Two students in the Applied Category Theory 2018 school have blogged about a classic paper in category theory:

• Daniel Cicala and Jules Hedges, Cartesian bicategories, The *n*-Category Café, 19 February 2018.

Jules Hedges is a postdoc in the computer science department at Oxford who is applying category theory to game theory and economics. Daniel Cicala is a grad student working with me on a compositional approach to graph rewriting, which is about stuff like this:

This picture shows four ‘open graphs’: graphs with inputs and outputs. The vertices are labelled with operations. The top of the picture shows a ‘rewrite rule’ where one open graph is turned into another: the operation of multiplying by 2 is replaced by the operation of adding something to itself. The bottom of the picture shows one way we can ‘apply’ this rule: this takes us from open graph at bottom left to the open graph at bottom right.

So, we can use graph rewriting to think about ways to transform a computer program into another, perhaps simpler, computer program that does the same thing.

How do we formalize this?

A computer program wants to be a morphism, since it’s a process that turns some input into some output. Rewriting wants to be a 2-morphism, since it’s a ‘meta-process’ that turns some program into some other program. So, there should be some bicategory with computer programs (or labelled open graphs!) as morphisms and rewrites as 2-morphisms. In fact there should be a bunch of such bicategories, since there are a lot of details that one can tweak.

Together with my student Kenny Courser, Daniel has been investigating these bicategories:

• Daniel Cicala, Spans of cospans, *Theory and Applications of Categories* **33** (2018), 131–147.

Abstract.We discuss the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.

• Daniel Cicala, Spans of cospans in a topos, *Theory and Applications of Categories* **33** (2018), 1–22.

Abstract.For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph)).

This stuff sounds abstract and esoteric when they talk about it, but it’s really all about things like the picture above—and it’s an important part of network theory!

Recently Daniel Cicala has noticed that some of the bicategories he’s getting are ‘cartesian bicategories’ in the sense of this paper:

• Aurelio Carboni and Robert F. C. Walters, Cartesian bicategories I, *Journal of Pure and Applied Algebra* **49** (1987), 11–32.

And that’s the paper he’s blogging about now with Jules Hedges!