15
votes

Accepted

### Inequivalent compact closed symmetric monoidal structures on the same category

A pretty interesting class of examples comes about by classifying compact monoidal groupoids. Given a group $G$, a $G$-module $M$, and a (normalized) 3-cocycle $a: G \times G \times G \to M$, one can ...

- 50.8k

15
votes

Accepted

### Trace in the category of propositional statements

To start with, I just want to make sure no one gets the impression that the categorical notion of trace was introduced by the paper you linked to; however "semi-famous" it might or might not be, it's ...

- 60.3k

14
votes

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### Exponentiation of vector spaces?

Okay, so you can get pretty close as follows: I still don't think $V^{\otimes W}$ makes sense, but riffing off of your comment, we can make sense of $(1 \oplus V)^{\otimes W}$ (where $1$ denotes the $...

- 109k

12
votes

### If a $\otimes$-idempotent object has a dual, must it be self-dual?

If $C$ is not assumed to be symmetric, then the answer to questions 1 and 2 is no. Let $p^* \dashv p_* \dashv p^!$ be a fully faithful adjoint triple with $p_* : A \to B$. (For instance, $A= \mathrm{...

- 60.3k

12
votes

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### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

The category $Func(C,C)$ is very rarely braided.
It's a bit like asking "when is the endomorphism algebra of a vector space commutative?"
For example, if $C=Vect\oplus Vect$, then $Func(C,C)$ is ...

- 40.5k

11
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### Reference for an unbiased definition of a symmetric monoidal category

This isn't quite an answer to your question either, but Appendix A of my 2003 book Higher Operads, Higher Categories contains something close.
First it defines commutative monoids in the style you ...

- 25.8k

10
votes

### Reference for an unbiased definition of a symmetric monoidal category

This is not quite what you mean, but relevant. Remember that a strictly associative and unital symmetric monoidal
category is called a permutative category. I observed ages ago (http://www.math....

- 29.2k

10
votes

Accepted

### What are the advantages of various "models" for the motivic stable homotopy category

The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model ...

- 22.6k

9
votes

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### The category of elements, enrichment, and weighted limits

My personal opinion is that the explanations that involve categories of elements (and generally, comma categories) are usually the most natural and intuitive explanations --- one just have to get used ...

- 3,986

8
votes

Accepted

### A question about the Tannaka-Krein reconstruction of finite groups

I don't think you've made an error. There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors. Namely, a finite dimensional Hopf algebra is the same thing as a ...

- 26.7k

8
votes

Accepted

### How is the morphism of composition in the enriched category of modules constructed?

I’ll change notation slightly, writing $\newcommand{\V}{\mathcal{V}}\V(A,B)$ and $\newcommand{\AV}{{}_{A}\! \V}\AV(M,N)$ respectively for the $\V$-hom-objects and $A$-module-homomorphism objects ...

- 16.4k

7
votes

### Inequivalent compact closed symmetric monoidal structures on the same category

The category $Cat$ of small categories admits two inequivalent tensor products: the Cartesian product and the funny product. This generalizes up to 2-cat, 3-cat, etc. By the way, a word of advise to ...

- 22.6k

7
votes

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### Symmetric monoidal structure on algebras

This is worked out in Higher Algebra, example 3.2.4.4.
Concretely, $\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})^\otimes$ is defined as follows: it is the simplicial set over $\mathrm{Fin}_\ast$ such that ...

- 15.7k

7
votes

### Krein's theorem in the Tannaka-Krein duality

In the comments you ask:
The question is the following: if I have an "abstract" category (or even a subcategory in the category of vector spaces), how can I understand that this is the category of ...

- 109k

7
votes

### Mac Lane's proof of coherence for symmetric monoidal categories

I think that this is a great question, since it is quite common to just say at this point "each relation between the transpositions follows from the relations in the group presentation of the ...

