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 ...
Carlo Beenakker's user avatar
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 ...
Neil Strickland's user avatar
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 ...
Maxime Ramzi's user avatar
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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 ...
Qiaochu Yuan's user avatar
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) \...
Qiaochu Yuan's user avatar
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 ...
Adrien's user avatar
  • 7,912
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,...
Tim Campion's user avatar
  • 56.6k
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 ...
Neil Strickland's user avatar
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) ...
AT0's user avatar
  • 1,367
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 ...
Tim Campion's user avatar
  • 56.6k
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 ...
Tim Campion's user avatar
  • 56.6k
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 ...
Gabriel C. Drummond-Cole's user avatar
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 ...
Mike Shulman's user avatar
  • 63.6k
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.
Kevin Arlin's user avatar
  • 2,716
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 ...
Cole Comfort's user avatar
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 ...
Noah Snyder's user avatar
  • 27.5k
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 ...
David White's user avatar
  • 24.8k
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{...
Mike Shulman's user avatar
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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 ...
Mike Shulman's user avatar
  • 63.6k
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 ...
Adrien's user avatar
  • 7,912
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. ...
Reid Barton's user avatar
  • 24.5k
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 ...
Tim Campion's user avatar
  • 56.6k
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 ...
Qiaochu Yuan's user avatar
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-...
Mike Shulman's user avatar
  • 63.6k
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 ...
Yonatan Harpaz's user avatar
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 ...
Noah Snyder's user avatar
  • 27.5k
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 $\mathbb k$, ...
Theo Johnson-Freyd's user avatar
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 $...
Geoff Robinson's user avatar
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 ...
Dave Penneys's user avatar
  • 5,162

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