On the identity map requirement in the definition of category

Question

One common definition of a category features objects, arrows, an associative composition law for arrows, and an identity map requirement: for every object there must be an self-pointing arrow that behaves as left-side and right-side identity for composition.

I have no problem seeing how one could use a formalism of objects and arrows between them to structure our thoughts on a wide variety of areas. Also, it's easy to see why one would want to have a basic building operation (i.e. the composition law) in such a formalism.

With a little more thought I can also see why one would stipulate that this composition law be associative: making composition associative has the effect of replacing the "pair of arrows" with the "sequence of arrows" as the formalism's basic building operation, and I can see that this change may make the formalism more flexible.

I can't come up with any rationale at all for requiring that there be an identity arrow for every object.

What is lost by dropping this requirement?

Note that dropping this requirement does not preclude the existence of identity arrows, and therefore, it does not preclude defining concepts (e.g. inverses, isomorphisms) that depend on the concept of an identity arrow.

In fact, AFAICT, if one were to drop the identity arrow requirement, one could still define functors, natural transformations, cones, etc. (except that the definitions, of course, would not have the usual stipulations for the identity).

EDIT: To sharpen the question a bit, and maybe make it less of a "philosophical issue", I note that several times I've come across remarks from Saunders Mac Lane to the effect that their aim (his and Eilenberg's) in coming up with category theory was to arrive at the concept of natural transformation. In fact, in Categories for the Working Mathematician, 2nd ed., Mac Lane writes (p. 18):

"category" has been defined in order to define "functor" and "functor" has been defined in order to define "natural transformation".

As I noted earlier, the definitions of functor and natural transformation don't seem to rely much on the identity law requirement for categories.

So I could make my question a bit more concrete: how does requiring that every object in a category have an identity contribute to what Eilenberg and Mac Lane were aiming for?

Answer 1

Isomorphisms are absolutely essential to category theory, and in particular the idea that isomorphic objects are "the same" is perhaps the single most important concept in all of category theory. As you observe, you can define isomorphisms without requiring the existence of identities. However, this definition is still a bit problematic in a few ways. First, if "isomorphic" means "the same", you would expect it to be an equivalence relation, which it is not if identities don't exist. Second and more importantly, functors do not have to take isomorphisms to isomorphisms if you do not require that they preserve identities. This is a very serious problem for the intuition that functors should be some kind of "natural" operation: intuitively, an operation that can send isomorphic objects to non-isomorphic objects seems like it would have to be highly unnatural. I suspect this had at least something to do with Eilenberg and Mac Lane's decision to require that categories have identities (and that functors preserve them).

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Here is another very basic thing that can go wrong in the absence of identities: you can have non-isomorphic terminal objects. Let $\mathcal{C}$ be a category with three objects $A$, $B$, and $C$ and exactly one map between any two objects, and let $\mathcal{D}$ be the non-unital category obtained by adjoining to $\mathcal{C}$ a new map $f:A\to C$ whose composition with any map of $\mathcal{C}$ is the unique map in $\mathcal{C}$ with the same domain and codomain. Then $A$ and $B$ are both terminal in $\mathcal{D}$, but they are not isomorphic (in fact, $A$ has no identity map).

Since any sort of universal property can be described as a terminal object in an appropriate category, this means that in non-unital categories, you cannot expect any universal property to define objects uniquely up to isomorphism. One way to fix this is to modify the definition of "terminal" to require that a terminal object have an identity map. Another way is to just require that all objects have identity maps, which holds in pretty much every example of interest.

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Finally, on a more philosophical note, I would say that the existence of identity elements is actually a corollary of the "correct" definition of associativity. I would say that an "associative operation" is an operation which given any finite ordered set of elements $(a_1,\dots,a_n)$ gives you a "product" $a_1\dots a_n$ such that you can always drop parentheses (so for instance, $(ab)c=abc$, where the left-hand side denotes the binary product of (the binary product of $a$ and $b$) and $c$, and the right-hand side denotes the ternary product of $a$, $b$, and $c$). The identity element is then just the product of the empty ordered set. 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.

Answer 2

When considering the 'one object case', this is basically the question of semigroups vs. monoids (or, in additive fashion, rings with or without units). And such a debate can go into philosophical deepness, where both sides have their own truth.

(Note that non unital categories do exist.)

So.. why to use identity elements?

1. Why not?
2. They can be freely joined to any given non-unital semigroup/ring/category in a straightforward way. Especially for categories, you don't have to picture as hook arrows, identities can be depicted as just the vertices themselves. [If arrows were to express some kind of 'moves' then identities were the moveless moments.]
3. They can make life easier, e.g. to talk about an inverse without unit element(s) is not that trivial and might raise further questions.