The Unapologetic Mathematician

Categories

Like groups, rings, modules, and other algebraic constructs, we define a category by laying out what’s in it, and how those things relate to each other.

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 $\mathcal{C}$ consists of two classes: the “objects” and the “morphisms”, or sometimes “points” and “arrows”. These are denoted ${\rm Ob}(\mathcal{C})$ and ${\rm Mor}(\mathcal{C})$, respectively.

Every morphism $m$ has a “source” and a “target” object: $s(m)$ and $t(m)$. If a morphism $m$ has source $a$ and target $b$ we often write $m:a\rightarrow b$. The class of all morphisms in $\mathcal{C}$ with source $a$ and target $b$ is written $\hom_\mathcal{C}(a,b)$, or just $\hom(a,b)$ 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 $a$, $b$, and $c$, we have an operation of “composition”: $\circ:\hom(b,c)\times\hom(a,b)\rightarrow\hom(a,c)$. We think of this as taking an arrow from $a$ to $b$ and one from $b$ to $c$ and joining them tip-to-tail to make an arrow from $a$ to $c$. This composition must be associative — the following diagram commutes:
$\begin{matrix}\hom(c,d)\times\hom(b,c)\times\hom(a,b)&\rightarrow&\hom(c,d)\times\hom(a,c)\\\downarrow&&\downarrow\\\hom(b,d)\times\hom(a,b)&\rightarrow&\hom(a,d)\end{matrix}$

Also, every object $a$ has an “identity” morphism $1_a:a\rightarrow a$ so that $1_a\circ m=m$ for all $m\in\hom(b,a)$ and $m\circ1_a=m$ for all $m\in\hom(a,b)$.

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 $\mathbf{Set}$: the category of sets. This has as objects the class of all sets (which can’t itself be a set). The morphisms $\hom_\mathbf{Set}(X,Y)$ are all functions $f:X\rightarrow Y$.

Similarly, we have the categories $\mathbf{Grp}$ — groups — $\mathbf{Ring}$ — rings with identity — $R-\mathbf{mod}$ — left $R$-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 $R$ with unit. We construct a small category $\mathbf{Mat}_R$ as follows: take as objects the set $\mathbb{N}$ of natural numbers. The morphisms $\hom_{\mathbf{Mat}_R}(m,n)$ are all $m\times n$ matrices with entries in $R$. The composition is regular matrix multiplication, and the identity on the object $n$ is the $n\times n$ identity matrix.

Another great example of a category is a preorder. Given a preorder $(P,\leq)$ we take the set of elements $P$ as the objects of our category. Then we say that there is a single morphism in $\hom_P(x,y)$ if $x\leq y$ and no morphisms in the hom-set otherwise. Reflexivity tells us that there is a morphism in $\hom(x,x)$ for every object $x$ which can serve as an identity, and transitivity tells us that if there’s a morphism in $\hom(x,y)$ and one in $\hom(y,z)$, then there’s one in $\hom(x,z)$ 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.

May 22, 2007 - Posted by | Category theory

1. Yay! Now we’re talking!

Are you going to cover derived categories and model categories too soonish? Or is it the wrong track?

Comment by Mikael Johansson | May 22, 2007 | Reply

2. Not for quiiiiite a while yet. I want to build up a good categorical vocabulary before I tackle topology.

Yes, you heard right. I want category theory under my belt as a prereqisite to topology.

Comment by John Armstrong | May 22, 2007 | Reply

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5. Has anyone ever written an idiot’s guide to the safe use of proper classes? cartesian products and `large functions’ such as source and target for arrows are OK, exponentials are not, but I’ve never seen it all laid out for beginners or lazy people. Quine’s book on the logic of set theory and Paul Taylor’s practical foundations book look to me like they might be useful sources, but I haven’t managed to muster up the time and determination to slog through them.

Comment by Avery Andrews | September 26, 2007 | Reply

6. Hi John!
Any chance that you stuff all the category theory material on your site in one old-fashioned pdf file?

thanks

Comment by Christian Hollersen | October 13, 2007 | Reply

7. Christian, I may eventually put these together into an old-fashioned PDF, or an older-fashioned book, even. But first I’m trying to get some original work out so I don’t mark myself as an expositor before I get my research program started.

Comment by John Armstrong | October 13, 2007 | Reply

8. That’s unfortunately all too realistic a concern! Actually I think there’s something to be said for the ‘small chunk at a time’ format here, tho it would be nice if there were an easier way to print them out without the comment-addition facility (what I do is use pdf creator to print the whole thing to pdf, then prinout out the useful stuff from that).

These course notes by Barr and Wells are pretty decent (and their book with 600 solved exercises is certainly worth the modest price, it seems to me).

Comment by MathOutsider | October 13, 2007 | Reply

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