# Tag Info

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### 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 ...

### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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 ...
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### Reference for "multi-monoidal categories"

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

### 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 ...

### 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 ...
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### 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 ...
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### 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 ...

### 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 ...

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### 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 ...