Re: Fundamental Theorem of Category Theory

Re: Fundamental Theorem of Category Theory

[Ross Street]

Dear All

The Fundamental Theorem of Category Theory, to my mind, encompasses
all the facts surrounding the fact that the presheaf category PA is
the bicategorically free small-cocompletion of a small category A.
With my students I have always called it:

"The Whole Kan Business".

It can be expressed something like this:

Theorem. For each small category A and small-cocomplete category X,
left Kan extension along the Yoneda embedding y_A : A --> PA provides
an equivalence of categories

	[A,X] --> Cocts[PA,X]

where the codomain is the full subcategory of the functor category
[PA,X] consisting of the small-colimit-preserving functors. Moreover,
the value of the equivalence at j : A --> X has a right adjoint given
by x |--> X(j-,x).

Ross


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]