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I am a physics student who is currently reading Szekeres's A Course in Modern Mathematical Physics. In the first chapter, he makes the tiniest bit of contact with category theory (presumably just to get the reader thinking about structure-preserving maps between structures of the same type), but he does lay down the definition of a category and therefore the morphisms which live therein. I am a bit baffled at how little the definitions seem to require (I suppose this is part of the great power of category theory). In particular, the fact that the class of morphisms is only (roughly, partly because I don't think I'm capable of being more precise) required to contain an identity as well as obey associativity seems like very little structure in the sense that one can in turn show (see the picture below) that the morphisms corresponding to a given class of objects are increasingly very restrictive in their nature (i.e. not just mappings). It seems indeed like the requirement of associativity foists upon us a huge restriction which is not a priori obvious. Is it indeed associativity which leads to this?

For example, suppose that I want to show that the class of all semigroups forms a category where in particular the morphisms are defined as (or rather must be?) semigroup homomorphisms. It is not clear that associativity should imply that the only morphisms one can define must be semigroup homomorphisms. I'm not sure I can prove this fact using just associativity though:

That a semigroup homomorphism can be a morphism is immediate given compositions of these maps. Conversely, suppose for the sake of contradiction that there is a morphism $\alpha$ between two objects which is not a semigroup homomorphism. Take two other morphisms $\beta$ and $\gamma$ with corresponding objects $A,B,C,D$ as the obvious targets and sources. Since $\alpha$ is not a semigroup homomorphism there are two elements $a_1, a_2 \in A$ such that $b_1 = \alpha(a_1)\alpha(a_2) \neq \alpha(a_1a_2) = b_2$. But I'm not sure how to proceed from here some perhaps my understanding of associativity being what "enforces" things is wrong...

Edit: As I think a little bit more and reflect on e.g. this question, perhaps I am misunderstanding that taking the morphisms of this category to be semigroup homomorphisms is a possible but not the only choice, just as we may in general choose various inner products on a given vector space. Is this understanding correct?

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  • $\begingroup$ just for the sake of giving an explicit example, the category whose objects are semigroups and whose arrows are semigroup isomorphisms (with composition defined as usual as function composition), is a perfectly good category, though not the one people mean when they say "the category of semigroups". $\endgroup$ Commented Feb 12, 2023 at 20:11
  • $\begingroup$ @spaceisdarkgreen Thank you, that is a very helpful comment. I will make a note of it in my book. $\endgroup$
    – EE18
    Commented Feb 12, 2023 at 20:19

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taking the morphisms of this category to be semigroup homomorphisms is a possible but not the only choice

Yes, this is correct. One cannot deduce what the morphisms in a category are simply by knowing the objects, though in many cases, this will be obvious. And one cannot deduce how the composition operator works simply by knowing the objects and morphisms, although there is typically an obvious choice.

Specifying a category requires that we specify the objects, morphisms, and composition law (which morphisms are the identity is determined by the composition law), just like specifying an inner product space requires giving both a vector space and a specific inner product. The point of category theory is that the definition of category is broad enough to apply to essentially every single area of mathematics.

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