We have introduced, in a previous paper, the fundamental lax 2-category of a ‘directed space’ X. Here we show that, when X has a T1-topology, this structure can be embedded into a larger one, with the same objects (the points of X), the same arrows (the directed paths) and the same cells (based on directed homotopies of paths), but a larger system of comparison cells. The new comparison cells a...

Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analyzing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2-category with one 0-cell, we investigate the free compact 2-category generated by a given category, and we describe its ...

For any 0-cell B in a 2-category B, we introduce the notion of adjoint algebra AdB. This is an center B. We prove that if C finite tensor category, this applied to CMod C-module categories coincides with one introduced by Shimizu. As consequence general approach, obtain new results on for categories.

We outline a definition of accessible and presentable objects in 2-category K endowed with “KZ context”, that is to say pair lax-idempotent monads interacting prescribed way; this perspective suggests unified treatment many “Gabriel-Ulmer like” theorems, asserting how arise as reflections generating ones. the notion (Gabriel-Ulmer) envelope for KZ context, sufficient concoct Gabriel-Ulmer duali...