Re: no fundamental theorems please

[claudio pisani <pisclau@yahoo.it>]

Of course, the proof that the object x:1->X (of Cat/X) has the slice  X/x -> X as a reflection in df/X (and its final object as reflecton map) is essentially the same of that of the standard Yoneda lemma, and the general case only requires a little more effort.
My point is that this formulation seems to me more in the "categorical spirit", stating a universal property that relates categories over X and discrete fibrations.
In fact the paradigm "categories, functors and natural transformations" can be in part replaced by "categories, functors and discrete (op)fibrations"; for instance a colimit x of the object p:P -> X of Cat/X is a reflection of p in slices over X, where the reflection map p -> X/x in Cat/X is the colimiting cone, and so on.
Furthermore, there is a clear analogy (which can be made precise with the proper choice of factorization system on posets) with the reflection of the subsets of a poset X in lower or upper sets of X (the principal sieves being a particular case).

Best regards

Claudio



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