alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
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So what's this category theory thing? Should I* Care? Is it the second coming of Euler?
*I care about physics and math. modeling, am fine with linear algebra and calculus
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the sort of minimalist case for it is that category theory is a really handy toolbox of ways to organize similar kinds of arguments that come up repeatedly in different branches of math. it was invented to be used in algebraic topology and is used heavily in algebraic geometry
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a tiny example from linear algebra: the determinant has the multiplicative property that det(AB) = det(A) det(B). it turns out to be a special case of what's called a functor (called "exterior power"), and this turns out to be a special case of what's called functoriality
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functoriality of the exterior power generalizes to an interesting statement about what's called the minors of a matrix, which it would be very difficult to guess is true if you didn't know that they came from exterior powers
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unfortunately the vast majority of people who use linear algebra will probably never need to think about minors, as far as i know. they have theoretical applications in various branches of pure math, e.g. algebraic geometry again, but mostly nobody else needs em i think
another tiny example from calculus: the chain rule for the jacobian also turns out to be an example of functoriality, this time for the functor called "taking the tangent space at a point." there is a particular kind of person that gets really excited about this kind of shit
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(it me, i'm that kind of person) but empirically there are many other kinds of people who don't give a shit that you can organize a bunch of stuff as a special case of a few categorical patterns and are happy to just take the stuff as it is, idk
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that's probably the most common place where people will see them (and can be explained using functoriality!) but e.g. if you wanted to calculate an inverse of a specific matrix in practice there are more efficient ways. that formula has mostly theoretical applications
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