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
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
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
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
Bill Lawvere writes:
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
As a followup to Bill's note, here is a slightly more recent positing by John Gray to another mailing list on this very topic. * To: types@theory.LCS.MIT.EDU * Subject: Re: Cobig, Coproduct, and Comma * From: gray@symcom.math.uiuc.edu (John Gray) * Date: Mon, 20 Mar 89 17:13:53 EST * Sender: meyer@theory.LCS.MIT.EDU Date: Mon, 20 Mar 89 15:32:11 CST
Cobig, Coproduct, and Comma Vaughan Pratt 3/19/89 Formally a comma category is most slickly described as a lax pullback. I've attempted an understandable account of this 2-category concept in an appendix below. I'd appreciate pointers to other accounts.
Comma categories are an ancient tool in category theory. They were introduced in F. W. Lawvere, Functorial Semantics of Algebraic Theories Thesis, Columbia University, 1963. He used them in --, The category of categories as a foundation for mathematics, Proceedings of the Conference on Categorical Algebra, La Jolla 1965, Springer-Verlag, New York. I discussed them in several places: J. W. Gray, Fibred and cofibred categories, same proceedings as above, 21-83. I gave a brief calculus of comma categories in: --, The categorical comprehension scheme, Category theory, Homology theory and their Applications III, Lecture Notes in Mathematics 99, Springer-Verlag, New York 1969, 242-312. They are described as "Cartesian quasi-limits" in the book: --, Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer-Verlag, New York 1974. which is the first place where the lax description of them can be found. I don't credit it to anybody there, since I assumed it was general knowledge. The name was changed to "lax limits" in: G. M. Kelly and R. Street, Review of the elements of 2-categories, Category Seminar, Lecture Notes in Mathematics 420, Springer- Verlag, New York 1974. The general theory of the properties of lax limits in 2-categories was discussed independently by Street and me in various publications. E. g., J. W. Gray, The existence and construction of lax limits, Cahiers Top. et Geom. Diff. 21 (1980), 277-304. --, Closed categories, Lax limits and homotopy limits, J. Pure Appl. Algebra 19 (1980), 127-158. --, The representation of limits, lax limits, and homotopy limits as sections, in Mathematical Applications of Category Theory, Contemporary Mathematics 30 (1984), AMS, 63-83. R. Street, Two constructions on lax functors, Cahiers Top. et Geom. Diff. 13, (1972), 217-264. --, Limits indexed by category-valued 2-functors, J. Pure and Applied Alg. 8 (1976), 149-181. It is of course very gratifying to see these ideas coming around again as useful tools in the semantics of programming languages. John Gray -- Bob
participants (5)
-
Bill Lawvere -
claudio pisani -
Robert L Knighten -
Uwe Egbert Wolter -
wlawvere＠buffalo.edu