Re: Comma categories

[claudio pisani <pisclau@yahoo.it>]

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
> 
> 
>