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
...

26
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 ...

23
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 ...

22
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) \...

21
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 (though ...

20
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,...

18
votes

### Abelian categories that are not monoidal

Let $\mathcal{A}$ be an additive category with a monoidal structure such that the maps
$$ \otimes \colon \mathcal{A}(A,B)\times\mathcal{A}(C,D) \to
\mathcal{A}(A\otimes C,B\otimes D)
$$
are ...

18
votes

### Abelian categories that are not monoidal

In the paper
Hovey, Mark, Additive closed symmetric monoidal structures on R-modules, J. Pure Appl. Algebra 215, No. 5, 789-805 (2011). ZBL1223.18005.
Hovey shows the following theorem (Theorem 3.3)
...

17
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 ...

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

### Proof that the unit of a Cartesian monoidal category is terminal

Here is a down-to-earth answer.
For ease of notation, let $\lambda: A\to I\times A$ be the component of the unitor and let $\pi_1$ and $\pi_2$ be the projections from $I\times A$ to $I$ and $A$.
Then ...

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 ...

14
votes

Accepted

### Is there a generalization of braided monoidal category without the isomorphism requirement?

Day, Panchadcharam, and Street have a paper on lax braidings, though I don't think it could be called a common notion. Anyway, "lax" seems to be the obvious terminology to try here.

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

### 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

### 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

Accepted

### When is the adjoint to a monoidal functor monoidal?

If $L$ and $R$ are a left and right adjoint, then doctrinal adjunction asserts that $L$ is oplax monoidal iff $R$ is lax monoidal. (I'm being a bit imprecise here, treating monoidality as if it were a ...

11
votes

Accepted

### Completeness of 2-category of Monoidal Categories

It depends on what you mean by "the 2-category of monoidal categories" and also what you mean by "complete".
The 2-category of monoidal categories and strict monoidal functors is complete as a Cat-...

11
votes

Accepted

### What is this symmetric simplex category, concretely?

$\Delta_+$ is the monoidal category generated from the associative operad, considered as a non-symmetric operad. Similarly, $(\Delta_+)_{{\rm sym}}$ is the symmetric monoidal category generated from ...

11
votes

Accepted

### A cohomology theory for fusion categories

There is no such cohomology theory known (in particular, this is not related to Davydov-Yetter cohomology which is about deformations and vanishes for finite groups). In my mind this is a very ...

11
votes

Accepted

### Monoidal category that is not spacial

One of the simplest examples of a non-spacial category is $\mathrm{End}(\mathrm{Vec}^{\oplus 2})$, the category of $2\times2$ matrices with vector space coefficients. Working over a field $\mathbb k$, ...

11
votes

Accepted

### Existence of a finite group with a given decomposition for a tensor square of one irreducible complex representation

I think that there is indeed no such finite group $G$, whether simple or otherwise. Note first that the representation $5_{1}$ can be assumed to be faithful ( for if $K$ is its kernel, then the group $...

10
votes

Accepted

### Mac Lane strictness theorem and categorifiability of fusion rings

@Todd Trimble is correct; a strict fusion category is just a fusion category that is strict as a monoidal category. (Being strict or being fusion is just a property of a monoidal category that can ...

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