37
votes

Accepted

### The two ways Feynman diagrams appear in mathematics

If I understand the question correctly, the search is for a calculation of the asymptotic expansion of Gaussian integrals using concepts and techniques from category theory. Here is one such ...

31
votes

### Proof that a Cartesian category is monoidal

There is a complete proof formalised by Scott Morrison in the Lean proof assistant here:
monoidal/of_has_finite_products.lean
[UPDATE] Here is some commentary. In the above file, we see
...

25
votes

Accepted

### Does every monoidal category admit a braiding?

No, sometimes there are even $x,y$'s with no abstract isomorphism $x\otimes y \cong y\otimes x$.
Here are two families of examples:
Monoids, viewed as discrete categories. The tensor product is just ...

22
votes

Accepted

### Categorical presentation of direct sums of vector spaces, versus tensor products

One way to think about what the monoidal structure on vector spaces is doing is that it is telling us that vector spaces do not really form a category, or not "just" a category: they form a ...

20
votes

Accepted

### Does projective imply flat?

I believe the following is a counterexample. Let $\mathcal{A}$ and $\mathcal{B}$ be closed symmetric monoidal abelian categories such that the unit object $1\in\mathcal{B}$ is projective and let $F:\...

20
votes

### Monoidal categories whose tensor has a left adjoint

If $\otimes : V \times V \to V$ has a left adjoint and $V$ has finite products then $\otimes$ preserves them in the sense that the natural map
$$(X \times Y) \otimes (Z \times W) \to (X \otimes Z) \...

19
votes

### What is the monoidal equivalent of a locally cartesian closed category?

In a certain sense a monoidal version of a slice category is a category of comodules over a cocommutative comonoid object. If $C$ is a cocommutative comonoid object in a monoidal category, then the ...

18
votes

### Does projective imply flat?

The paper:
When projective does not imply flat, and other homological anomalies, Theory and Applications of Categories, Vol 5, pp. 202-250, 1999, available here
by Gaunce Lewis shows that this ...

18
votes

Accepted

### Monoidal categories whose tensor has a left adjoint

Just to clean up the $\epsilon$ of room left after Qiaochu's answer -- we can get rid of the extra hypotheses. I'll write $I$ for the monoidal unit and $1$ for the terminal object.
Assume that $(\ell,...

17
votes

Accepted

### Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

In a nonabelian setting the correct notion of kernel is given by the kernel pair, and the correct notion of cokernel is given by the cokernel pair. For example, in any category, a morphism $f : a \to ...

17
votes

Accepted

### How are the Walker-Wang TQFT and the Crane-Yetter TQFT related?

Yes, the Walker-Wang model is related to the Crane-Yetter-Kauffman TQFT in the same way the Levin-Wen model is related to the Turaev-Viro TQFT. See, for example, the table on page 14 of the notes ...

17
votes

### Kan extensions in the $2$-category of monoidal categories

I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\...

17
votes

### Proof that a Cartesian category is monoidal

In Categories for the Working Mathematician, the first thing that Mac Lane does after defining a monoidal category in Ch. VII.1 is to verify that a category with finite products is monoidal. The ...

16
votes

Accepted

### Modular Tensor Categories: Reasoning behind the axioms

Interesting question ! As far as I know, there are at least two secretly equivalent answers.
You somehow already gave the first one: a modular tensor category is the same as a modular functor (...

15
votes

Accepted

### Uniqueness of dualizing objects

If a dualizing object exists, there is a bijection between isomorphism classes of dualizing objects and isomorphism classes of $\otimes$-invertible objects (i.e. the Picard group), given by tensoring ...

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 ...

13
votes

Accepted

### Reference for "multi-monoidal categories"

Look at Section 3 of Leinster's Higher Operads, Higher Categories, where the term used is "unbiased monoidal category."

13
votes

### String diagrams for bimonoidal categories (a.k.a. rig categories)?

This question is answered in the affirmative in the following preprint:
Cole Comfort, Antonin Delpeuch, Jules Hedges, Sheet diagrams for bimonoidal categories, arXiv:2010.13361
The morphisms are ...

13
votes

### What is a tensor category?

There is no single accepted definition of “tensor category” that matches all uses. Almost always it means abelian (or a similar cocomplete condition) and k-linear. Usually it also means rigid. Often ...

13
votes

Accepted

### What is a tensor category?

There seem to be many different definitions in the literature, based on individual papers. But, I think that might change, now that the textbook Tensor Categories, by Etingof, Gelaki, Nikshych, and ...

12
votes

Accepted

### Does every commutative variety of algebras have a cogenerator?

The answer is no.
Let $A$ be the algebra with universe $\{0,1\}$ and fundamental operations $f(x,y,z)=x+y+z \pmod{2}$ and $g(x)=x+1\pmod{2}$. Then $f$ and $g$ commute with each other and with ...

12
votes

### Why is a braided left autonomous category also right autonomous?

(Apologies for problems with the LaTeX; I’m on a dodgy internet connection and having trouble previewing to fix it. Will attempt to fix it later when home.)
With any question like this — deriving an ...

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

### What do you call $C$ if $[D,C] = D^\vee \otimes C$ for all $D$?

First of all, a nitpick: the condition "$[D,C] = D^\vee\otimes C$" should be stated more precisely as "the canonical map $D^\vee\otimes C \to [D,C]$ is an isomorphism". Now as you mentioned in a ...

12
votes

Accepted

### Is a Hopf algebra a group object of some category?

Not with their definition, where they assume the underlying category to be cartesian. You can define a notion of "Hopf object" in arbitrary symmetric monoidal categories, where you also need ...

12
votes

Accepted

### Taking the category of sheaves is symmetric monoidal

Provided at least one of $M$ and $N$ is locally compact, the $\infty$-topos $\mathrm{Sh}(M \times N)$ is the product of $\mathrm{Sh}(M)$ and $\mathrm{Sh}(N)$ in $\mathrm{RTop}$. This is HTT 7.3.1.11.
...

11
votes

Accepted

### Why is a braided left autonomous category also right autonomous?

Here's a hint: what you should probably use here is the method of string diagrams (due to Joyal and Street, but by now ubiquitous). In other words, draw a picture in terms of tangles; you will see two ...

11
votes

### What is the monoidal equivalent of a locally cartesian closed category?

I think there is something intriguing and slightly mysterious going on here.
First, my proposed definition would be slightly different from Dimitri Chikladze's. I agree that the natural ...

11
votes

### 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 ...

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