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 ...
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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 ...
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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 $\mathbb k$, ...
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10
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 ...
9
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:
...
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9
votes
The 'correct' way to draw a diagram
How to draw diagrams is largely artistic choice.
The one mathematically based rule I can think of is that pullbacks should always be parallelograms.
The point is that pullbacks capture substitution, ...
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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 ...
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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 $S_0\subset S_1\subset S_2\subset\cdots$ ...
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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 ...
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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 $...
- 1,089
3
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 ...
- 627
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 ...
- 345
2
votes
Singularity-free isotopies between string diagrams for monoidal categories
This conjecture was confirmed recently, with two different methods:
By Jamie and I, as a corollary of our work on the word problem for monoidal categories (in which we give algorithms to detect ...
- 345
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 ...
- 345
1
vote
On the correspondence between proof nets and sequents
First, it might be of interest to you to know that Table 1 in this paper by Dominic Hughes compares various diagrammatic presentations of the free *-autonomous category (including the paper you've ...
1
vote
Accepted
Why is 'every braided monoidal category spacial'?
As shown at the end of the question, the statement is equivalent to another post which proves the relations discussed here. It has now been answered by Peter LeFanu Lumsdaine.
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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 ...
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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}\...
- 3,137
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, ...
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