importance of "Identity" in Category theory

Question

Reading about category theory a category is a collection of objects "types" that have morphisms "transformations" with two properties:

Associativity: for every object all its morphisms should be associative when composed together. Identity: for every object must have one identity morphism which doesn't change any of his other morphisms when it is composed with them. So associativity seems like a reasonable property that ensures correct composition.

but why is identity important? what purpose does it serve? and why is it called "identity" and not just "neutral morphism"? And why do some categories like "magma, semigroup" don't have identity or associativity but are still called categories?

for anyone else that comes here: ignore this link purpose of identity morphism and go directly to this link The Answer

it is possible to define isomorphisms without identities, but it is a bit more complicated and less elegant. I think you're reading far too much into Eilenberg and Mac Lane: they weren't trying to make some "universal" definition that applies to every situation imaginable; they were just making a natural-seeming definition that works in all the relevant examples they could think of."

Considering "associative" operations which don't have identities is thus analogous to considering finite sets but not allowing the empty set. Of course, this is sometimes useful to do, but unless you have a good reason to, it is probably not the natural thing to do.

Thank you PrudiiArca for your help and special thanks to Eric Wofsey for the answer

Answer 1

I think you do confuse things here.

A category is a mathematical structure, which generalizes the well known pattern „sets with structure + structure preserving maps“. In all the instances e.g. vector spaces + vector space homomorphisms/ groups + group homomorphisms/ measure spaces and measurable functions etc. etc. the composition of functions is associative, so it is natural to have associativity as an axiom. Moreover in all instances, the easiest structure preserving map you can think of is the identity function on your object, which literally does nothing and hence is structure preserving. Since all examples, which we want to generalize come with identities, we introduce an axiom of having identity morphisms for each object. You are wrong in saying that categories of magmas or semigroups dont have identities. They do have them, since as I explained the identity function on a magma/semigroup does not interfer with the structure and hence is structure preserving.

Now one notices that the concept of a category is more powerful than just describing these examples. For example one notices that certain structures like groups or monoids can be expressed in purely categorical terms as one object category, whose morphisms represent the elements and satisfy some axioms with respect to composition, which is given by the group operation. This is the point, were you cannot express magmas or semigroups in categorical terms, because a magma needs not be associative and a semigroup doesn’t have to have identities.

Answer 2

Let's consider how the definitions reduce in the simple case when the category has exactly one object. Then the axioms of a category are equivalent to the axioms of a monoid on the set of morphisms of the category. This is because monoids have only two axioms: associativity and existence of identity element. This identity element corresponds precisely to the identity morphism of the one object in the category. Thus, the purpose served by the identity morphism is precisely the same as the purpose served by the identity element in a monoid. In this sense, one can say that category theory is a generalization of the theory of monoids.

You mentioned magmas and semigroups in your question - these are what we get from a monoid by first removing the identity axiom (semigroups) and then removing the associativity axiom (magmas) leaving something that is very unstructured. We can consider categories whose objects are magmas or semigroups, with the appropriate notion of morphism. And we can also say that a category with a single object has morphisms which form a monoid, and therefore is a special kind of semigroup and magma. But we cannot say that an arbitrary semigroup or magma is equal to the morphism set of some category with one object.

Note that one could generalize the definition of a category to be built upon semigroups or magmas instead of monoids, simply by removing axioms. We could call the resulting generalizations a "catsemigroup" and a "catmagma" for example. Then categories would be special kinds of "catsemigroups" and "catsemigroups" would be special kinds of "catmagmas". But not the other way around.