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