Topos Theory (Part 5)

Azimuth 2020-01-27

It’s time to understand why the category of sheaves on a topological space acts like the category of sets in the following ways:

• It has finite colimits. • It has finite limits. • It is cartesian closed. • It has a subboject classifier.

We summarize these four properties by saying the category of sheaves is an elementary topos. (In fact it’s better, since it has all limits and colimits.)

As a warmup, first let’s see why that the category of presheaves on a topological space is an elementary topos!

It’s actually just as easy to see something more general, which will come in handy later: the category of presheaves on any category is an elementary topos. Remember, given a category $\mathsf{C},$ a presheaf on $\mathsf{C}$ is a functor

$F \colon \mathsf{C}^{\textrm{op}} \to \mathsf{Set}$

A morphism of presheaves from $F$ to another presheaf

$G \colon \mathsf{C}^{\textrm{op}} \to \mathsf{Set}$

is a natural transformation

$\alpha \colon F \Rightarrow G$

Presheaves on $\mathsf{C}$ and morphisms between them form a presheaf category, which we call

$\mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

or $\widehat{\mathsf{C}}$ for short.

Presheaves on a topological space $X$ are just the special case where we take $\mathsf{C}$ to be the poset of open subsets of $X.$ So if you want a ‘geometrical’ intuition for presheaves on a category, you should imagine the objects of the category as being like open sets. This will come in handy later, when we talk about sheaves on a category.

But there is also another important intuition regarding presheaves. This is more of an ‘algebraic’ intuition. Starting from a category $\mathsf{C}$ and building the presheaf category

$\mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

is analogous to taking a set $S$ and building the set

$\mathbb{Z}^S$

of all functions from $S$ to the integers. $\mathbb{Z}^S$ is a commutative ring if we use pointwise addition and multiplication of functions as our ring operations. $\mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$ is an elementary topos—but this means we should think of an elementary topos as being a bit like a commutative ring. More precisely, it’s like a ‘categorified’ commutative ring, since it’s a category rather than merely a set.

There’s a one-to-one function

$S \hookrightarrow \mathbb{Z}^S$

sending any element of $S$ to the characteristic function of that element. Similarly, there’s a full and faithful functor

$\mathsf{C} \hookrightarrow \mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

sending any object $c \in \mathsf{C}$ to the presheaf $\mathrm{hom}(-,c).$ This is called the Yoneda embedding. So, presheaf categories are a trick for embedding categories into elementary topoi!

In fact presheaf categories have all limits and colimits, so they are better than just elementary topoi: we will someday see they are examples of ‘Grothendieck topoi’. There are also other ways in which my story just now could be polished, but it would be a bit distracting to do so. Let’s get to work and study presheaf categories!

Today I’ll just talk about colimits and limits.

Colimits in presheaf categories

Presheaf categories have all colimits, and these colimits can be ‘computed pointwise’. What does this mean? Colimits are a lot like addition, and taking colimits of presheaves is a lot like how we add functions from a set to $\mathbb{Z}.$ We just add their values at each point.

More precisely, say we have a diagram of presheaves on $\mathsf{C}.$ This is a functor

$F \colon \mathsf{D} \to \mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

where $\mathsf{D}$ is any small category, the ‘shape’ of our diagram. The colimit of $F$ should be a presheaf, and I’ll call this

$\mathrm{colim} F \in \mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

How do we compute it? Well, notice that $F$ is a functor from $\mathsf{D}$ to the category of functors from $\mathsf{C}^{\mathrm{op}}$ to $\mathsf{Set}.$ So, we can change our viewpoint and think of it as a functor

$F \colon \mathsf{D} \times \mathsf{C}^{\mathrm{op}} \to \mathsf{Set}$

I’m using the same name for this because I’m lazy! Note that for each object $c \in \mathsf{C}^{\mathrm{op}}$ we get a functor

