My masters degree involved a good bit of category theory. Personally, I don’t see how it has any use outside of mathematics. (Note ‘maths’ includes ‘mathematical logic’ - so it’s still a broad field of applicability).
I am highly motivated to be persuaded otherwise, and hence will be watching this series of posts with keen interest.
--- <disclaimer> I am not a working mathematician, and have not published any papers. My masters thesis involved a lot of category theory—but only relatively simple category-theoretic concepts (it was an application of category theory to a subfield of mathematical logic).
Limits, free objects, adjunctions, natural transformations etc. but not higher-order categories, topoi/toposes or anything fancy like that. </disclaimer>
<handwavy discussion of technical math> As I understand it, the usual application of category theory is mostly to things involving natural transformations (it is said the need for a way to formalize natural transformations is what led to the invention of category theory) - and even then, it seems mostly to be applied to nice algebraic objects with (category theoretic) limits, and slight generalizations of these. So, to groups and rings and modules, and then to some categories made of stuff kinda like those things.
There’s also the connection to topology and logic, via simplicial complexes, homotopies, toposes, type theory etc. which seems very interesting to me. It seems useful if you want to think about constructive mathematics (i.e., no law of excluded middle) - which is promising for maths involving some notion of ‘computation’ (for a given abstraction of computation) which has obvious applications to computing (especially automated theorem provers / checkers).
In these senses, I can certainly see its use as another mathematical field, and a good way of reasoning IN MATHS or ABOUT MATHS. But I don’t quite understand its tremendous reputation as this amazing mathematical device, and less so its applications outside of what I’ve mentioned above. </handwavy discussion of technical math> ---
In particular, from my admittedly limited knowledge, category theory only seems useful: a) If you already have a bunch of different fields and you want to find the connections. b) If you want to start up a new field, and you need a grounding (after which the useful stuff will be specifically in the field being developed, and not a general category theory result). c) For good notation / diagrams / concepts for a few things.
EDIT: Interested to hear the opinion of someone who actually works with category theory on a regular basis.
My masters degree involved a good bit of category theory. Personally, I don’t see how it has any use outside of mathematics. (Note ‘maths’ includes ‘mathematical logic’ - so it’s still a broad field of applicability).
I am highly motivated to be persuaded otherwise, and hence will be watching this series of posts with keen interest.
---
<disclaimer>
I am not a working mathematician, and have not published any papers. My masters thesis involved a lot of category theory—but only relatively simple category-theoretic concepts (it was an application of category theory to a subfield of mathematical logic).
Limits, free objects, adjunctions, natural transformations etc. but not higher-order categories, topoi/toposes or anything fancy like that.
</disclaimer>
<handwavy discussion of technical math>
As I understand it, the usual application of category theory is mostly to things involving natural transformations (it is said the need for a way to formalize natural transformations is what led to the invention of category theory) - and even then, it seems mostly to be applied to nice algebraic objects with (category theoretic) limits, and slight generalizations of these. So, to groups and rings and modules, and then to some categories made of stuff kinda like those things.
There’s also the connection to topology and logic, via simplicial complexes, homotopies, toposes, type theory etc. which seems very interesting to me. It seems useful if you want to think about constructive mathematics (i.e., no law of excluded middle) - which is promising for maths involving some notion of ‘computation’ (for a given abstraction of computation) which has obvious applications to computing (especially automated theorem provers / checkers).
In these senses, I can certainly see its use as another mathematical field, and a good way of reasoning IN MATHS or ABOUT MATHS. But I don’t quite understand its tremendous reputation as this amazing mathematical device, and less so its applications outside of what I’ve mentioned above.
</handwavy discussion of technical math>
---
In particular, from my admittedly limited knowledge, category theory only seems useful:
a) If you already have a bunch of different fields and you want to find the connections.
b) If you want to start up a new field, and you need a grounding (after which the useful stuff will be specifically in the field being developed, and not a general category theory result).
c) For good notation / diagrams / concepts for a few things.
EDIT: Interested to hear the opinion of someone who actually works with category theory on a regular basis.