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