These notes are a current snapshot of the development of the theory of collagories, which are
defined essentially as “distributive allegories without zero morphisms” and named for their close
relationship with the adhesive categories currently popular as a foundation for the categorical
double-pushout approach to graph transformation.
We argue that, thanks to their relation-algebraic flavour, collagories provide a more accessible
and more flexible setting. One contributing factor to this is that the universal characterisations
of pushouts and pullbacks in categories can be replaced with the local characterisations
of tabulations and co-tabulations in collagories.
We document accessibility by showing ways to construct collagories of semi-unary algebras,
which allow natural representations in particular of graph structures, also with fixed label sets.
Via the local ordering on homsets, collagories have a simple 2-categorical structure, and we
use this to show that co-tabulations are equivalent to lax colimits of difunctional morphisms,
and co-tabulations arising from spans of mappings are equivalent to bipushouts, which satisfy
stronger conditions than just pushouts of mappings.
Finally, we consider Van-Kampen squares, the central ingredient of the definition of adhesive
categories, and obtain an interesting characterisation of Van-Kampen squares in collagories.