Accessible and locally presentable categories are particular classes of categories which can be neatly characterized in (at least) three different ways: Through category theoretic properties, or as categories of models of certain types of sketches or first order theories. Given a category whose objects are logics and whose morphisms are some sort of translations, one can ask whether this category is accessible. An affirmative answer has at least two interesting aspects: First, one has at one's disposal a bunch of results on accessible categories (for example, in such a category, for each kind of colimit which exists, the corresponding kind of limit also exists and vice versa). Second, by the categorial characterization, each object in an accessible category is a colimit of a diagram of well-behaved, so-called presentable, objects, which gives an interesting point of view towards an important question of universal logic; namely whether every logic can be gained by fibring several simpler logics and whether there are fundamental building bricks ("prime logics") for such a process: In an accessible category these are to be sought for among the presentable logics.

In this tutorial we will outline the basic idea of accessible categories and explain in more detail, and with examples, their significance for universal logic as indicated above. People who wish to follow this tutorial are highly recommended to also attend the lectures of Marcelo E. Coniglio on Category theory and Logic.

References:

J. Adamek and J. Rosicky: Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge,1994

P. Arndt, R. de Alvarenga Freire, O.O. Luciano and H.L. Mariano: A global glance on categories in logic. Logica Universalis, Vol.1, Nr.1, Birkhäuser 2007