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Adjunctions
In this chapter we will continue with this leit motif that we developed in the previous two chapters - to begin each of them by introducing a new concept of equality between categories (and furthermore, for each new type of equality to be more relaxed than the previous one.)
We started the chapter about functors by reviewing categorical isomorphisms, which are invertable functions between categories.
Then in the chapter on natural transformations we saw categories that are equivalent up to an isomorphism.
And now we will relax the condition even more and will review a relationship that is not exactly an equality, , but it is not non-equality either. It is not two-way, but at the same time it is not exactly one-way as well. A relationship called adjunction.
As you can see, I am not very good at explaining, so I got some examples alligned. But before we proceed with them, we will go through some of the basic terminology.
Like equivalence, adjunction is a relationship between two categories. Like equivalence, it is composed of two functors.
However, unlike equivalence, an adjunction is not symmetric i.e. although the two functors are two way and we
F \bullet G \cong id_{D}
G \bullet F \cong id_{C}
F \bullet G \to id_{D}
G \bullet F \to id_{C}
Free-forgetful adjunctions
Formal concept analysis
Tensor-Hom adjunction
The tensor-hom adjunction is actually a codename for the curry-uncurry function that we saw earlier.
Adjunctions
https://github.com/adamnemecek/adjoint/
https://chrispenner.ca/posts/adjunction-battleship
-- Left adjoints preserve joins and right adjoints preserve meets.
-- Free/forgetful adjunctions are ones where the unit is Id.