(Joint work with P. LeFanu Lumsdaine.)
Lawvere theories and (coloured) operads provide particularly nice representations for suitable algebraic theories with a given set of sorts, as monoids in certain categories of collections.
We extend this to dependent type theories: For an inverse category C, we show how suitable “C-sorted type theories” may be viewed (1) as monoids in a category of collections, and (2) as generalised Lawvere theories in the sense of Berger–Melliès–Weber. Moreover, (essentially) every dependent type theory arises in this way.
Inverse categories are known from homotopy theory, where they (or their opposite categories) provide a good notion of a category of ``cells''. Examples are the category of semi-simplices, the category of globes, the category of opetopes, etc.