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As you may have guessed from the title I'm talking about category theory (CT). It's a fascinating subject to me. It can beautifully describe the essence of what it means to "have a structure" and it's a wonderful tool for studying mathematics itself.

The definition of a category is really really simple (objects, morphisms, and some rules of composition and associativity). Nonetheless you can have very sophisticated theorems when using multiple nested layers (so called n-Category).

So my question is: Since it seems arbitrary to use composition of morphisms between objects to describe what it means to have some kind of structure, have we come to this logically? Or was it an arbitrary choice (that fortunately turned out to work very well)? Are there any attempts to go deeper in abstraction or is there a good reason to think that CT is really fundamental? Is there a way of building complexity from simplicity with something other than morphisms and compositions, and which would be at least as useful as CT?

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    You seem to operate under the assumption that something is either "really fundamental" or an "arbitrary choice". Of course categories do not describe "any" kind of structure, of course we did not come to them "logically", history is messier than that, of course the formalism involves some "arbitrary" choices (e.g. there are multiple different axiomatizations of CT), of course there are alternatives (see e.g. univalent foundations), and of course this does not mean it is "arbitrary" or unmotivated. – Conifold Jun 1 '18 at 23:59
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Fundamentally, category theory does not seek to model "having a structure". It rather starts from the realization that the way how we use the structure is typically via the structure-preserving or reflecting maps; and that it is only the maps that really matter.

For example, in topology the purpose of the topology is just to define what continuous functions are. Everything else can be expressed in terms of continuous functions (not at least because we can recover the topology on X by looking at the continuous functions from X to Sierpisnki space).

Sticking with topology, one can then see that for many results in topology, the precise nature of the continuous functions is irrelevant. For example: Most theorems have natural counterparts in the setting of computable functions. This is because the relevant categories have very similar properties.

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