Re: Fundamental Theorem of Category Theory?

[claudio pisani <pisclau@yahoo.it>]

The Yoneda Lemma is in fact a particular case of the reflections of Cat/X in discrete (op)fibrations over X (the reflection of an object x:1->X gives the slice X/x -> X, which corresponds to the representable X(-,x));
another particular case (X=1), gives the components of a category.
The above reflections are a consequence of the "comprehensive" factorization systems (final functors, discrete fibrations) and (initial functors, discrete opfibrations) on Cat.
It turns out that several aspects of category theory can be developed in any finitely complete category C with two factorization systems properly related (the main axiom is "reciprocal stability").
Thus category theory can be indeed founded on (a generalization of) the Yoneda Lemma; in particular, in this perspective, universal properties inside C depend on the universal properties which follow from the factorization systems.

Best regards

Claudio Pisani







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