Re: Comma categories

[Bill Lawvere <wlawvere@buffalo.edu>]

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
>>
>>
>>
>
>
>
>