3 - Operations on Presentations

Summary

The usefulness and richness of 2-polygraphs is confirmed by the large number and variety of categories they present. In order to show that a given polygraph is a presentation of a given category, one can either tackle the issue directly, by using rewriting tools, or take a modular approach, by combining already known presentations: this is the route taken in the present chapter. Three significant applications are given. First addressed is the presentation of limits and colimits by means of given presentations of the base categories, and precisely shown is how to systematically build presentations of products, coproducts, and pushouts. Next, it is shown how to add formal inverses to some morphisms of a category at the level of presentations. Finally, distributive laws are investigated in relation to factorization systems on categories. A notion of composition along a distributive law between two small categories sharing the same set of objects is introduced, and it is shown how to derive a presentation of this composite from presentations of the components.