One way to explain categories is to say that the morphisms of a category represent relationships between the objects. I think this is too optimistic. Thinking more negatively, we may ask: “what sort of structure has norphisms between objects representing negative relationships?”. In this paper the authors define norphisms using enriched category theory. In this post I will do the same, but using a simpler enrichment.

For the already indoctrinated, this post is about (\mathsf{Bool}^{op},\vee, \bot)-enriched categories where \mathsf{Bool} is the category \bot \implies \top i.e. the category with two objects (\bot and \top) and a unique non-identity arrow from \bot \to \top. Before unpacking what this means, I will tell you how they are interpreted. A (\mathsf{Bool}^{op},\vee, \bot)-category has an underlying graph. The graph

is a \mathsf{Bool}^{op}-enriched category. The edges of this structure represent unavoidable obstructions. The unavoidableness pops right out of the definition of enrichment. To see that we need to unpack the definition of \mathsf{Bool}^{op}-enrichment.

In a V-category for each object x there is a V-morphism e_x : I \to G(x,x) where I is the monoidal unit of V. In our case, for a \mathsf{Bool}^{op}-category G, we have a morphism in \mathsf{Bool} denoted G(x,x) \implies \bot. Because \bot is the bottom element of \mathsf{Bool}, this means that G(x,x)=\bot so you can’t have an obstruction from an object to itself. If you don’t go anywhere, then you can’t get stuck! You may obtain the composition law by again reversing the arrows and the monoidal structure. This gives

G(x,z) \implies G(x,y) \vee G(y,z)

So if you can’t get from x to z, then either you can’t get from x to y or you can’t get from y to z. Note that this law is the contrapositive of the transitive law of posets G(x,y) \and G(y,z) \implies G(x,z). For this reason my collaborator named these structures contraposets.

The cool thing about contraposets is that when you glue contraposets together they get smaller! Let’s take the example of the contraposet from before and add another vertex without any obstruction.

This is no longer a contraposet because now the obstructions can be avoided by going to the extra vertex. The smallest contraposet containing the above edges is the following:

It’s magical to me that all of the obstructions dissappear. The lesson here is clear: we can avoid our failures when we work together!