Title: Pictures of finite limits

Abstract: In (classical) Lawvere theories, the central role is played by categories with finite products. For example, the free category with finite products on one object (FinSet^op) is the Lawvere theory of the empty algebraic theory, and the free category with finite products on a signature of an algebraic theory has a concrete description as a category of terms.

In recent joint work with Ivan di Liberti, Fosco Loregian and Chad Nester, we developed a Lawvere-style approach to algebraic theories with partially defined operations. It turns out that in this setting, instead of categories with finite products, the relevant concept is discrete cartesian restriction categories. We developed the technology to describe free such categories. It is known that splitting idempotents yields categories with finite limits. After describing the main steps of the above narrative I will focus on giving examples of the resulting string diagrammatic calculus for free categories with finite limits.