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I have recently been introduced to Category Theory as a mean to relate objects belonging to different categories via functors. Trough this one has the ability to connect and discover structures and similarities between mathematical object which belong to very different categories of mathematical objects.

I was wondering if one could construct an extension to this by including not just different branches of mathematics but making it a pluridisciplinary one.

So a concrete example of what I am trying to ask would be:

Suppose I am working with topological retractions and I am investigating how this relates to the Holographic Principle, can I make use or construct an extended version of Category Theory?

If this is not the case then what restrictions cause it to be impossible?

If on the other hand this is possible, could someone provide an example of such situation which has mathematical rigour?

Thank you for the help!

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    $\begingroup$ To clarify: you're asking whether category theory, or some "extension" of category theory, can be used to apply connect certain areas of math with certain areas in other disciplines? $\endgroup$ May 4, 2020 at 16:44
  • $\begingroup$ @NoahSchweber exactly right! $\endgroup$
    – Morettin
    May 4, 2020 at 16:46
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    $\begingroup$ There's nothing in principle preventing you from making categories whose objects and morphisms model, or are somehow based on, the subject matter of traditionally non-mathematical disciplines. You don't need to change the notion of a category to do that, nor is it excluded in any way by current category theoretic definitions or practice. $\endgroup$ May 4, 2020 at 18:08
  • $\begingroup$ @MaliceVidrine okay I understand thank you! do you think you’d be able to give me an example, or point me to a link where I could find one $\endgroup$
    – Morettin
    May 4, 2020 at 18:51
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    $\begingroup$ You might be looking for this: math.mit.edu/~dspivak/teaching/sp18/7Sketches.pdf $\endgroup$ May 5, 2020 at 21:51

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Category theory is not about the objects in itself but about the relation between objects (aka. morphisms). You may think of the definition of a category as a relation in which the same two objects can relate in multiple different ways and where you can compose these relations. As such it is so vastly general that it may be applied to other branches of sciences. To be honest I am not educated universally enough to give specific examples but I don’t think it is hard to come up with ones sounding like the category of foldings of DNA or the category of bindings of molecules (I really have no idea what I am talking about), at least in such a way that objects are configurations and morphisms are transitions between configurations. Admittedly this is more like pushing other domains into mathematics by means of abstraction, but this is exactly what math is supposed to do.

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  • $\begingroup$ Yes I understand and that was my thinking behind this questions. I was looking for a much more rigorous and concrete answer to see where and how such construction would be made, since I have always seen Category Theory as part of mathematics (not above it, ie. independent of mathematics). $\endgroup$
    – Morettin
    May 4, 2020 at 17:30
  • $\begingroup$ What is mathematics to you? Imo it is something like „description and deduction by abstraction“, which definitely applies to the definition of a category. So in my mind categories will always be a thing of mathematics. This is why my answer bends more in the direction of „of what use are categories in mathematical analysis of non-pure-math subjects“... $\endgroup$ May 4, 2020 at 18:02

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