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Adjunction

I am learning about adjunction by:

• Understanding and relating the three definitions of adjunction.
• Collecting examples of adjunction.
• Classifying adjunctions.
• Relating adjunction to the Yoneda lemma.

I am currently focusing on The Fundamental Theorem of Covering Spaces.

The Significance of Adjunction

Adjunctions describe how information stays the same though context changes.

A pair of adjoint functors {$F\dashv G$} is synchronized so that the same information (a set of morphisms) is presented in different contexts (different categories, thus different worlds). The two worlds may be very different and in many respects not comparable, yet each world provides a context by which the asserted information carries over.

Modeling chains of perspectives

A pair of adjoint functors may thus very well model a perspective in one world upon another world. This is all the more plausible given that adjunction is an asymmetric relationship whereby the left functor is qualitatively distinct from the right functor. There are many examples where {$F$} is a "forgetful functor" and {$G$} is a "free functor". But there are other patterns as well, for example, {$F$} may be a tensor product functor and {$G$} may bė a homset functor.

Furthemore, there can be chains of adjoint functors. A left adjoint functor {$F$} may itself have a left adjoint functor, in which case that adjoint functor is unique up to isomorphism. Or it may not have a left adjoint functor. Similarly, on the right. Thus any adjunction sits within a chain of adjoint functors, known as an adjoint string.

A key goal of "math for wisdom" is to use math to model perspectives, and in particular, divisions of everything into perspectives. Such structures are fundamental for absolute truth because they describe a small set of perspectives (two, three, four or more) in terms of their relationships with each other. One way to think of a division of everything is as a chain of perspectives between everything and everything.

Finite strings of adjoint functors may very well model such chains of perspectives. I am thus collecting and classifying examples of adjunctions, and especially, adjoint strings. I am hoping to clarify the qualitative aspects and chase down their origins in the details of category theory.

The challenge is similar to that by which homology and cohomology describe "holes". How do you define and describe that which is not there?