Isoregular theories, accessible 2-categories, and
free constructions

The importance of free constructions in 2-dimensional category theory is widely recognised; it is enough to think for instance about free completions under limits or colimits of some shape, as well as free regular and exact completions. A way to show that such free constructions exist is usually provided by an adjoint functor theorem, an instance of which is proved in [1] in the context of accessible 2-categories with flexible limits.

The purpose of this talk is to present a notion of 2-dimensional theory, in the sense of logic, whose 2-categories of models are indeed accessible with flexible limits, and for which the forgetful 2-functors that naturally arise between them are accessible and flexible-limit preserving. This, together with the result of [1], implies that free constructions, in form of left biadjoints, always exist in such a framework.

We call the 2-dimensional theories in question isoregular. The idea being that, just like in ordinary cartesian logic one is allowed to express properties defined by unique existential quantification, within isoregular logic one can express properties defined by an existence which is unique up to (unique) isomorphism.

Examples of 2-categories arising this way include those whose objects are: Categories with (co)limits of some shape, Grothendieck fibrations, Clans, Regular and Exact categories, as well as Protomodular, and Semiabilian categories.