Bridges and Profunctors

The main aim of the dissertation is to present a way for de ning bicategories, double categories, lax and colax functors between them, where the coherence pentagon and the lax comparison cells are implicitly encoded in the system. Besides this, we also study related structures such as Morita contexts. For bicategories we follow Tom Leinster's unbiased approach, and we use profunctors regarded as categories (by their collage) and re ections therein. The main theses of the dissertation, section by section: 1 Bicategories In def. 1.1. we give an elementary interpretation of Tom Leinster's `unbiased bicategory ', [Leinster]. (We will mostly omit the pre x `unbiased'.) According to this, an (unbiased) bicategory is given by its objects, arrows between objects, and 2-cells between arrows ( · 88 ⇓ · ) which can be composed `vertically' (for each pair of objects, the arrows between them and their 2-cells form a category). Also, the arrows can be composed `horizontally', though this composition is only weakly associative (associative up to coherent isomorphisms). This horizontal composition of arrows is given by a weakly associative family of operations: an arrow is assigned to each path of the underlying graph, in a functorial way. Paths of length 0 are also considered, as objects. That way, the composition and the unit can be handled together. Moreover, the coherence axiom requires commuting squares only instead of pentagons. Then some bicategorical notions are introduced, such as adjoint pair of arrows, internal monoid, its action, internal bimodule. In example 1.1.7., among others, we present the bicategory Span. Its objects are the sets, and an arrow from set A to set B is a span A← E → B of functions, considered as a bipartite graph, E being the set of edges. In this reading, a 2-cell of Span is a morphism of graphs which xes all the vertices. Example 1.1.14: Internal monoids in the bicategory Span are just the categories. 2 Profunctors De nition 2.1. We call a category H a bridge between categories A and B if A and B (or their isomorphic copies) are disjoint full subcategories of H, and ObH = ObA ∪ObB. In notation: H :A B. We call the arrows of H \ (A ∪ B) heteromorphisms. A bridge H is called directed from A to B (in notation H :A9 B), if all its heteromorphisms are of the form A→ B.