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 ...
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 ...
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 $\mathbb k$, ...
8 votes
Accepted

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

There isn't anything like a graphical calculus really expressly dedicated to rig categories, and well-documented and proved to be coherent, but there are 'approximations' to what you are looking for: ...
  • 5,833
8 votes

The two ways Feynman diagrams appear in mathematics

Before interpreting them in more advanced language like "string diagrams" or "monoidal closed categories" it might be good to stress that Feynman diagrams are very elementary combinatorial objects ...
7 votes

Monoidal category that is not spacial

If your monoidal category is the fundamental 2-groupoid of a space, then this is exactly asking whether $\pi_1$ acts trivially on $\pi_2$ or not (or more precisely, that equality is saying that $A \in ...
  • 26.7k
5 votes
Accepted

String diagrams for bimodules over noncommutative algebras?

In general, the correct generalization of the "stringy" notation for bicategories — in which objects label regions in $\mathbb R^2$, 1-morphisms label codimension-1 defects (whose projection to $\...
4 votes

Singularity-free isotopies between string diagrams for monoidal categories

If I understand your question correctly, it seems that there is a problem of winding numbers. Your diagrams seem to allow ending on univalent vertices. Consider, then, a string that starts at a ...
3 votes

Monoidal category that is not spacial

An example I learned about in Khovanov's "Heisenberg algebra and a graphical calculus", 2010, is the restriction and induction functors for the infinite chain $S_0\subset S_1\subset S_2\subset\cdots$ ...
3 votes

How can one represent vector addition diagrammatically in categorical quantum mechanics?

A more interesting way to go, which we haven't even fully developed yes, is to use W-spiders (a.k.a. anti-special ones), which are introduced here: https://arxiv.org/abs/1002.2540 since they allow ...
3 votes

How can one represent vector addition diagrammatically in categorical quantum mechanics?

The short answer is yes, you put $A+B$ in the box. To be more specific, in the graphical calculus for monoidal categories strings correspond to equivalence classes of objects (in your case $X$ and $...
3 votes

Branching behavior in string diagrams/monoidal categories?

You can represent such a branching behaviour in bimonoidal categories (also known as rig categories). In addition to your multiplicative monoidal structure $\otimes$, which is used to represent ...
  • 315
2 votes

String diagrams for bimodules over noncommutative algebras?

I was doing something like that when I was working on my PhD thesis. See string diagrams in the end of this paper http://www.tac.mta.ca/tac/volumes/25/1/25-01abs.html The idea is to put strings into ...
2 votes
Accepted

Temporal semantics for string diagrams

As you correctly note, the overall duration alone is not compositional data on your processes/diagrams. However, the algorithm you describe gives a clue as to how one could obtain duration data on ...
2 votes

How can one represent vector addition diagrammatically in categorical quantum mechanics?

You can use sheet diagrams for bimonoidal categories for this. The tensor product and the direct sum give a bimonoidal structure on $\text{FVect}$ (meaning that one distributes over the other). On top ...
  • 315
1 vote

Automated rewriting of string diagrams in symmetric monoidal categories

Double-pushout graph rewriting is actually the basis of a long line of work on automated rewriting techniques for string diagrams: for a very thorough introduction I recommend Aleks Kissinger's PhD ...
1 vote

How can one represent vector addition diagrammatically in categorical quantum mechanics?

Btw, are you reading CQM I and CQM II? Since the actual meat is in the forthcoming CQM III, and in our forthcoming book. When one starts to use the categorical algebra of complementary observables, ...
1 vote

How can one represent vector addition diagrammatically in categorical quantum mechanics?

Clearly either of $$\overset{X}{\longrightarrow}\fbox{$A$}\overset{Y}{\longrightarrow}\\+\\ \overset{X}{\longrightarrow}\fbox{$B$}\overset{Y}{\longrightarrow}$$ or $$\overset{X}{\longrightarrow}\...

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