16
votes

Accepted

### 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 $...

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 ...

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 ...

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{...

12
votes

Accepted

### 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 ...

11
votes

Accepted

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

The $\newcommand{\V}{\mathcal{V}}\newcommand{\D}{\mathcal{D}}\newcommand{\C}{\mathcal{C}}$universal property of being an “initial/terminal cowedge” — even strengthened to “$\V$-initial/-terminal” — is ...

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 ...

10
votes

Accepted

### 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 ...

9
votes

Accepted

### A result on symmetric closed monoidal categories

This is a special case of:
Let $F\dashv G$ be an adjunction. If there exists an isomorphism $id \cong GF$, then the unit $id \to GF$ is an isomorphism.
Indeed, $[-,A]$ is left adjoint to itself as ...

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 ...

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 ...

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 ...

7
votes

Accepted

### 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 ...

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 ...

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 ...

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 ...

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.
...

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

### 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 ...

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 ...

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 ...

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 ...

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 ...

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" ...

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 ...

5
votes

Accepted

### Tensor product of unit and co-unit in a closed compact category

The definition of compact closed category merely says that for every object $A$ there exists an object $A^\star$ and a unit $\eta_A$ and counit $\varepsilon_A$ satisfying the snake equations. That is, ...

5
votes

Accepted

### Formal completion of a quotient stack

I will assume $X$ is smooth for simplicity, but it is probably not needed. Given the stack $X/G$, there are two completions one may consider:
Completing along $\mathrm{B}G\rightarrow X/G$ one obtains ...

5
votes

Accepted

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

Certainly overkill, but this result is a special case of `change of operads' since the given morphism of monoids induces a morphism from the operad $O$ for left $A$-modules to the operad $P$ for left $...

4
votes

### Symmetric monoidal category with trivial switch morphisms

In my thesis I have named objects $x$ whose switch map $x \otimes x \to x \otimes x$ is the identity symtrivial (since I could not find any term in the literature). It was then used by others as well, ...

4
votes

Accepted

### Symmetric monoidal category with trivial switch morphisms

Coincidentally, terminology for such categories has been introduced very recently:
John C. Baez and Jade Master. Open Petri Nets. Nov 2018. arXiv:1808.05415
More precisely, the authors refer to a ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

symmetric-monoidal-categories × 83ct.category-theory × 61

monoidal-categories × 37

reference-request × 12

higher-category-theory × 8

braided-tensor-categories × 6

enriched-category-theory × 6

at.algebraic-topology × 5

infinity-categories × 5

adjoint-functors × 5

homotopy-theory × 4

operads × 4

topological-quantum-field-theory × 4

fusion-categories × 4

tannakian-category × 4

string-diagrams × 4

ag.algebraic-geometry × 3

rt.representation-theory × 3

homological-algebra × 3

modules × 3

duality × 3

higher-algebra × 3

linear-algebra × 2

ra.rings-and-algebras × 2

model-categories × 2