I collect hereby some of my studies on profunctors.
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- Ok, but what is a profunctor?
Plenty faces of a profunctor - We can consider profunctors as categories and profunctor morphisms as functors between them.
- And, actually, every adjunction is a single profunctor.
On the two definitions of adjunction - Of course, the Yoneda lemma can also be rephrased in several ways.
- Ok, but what is a profunctor?
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Moreover, every functor can be seen as a fibration of categories glued together by profunctors:
Functor as a fibration - One definition of profunctors pops up a possibility for a ‘2-way profunctor’, a bridge, which is, by the way, a Morita context.
A Bridge Construction
Morita equivalence by categorical bridges - A two dimensional analogue (‘double profunctor‘) is capable to recapture colax and lax functors and their adjointness via [co]reflection properties.
Co/lax functors without apriori comparison cells
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- Additionally, we can define an abstract logic framework for an arbitrary profunctor:
Situation Tree Logic on a Profunctor - This can be used to prove abstract preservation theorems:
An abstract infinitary preservation theorem