The first difference that gives some people pause is that we don’t start with a set, but a class. Classes are pretty much like sets, but they can be “bigger”. In particular, we sometimes run into technical problems with sets containing other sets, so we introduce classes as things that can hold any sort of sets with no problem. Of course we’ve only pushed back the problem to when we might want to collect classes together, but we’ll burn that bridge when we come to it.
Anyhow, there’s really nothing that bad about basing an algebraic structure on a class. There are perfectly good reasons (we’ll see) for putting a ring structure on a class. In this case we call the result a “large ring”. On the other hand, when every class involved in a category is a set, we call it a “small category”. Seriously, it’s not as big a deal as people seem to think.
Okay, that out of the way; a category consists of two classes: the “objects” and the “morphisms”, or sometimes “points” and “arrows”. These are denoted and , respectively.
Every morphism has a “source” and a “target” object: and . If a morphism has source and target we often write . The class of all morphisms in with source and target is written , or just if the category is understood. If all these “hom-classes” are actually sets, we say the category is “locally small”. Most of the categories we consider will be locally small, and I’ll just use this assumption without mentioning it explicitly.
Given any three objects , , and , we have an operation of “composition”: . We think of this as taking an arrow from to and one from to and joining them tip-to-tail to make an arrow from to . This composition must be associative — the following diagram commutes:
Also, every object has an “identity” morphism so that for all and for all .
We can see that this looks a lot like the definition of a monoid, and for good reason: a monoid is “just” a (small) category with a single object. Walk through the definitions and say that there’s only one object. You’ll see that every morphism has the same source and target, so they can all be composed with each other. Then we’ve got a set of morphisms equipped with an associative composition with an identity element — a monoid!
The most commonly seen use of categories is to describe other algebraic structures. The standard example here (which will motivate much of our later definitions) is : the category of sets. This has as objects the class of all sets (which can’t itself be a set). The morphisms are all functions .
Similarly, we have the categories — groups — — rings with identity — — left -modules — and so on. Each of these categories has as objects the class of all the apropriate algebraic structures, and as morphisms all homomorphisms of those structures.
As a more concrete example, consider a ring with unit. We construct a small category as follows: take as objects the set of natural numbers. The morphisms are all matrices with entries in . The composition is regular matrix multiplication, and the identity on the object is the identity matrix.
Another great example of a category is a preorder. Given a preorder we take the set of elements as the objects of our category. Then we say that there is a single morphism in if and no morphisms in the hom-set otherwise. Reflexivity tells us that there is a morphism in for every object which can serve as an identity, and transitivity tells us that if there’s a morphism in and one in , then there’s one in which can serve as their composite.
For a good while we’ll be giving a lot of definitions of concepts in the language of categories, usually motivated from the category of sets. Category theory gets a bad rap as involving a lot of definitions, but the language really does streamline a lot of thought about mathematics, so it’s worth picking up a basic fluency. Everything I’ll define in this first series I’ve actually already given good examples of in special cases, so the motivation should be apparent. We’ll see them coming up again and again in later work, which (I hope) will help lead to a comprehension of later mathematical concepts by analogy from the simpler concepts in algebra.