- 57.2k

6
votes

Accepted

### Left adjoint for categories of commutative monoids?

The left adjoint is the functor Sym, defined as $Sym(X) = I \coprod X \coprod (X \otimes X)/\Sigma_2 \coprod (X\otimes X\otimes X)/\Sigma_3 \coprod \dots$, where $I$ is the unit. I only ever work in ...

- 22.6k

6
votes

### The category of elements, enrichment, and weighted limits

The question is a bit imprecise, so I'm not quite sure whether this will really answer it, but let me try to paint the reduction of weighted colimits to conical colimits in a more "conceptual" light.
...

- 51.1k

6
votes

Accepted

### Adding inverses to a symmetric monoidal category (Reference?)

Maybe you could be interested in the recent PhD thesis:
Une introduction élémentaire au 2-groupe de Grothendieck
by C. Drugmand, 2016, UCL, Louvain-la-Neuve.
http://hdl.handle.net/2078.1/176774

6
votes

Accepted

### Is there something like "Noncommutative geometry internal to a category"?

I'll leave open how this might connect to "internal noncommutative geometry", but you can say something about internal $*$-algebras.
You can formulate $*$-algebras, anti-linear involution and all, ...

- 3,779

6
votes

### When is the endofunctor category of a monoidal category braided? When is it ribbon? Fusion? Modular?

This is in some sense an unnatural question to ask. Endofunctors only form a monoidal category in general, and if you want a braiding, that's not just an extra property: it's extra structure. Where ...

- 109k

6
votes

Accepted

### Is there a quotient or exact sequence of symmetric, premodular (ribbon fusion) and modular categories?

Short answer: Look for papers on "deequivariantization". (I think the original references are by Müger and Brugières, but I am not sure whether they used the term "deequivariantization".)
Longer ...

- 12k

6
votes

Accepted

### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

If the dimension of $Z(S^{n-1})$ is greater than 1, then the TQFT is not even approximately multiplicative under connect sum.
If $Z(S^{n-1})$ is 1-dimensional, then a simple cut and paste argument ...

- 12k

6
votes

Accepted

### If a $\otimes$-idempotent object has a dual, must it be self-dual?

EDIT: It looks like I was too hasty to delete this answer. I think I have fixed the gap that Anton noted in the comments and in his answer. As Anton pointed out in his answer, the trace loop that I ...

- 1,563

6
votes

Accepted

### Categorical definition of infinite symmetric product

This seems entirely straightforward unless I'm missing something. For any $n\in\mathbb{N}$, $\Sigma_n$ acts on $X^{\otimes m}$ for any $m\geq n$ (on the first $n$ coordinates), and this action ...

- 30k

6
votes

### Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?

Yes, this is proved in https://arxiv.org/abs/1506.01475 by Thomas Nikolaus and Steffen Sagave.
If you need a specific monoidal left Quillen equivalence, you can upgrade Dugger's result to the ...

- 31.4k

6
votes

### A question about the Tannaka-Krein reconstruction of finite groups

Although you already have one sufficient answer, I thought I would add here that Etingof and Gelaki, "Isocategorical groups" considered exactly this question. For finite groups, they gave a complete ...

- 2,700

6
votes

Accepted

### Functors that preserve monoids

A simple class of counterexamples can be found among thin small strict monoidal categories, i.e. preordered monoids. These are just sets equipped with a preorder and a monoid structure such that the ...

- 57.2k

5
votes

Accepted

### Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

I think that the best thing in this direction is the paper "Permutative categories, multicategories and algebraic K-theory" by Elmendorff and Mandell.
I have only skimmed this so I may well not be ...

- 49.8k

5
votes

Accepted

### Monoidal tensor product which preserves directed limits

Public Service Announcement!
It's very confusing -- I'd daresay incorrect -- to say "directed limit" to mean "limit indexed by a cofiltered diagram". Actually, historically the term "directed limit" ...

- 51.1k

5
votes

Accepted

### String diagrams for bimodules over noncommutative algebras?

In general, the correct generalization of the "stringy" notation for bicategories — in which objects label regions in $\mathbb R^2$, 1-morphisms label codimension-1 defects (whose projection to $\...

- 49.9k

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