Sometimes in math things sound obviously true and that’s exactly how it turns out. Other times, you’re less lucky. Today was one of those days. In this post we’ll see that, although \(\mathsf{Grph}\) is adhesive, \(\mathsf{Grph}^{op}\) isn’t.

For a while now, Will Turner and I have been working both with adhesive categories and with \(\mathsf{Grph}^{op}\). As is often the case, if you’re holding a nail-shaped thing in one hand and hammer-shaped thing in the other, you might think that they might go together well. That’s what we did. In the back of our minds there was this little harmless assumption that \(\mathsf{Grph}^{op}\) was adhesive. The proof of this “fact” was one of those “to-dos” that I was saving for a rainy day. The trouble is that there aren’t that many rainy days in Florida, so I never checked it until today when Will found a little counterexample to a conjecture we had made. It turns out that \(\mathsf{Grph}^{op}\) isn’t adhesive and in fact neither is \(\mathsf{Set}^{op}\) (and hence no copresheaf category is!).

So what’s the plan? Well, we’re going to think about co-adhesive categories: i.e. any category \(\mathsf{C}\) such that \(\mathsf{C}^{op}\) is adhesive. I’ll remind you what an adhesive category is and then I’ll prove that \(\mathsf{Set}\) isn’t co-adhesive. It’ll be real short and easy. I’m writing this mostly for me so that I never make the same mistake twice.

Df: an adhesive category is a category \(\mathsf{C}\) in which:

• all pushouts of monomorphims exist,
• all pullbacks exist and
• all monic pushout squares as Van Kampen.

What does Van Kampen mean? A pushout square

is Van Kampen if, whenever we are given a commutative square sitting above it (i.e. the given pushout square is the bottom face) as in the following diagram

then, if back faces are pullback squares, then the front faces are pullbacks if and only if the top face is a pushout.

If adhesive categories seem mysterious to you, don’t worry. The definition scares a lot of people at first sight. But adhesive categories are very nice and in fact most of the nice categories you know and love are adhesive: \(\mathsf{Set}\) is adhesive, any copresheaf category is adhesive (such as graphs, Petri nets, simplicial sets, hypergraphs, databases) and any small topos is adhesive too. Surprisingly (at least for me) it turns out that \(\mathsf{Set}^{op}\) isn’t adhesive.

Prop: \(\mathsf{Set}\) is not coadhesive.

Proof: we have to show that \(\mathsf{Set}^{op}\) is not adhesive. Since \(\mathsf{Set}\) has all pushouts and pullbacks, the problem must lie with the Van Kampen squares. So take the following pushout square in \(\mathsf{Set}^{op}\) (I’ll draw it as a pullback square in \(\mathsf{Set}\) because it’s easier to think about that way).

And take any cube sitting above it as in the definition of a Van Kampen square (again notice that everything will look upside-down because we’re dualizing everything).

This cube has the pullback square we started with as its bottom face and it’s generated by taking pushouts starting with the red function \(f\) defined as \(f = \{(xa,a_x),(xb, b_x), (ya, y’), (yb, y’)\}\). It is easy to verify that, although all of the side faces of the cube are pushouts, the top face is not a pullback. Thus \(\mathsf{Set}\) is not co-adhesive, as desired.

So there you go. It turns out as usual that it’s the “easy things” that bite you. It turns out that categorical duality continues to be difficult to think about. And it turns out that I have little intuition as to which categories can be both adhesive and co-adhesive! If you know of any examples please do get in touch with me!