$F(-, c) \colon \mathsf{D} \to \mathsf{Set}$

Since $\mathsf{Set}$ has colimits, we can take the colimit of this functor and get a set

$\mathrm{colim} F(-, c)$

But you can check that this set depends functorially on $c,$ so it defines a functor from $\mathsf{C}^{\mathrm{op}}$ to $\mathsf{Set},$ which is our desired functor

$\mathrm{colim} F \in \mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

Of course, you also have to check that this really is the colimit of our diagram of presheaves on $\mathsf{C}.$

Mac Lane and Moerdijk refer the reader to Section V.3 of Categories for the Working Mathematician for a proof of this result, but I think it’s better to prove it yourself. That is, it seems less painful to follow your nose and do the obvious thing at each step than to wrap your brain around someone else’s notation. I guess this is only true if you’ve go the hang of the subject, but anyway:

Puzzle. Show in the above situation that $\mathrm{colim} F(-, c)$ depends functorially on $c \in \mathsf{C}^{\mathrm{op}}$ and that the resulting functor is the colimit of the diagram $F.$

By the way, I can’t resist mentioning an important fact here: the category of presheaves on $\mathsf{C}$ is the free category with all colimits on $\mathsf{C}.$ In other words, it not only has all colimits, it’s precisely what you’d get by taking $\mathsf{C}$ and ‘freely’ throwing in all colimits. This is why I mentioned colimits first.

This fact is analogous to the fact that when $S$ is a finite set, $\mathbb{Z}^S$ is the free abelian group on $S.$ The fact that we don’t need a finiteness condition when working with presheaves is one of those ways in which categories are nicer than sets.

Limits in presheaf categories

Presheaf categories also have all limits, and these too can be ‘computed pointwise’. The argument is just like that for colimits, but now we use the fact that $\mathsf{Set}$ has all limits.

Puzzle. Given a diagram of presheaves on $\mathsf{C}$

$F \colon \mathsf{D} \to \mathsf{Set}^{\mathsf{C}^{\mathrm{op}}}$

show that $\mathrm{lim} F(-, c)$ depends functorially on $c \in \mathsf{C}^{\mathrm{op}}$ and that the resulting functor is the limit of the diagram $F.$

The category of graphs

It helps to understand some examples of what we’re doing here, and a nice example is the category of graphs. There’s a category $\mathsf{G}$ that has two objects $v$ and $e$ , and just two morphisms

$s, t \colon v \to e$

besides the identity morphisms. A presheaf on $\mathsf{G}$ is what category theorists call a graph.

Why? Well, say we have such a presheaf

$F \colon \mathsf{G}^{\mathrm{op}} \to \mathsf{Set}$

It consists of a set $V = F(v)$ called the the set of vertices and a set $E = F(e)$ called the set of edges, along with two functions

$F(s) \colon E \to V$

$F(t) \colon E \to V$

assigning to each edge its source and target. This is just what category theorists call a graph; graph theorists might call it a ‘directed multigraph’, and some other mathematicians would call it a ‘quiver’.

Note how the ‘op’ in the definition of presheaf turned the morphisms $s,t \colon v \to e$ into functions $F(s), F(t) \colon E \to V.$ The ‘op’ is more of a nuisance than a help in this example (at least so far): we had to make $s$ and $e$ go from $v$ to $e$ precisely to counteract the effect of this ‘op’.

Since colimits and limits are computed pointwise in a presheaf category, we can compute the colimit or limit of a diagram of graphs quite easily: we just take the colimit of the sets of vertices and the sets of edges separately, and the rest goes along for the ride.

Puzzle. Let the graph $G_V$ be the walking vertex: that is, the graph with just one vertex and no edges. Let $G_E$ be the walking edge: that is, the graph with one edge and two vertices, the source and target of that edge. Show that there are exactly two different morphisms

$f, g \colon G_V \to G_E$

Compute the equalizer and coequalizer of

$f, g \colon G_V \to G_E$