Morita contexts as lax functors

TL;DR

This paper demonstrates that Morita contexts can be understood as lax functors from a specific category, generalizing the monad-lax functor correspondence and providing a unified framework for their theory.

Contribution

It introduces the perspective of viewing Morita contexts as lax functors from the chaotic category, extending the monad-lax functor analogy to a broader categorical setting.

Findings

Morita contexts correspond to lax functors from the chaotic category with two objects.

General results about lax functors apply to Morita contexts, unifying their theory.

The approach offers an accessible introduction to lax functors for those familiar with monads.

Abstract

Monads are well known to be equivalent to lax functors out of the terminal category. Morita contexts are here shown to be lax functors out of the chaotic category with two objects. This allows various aspects in the theory of Morita contexts to be seen as special cases of general results about lax functors. The account we give of this could serve as an introduction to lax functors for those familiar with the theory of monads. We also prove some very general results along these lines relative to a given 2-comonad, with the classical case of ordinary monad theory amounting to the case of the identity comonad on Cat.