I've been curious about the concept of categories in physics and tried finding some more information about the subject. There are lots of textbooks about category theory, some from mathematicians and some written specifically for physicists (example), but I'm having a hard time seeing in which areas of physics the topic comes up and how they are connected (if they are).

For example, I stumbled up on Topos Physics, which seems to make use of the language of categories. But I also saw something called "Categorical quantum mechanics", which seems to have relations to Category theory and Topos Physics. Category theory seems also to make an appearance in Quantum infromation theory (see here), so there seems to be once again a link to Topos Physics (over Catgeorical QM).

As mentioned above, I'm interested in the topic, but I'm only a graduate student (currently masters), whose studies up to now mostly focused on QFT, GR and Lie theory.

So my question boils down to:

  • Does Category theory in physics mostly appear in connection to Topos Physics?
  • Are there any topics close to QFT or GR that involve Category theory and are not completely off limits to a graduate student in physics (it seems that much of what I could find assumed extensive knowledge of String theory, which I don't have).

P.S. I saw this question, but I was hoping to find some more examples and maybe bit of a clearer structure in the overview.

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    $\begingroup$ This post (v2) seems too broad as one may argue that category theory applies to almost any topic of physics. $\endgroup$ – Qmechanic Apr 26 '20 at 21:15
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    $\begingroup$ I'm sure not that there is a way to write this question that will make it answerable on this site. It is like asking "Where does topology appear in physics?" The answer to such a question would have to be a textbook. In fact, it's worse because of the fundamental role of category theory in providing the abstract semantics for all of mathematics. One jumping-off point could be Baez's "Quantum Quandries": arxiv.org/pdf/quant-ph/0404040.pdf $\endgroup$ – d_b Apr 26 '20 at 21:44
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    $\begingroup$ Another would be to try to understand the role of category theory in topological phases of matter, and the connections with anyons and topological field theories. One starting point for this is Steve Simon's (proto)book, "Topological quantum" www-thphys.physics.ox.ac.uk/people/SteveSimon/topological2019/… $\endgroup$ – d_b Apr 26 '20 at 21:46
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    $\begingroup$ Finally, I will say that even most physicists who spend a lot of time thinking about the role of category theory in physics don't think too much about topoi. So the majority of work involving category theory in physics doesn't say anything (explicitly) about topoi. This will probably change in the near future as more categorical and topos-theoretical ideas make their way from mathematics to physics. $\endgroup$ – d_b Apr 26 '20 at 21:49
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    $\begingroup$ In contrast to the “it can appear everywhere” point of view, I will just say that when I got my physics Ph.D. several decades ago category theory was never once mentioned. I doubt that any of my professors knew it existed, although it had been around for about 40 years. $\endgroup$ – G. Smith Apr 27 '20 at 4:02