I recently noticed that in Abstract no. 652-4 in the Notices of the AMS volume 14 (1967) page 937, John Gray advocates a systematic treatment of the calculus of comma categories and lists five operations which should be explicitly accounted for in such a calculus. He also mentions that Jon Beck contributed to that discussion. Probably John Gray's notes, if they still exist, would be a helpful guide to someone planning to write a systematic treatment as suggested recently Uwe Wolters. Bill On Mon, 5 Nov 2007, claudio pisani wrote:
The following facts about slice categories may be worth noticing:
1 In the equivalence between df/X (discrete fibrations over a category X) and presheaves on X, the slices X/x -> X correspond to the representable presheaves.
2. (Yoneda Lemma) The reflection of x:1->X (as an object of Cat/X) in df/X is (isomorphic to) X/x (with its terminal object as reflection map). In particular, the full subcategory sl/X of df/X generated by the slices over X is isomorphic to X.
3. For any functor p:P->X, a morphism p->X/x in Cat/X is a cone of base p and vertex x.
4. So, a reflection of p->X/x of p in sl/X is a colimiting cone.
5. A functor f:X->Y has a right adjoint iff the pullback f*Y/y of any slice of Y is (isomorphic to) a slice of X.
6. If ex_f -| f* : df/Y -> df/X is the "left Kan extension" along f, then the counit e: ex_f f* Y/y -> Y/y is an iso for any y iff f is "dense" (aka "connected") while it is a colimiting cone for any y iff f is "adequate" (aka "dense"). Using instead the adjunction f_! -| f* : Cat/Y -> Cat/X the counit is a colimiting cone for any y iff f is adequate (as before), while it is an absolute colimit iff f is dense.
Best regards.
Claudio
--- Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no> ha scritto:
Dear all,
I'm looking for a comprehensive exposition of definitions and results around comma/slice categories. Especially, it would be nice to have something also for non-specialists in category theory as young postgraduates. Is there any book or text you would recommend?
Best regards
Uwe Wolter