Re: Comma categories

[wlawvere@buffalo.edu]

Dear Uwe
You are right in thinking that there should be such 
an exposition because the construction is explicitly 
or implicitly involved in so many contexts that a 
formal summary would be useful. Unfortunately, 
I know of no such exposition though Hugo Volger 
started one many years ago.

As you can see from the TAC Reprint of my 
thesis, the original motivation was to 
be able to state the definition of adjointness in a 
wholly elementary way for arbitrary categories 
without involving enrichments in some fixed
category of sets. If A is a reflective subcategory in 
some X and if B is coreflective in the same X, then 
composing the implicit functors yields an adjoint 
pair between A and B. The point is that conversely 
any adjoint pair can be so factored through a third 
"adjunction" category X and the universally available 
choice has this simple construction as a pullback.
It proved to be the appropriate tool for calculating
Kan extensions, adequacy comonads, fibrations,etc.
Grothendieck defined slice categories and Artin the 
gluing, both of which are special cases of this
construction.

Although inserters are interdefinable (like equalizers
vs pullbacks), some consider inserters more basic: 
given x:A->C and y:B->C, one can take the 
inserter of the two composites AxB->C to obtain 
the construction under discussion. 

In the special case A=B=1 (when the inserter and the 
"comma" category are the same) we obtain the homset 
(x,y) of two objects of C. The latter was the reason 
for my notation: it generalizes a frequent notation for 
hom.[Recall that every object belongs to a unique 
category; thus the standard notation C(x,y) is
actually redundant (if C is not enriched), though easier
to understand. Either notation is preferable to the 
excessive HomsubC, a back formation not be confused
with the informative HomsubR when C arises from 
adjoining some additional structure R to a given base.]
 
Concerning the bizarre name:
(1) I had neglected to give the construction any name, 
so (2) one started giving it a name based on reading 
aloud the notation: x comma y; (3) some continued
the "name" but changed the notation to a vertical arrow.

Since it is well justified to name a category for its 
objects, and since the effect of insertion is to create 
objects with one ingredient more of structure, recent 
discussions here have proposed the name/notation
             Map(x,y)
[or for emphasis Map(subC)(x,y)]
for the category with its faithful functor to AxB.

Although I often use the word "map" interchangeably
with "morphism", note that the above suggests a more
concrete content: philosophically, in order to confront 
objects in two categories A and B, it is necessary to 
first functorially transport them into a common 
category C. For example to map a 2-truncated simplicial 
set to a diffentiable manifold (such as a piece of
paper) one first interprets each in appropriate ways as 
topological spaces, and the resulting objects form a 
category (having full subcategories of "cartographical" 
interest).  

I would be happy to offer a prize for the best exposition!

Bill

Quoting Uwe Egbert Wolter <Uwe.Wolter@ii.uib.no>:

> 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
> 
> 
> 
>