The Azimuth Project
Cospan notations

Idea

This is a page for listing notational conventions regarding decorated and structured cospans, fibrations, etc. in the papers written by John Baez, Daniel Cicala, Kenny Courser, Jade Master, Joe Moeller and Christina Vasilakopoulou.

For examples of papers obeying these conventions see:

• John Baez and Jade Master, Open Petri nets: PDF and LaTeX.

• John Baez and Kenny Courser, Structured cospan categories (draft): PDF and LaTeX.

Conventions

We use \mathsf for categories, \mathbf for bicategories and use \mathbb in the first letter for double categories, boldface for the rest. So, for example, in applications to Markov processes we have a category $\mathsf{Mark}$, a bicategory $\mathbf{Mark}$ and a double category $\mathbb{M}\mathbf{ark}$. They look better in an actual paper!

We use $L : \mathsf{A} \to \mathsf{X}$ as our general example of a left adjoint in the theory of structured cospans, and $R : \mathsf{X} \to \mathsf{A}$ for its right adjoint. We call the resulting double category of structured cospans ${}_L \mathbb{C}sp$. This in turn gives a bicategory ${}_L \mathbf{Csp}$ and a category ${}_L \mathrm{Csp}$.

In the theory of decorated cospans $R : \mathsf{X} \to \mathsf{A}$ is an opfibration corresponding to some pseudofunctor $F: \mathsf{A} \to \mathsf{Cat}$. We call the resulting double category of decorated cospans $F\mathbb{C}sp$. This in turn gives a bicategory $F\mathbf{Csp}$ and a category $F\mathrm{Csp}$.

Macros

Placing these macros in your LaTeX preamble can help simplify following the conventions.

• Categories

: \newcommand{\cat}[1]{\mathsf}

• Bicategories

: \newcommand{\bicat}[1]{\mathbf}

• Double categories

: \newcommand{\doublecat}[1]{\mathbf{\mathbb #1}}