...
• 435
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 ...
• 941
154 views

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 ...
• 11.5k
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)} ...
• 11.5k
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 ...
• 63.1k
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 ...
• 489
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. ...
• 2,074
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 $\otimes_{H}$ ?
• 69
376 views

• 14.4k
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 ...
• 2,931
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. $$X\otimes Y\cong A\oplus B \oplus C$$ (where $A,B, C, X$ and $Y$ ...
• 309
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, ...
• 2,931
1 vote
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 ...
• 14.4k
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 ...
• 51
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 ...
• 2,931
338 views

• 7,386