Co-tabulations, Bicolimits and Van-Kampen Squares in Collagories

We previously defined collagories essentially as “distributive allegories without zero morphisms”. Collagories are sufficient for accommodating the relation-algebraic approach to graph transformation, and closely correspond to the adhesive categories important for the categorical DPO approach to graph transformation. Heindel and Sobocinski have recently characterised the Van-Kampen colimits used in adhesive categories as bicolimits in span categories. In this paper, we study both bicolimits and lax colimits in collagories. We show that the relation-algebraic co-tabulation concept is equivalent to lax colimits of difunctional morphisms and to bipushouts, but much more concise and accessible. From this, we also obtain an interesting characterisation of Van-Kampen squares in collagories