The Bicategory of Open Functors

TL;DR

This paper introduces a bicategory of open functors, extending traditional categories and functors to include external influences, with detailed proofs and plans for future comparisons to existing structures.

Contribution

It defines a bicategory of open functors using minimal auxiliary constructions, providing detailed proofs and setting the stage for future comparative analyses.

Findings

Defines a bicategory of open functors with explicit details.

Provides comprehensive proofs for the bicategory structure.

Plans to compare with other categorical constructions in future work.

Abstract

We want to replace categories, functors and natural transformations by categories, open functors and open natural transformations. In analogy with open dynamical systems, the adjective open is added here to mean that some external information is taken into account. For the particular use of the authors, such an open functor is described by two components: a presheaf representing the possible external influences for each input, and a classical functor from the category of elements of this presheaf to the category of results. Considering the appropriate notion of composition then leads to a bicategory. This report describes this bicategory with as little auxiliary constructions as possible and gives all the details of all the proofs needed to establish the bicategory, as explicitly as possible. Subsequent reports will give other presentations of this bicategory and compare it to other existing constructions, e.g. spans, fibrations, pseudoadjunctions, Kleisli bicategories of pseudo-monads, and profunctors (or distributors).