Discrete fibrations vs. functors into Set
CAUTION: The Sender of this email is not from within Dalhousie. Dear all, We consider two categories. The first category with objects given by a small category B and a functor F:B->Set and morphisms (H,alpha):(B,F)->(C,G) given by a functor H:B->C and a natural transformation alpha:F=>H;G. The second category has as objects discrete fibrations p:E->B and morphisms (H,phi):(E,p)->(D,q:D->C) are given by functors H:B->C and phi:E->D such that phi;q=p;H. 1. Are there any "standard" terms and notations for these categories? 2. For both categories we do have projection functors into Cat! Are these functors kind of (op)fibrations? 3. We know that the Grothendieck construction establishes equivalences between corresponding fibers of the two projection functors into Cat. Do these fiber-wise equivalences extend to an equivalence between the two categories? Thanks Uwe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
CAUTION: The Sender of this email is not from within Dalhousie. See pp.16-17 of my notes on fibered cats available from the arxiv. There is an obvious functor Set^(_) : cat^op -> Cat to which one can apply the Grothendieck construction. Moreover, a cartesian functor is a fibered equivalence iff all its fibers are ordinary equivalences. All this is folklore and just documented in my notes. Thomas
We consider two categories. The first category with objects given by a small category B and a functor F:B->Set and morphisms (H,alpha):(B,F)->(C,G) given by a functor H:B->C and a natural transformation alpha:F=>H;G. The second category has as objects discrete fibrations p:E->B and morphisms (H,phi):(E,p)->(D,q:D->C) are given by functors H:B->C and phi:E->D such that phi;q=p;H.
1. Are there any "standard" terms and notations for these categories? 2. For both categories we do have projection functors into Cat! Are these functors kind of (op)fibrations? 3. We know that the Grothendieck construction establishes equivalences between corresponding fibers of the two projection functors into Cat. Do these fiber-wise equivalences extend to an equivalence between the two categories?
Thanks
Uwe
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
CAUTION: The Sender of this email is not from within Dalhousie. Answer to Uwe, The notion of a discrete fibration, and its equivalence with a functor to Sets as well as with the action of a category on a set,?? were initially introduced by Charles Ehresmann in "Gattungen von Lokalen Strukturen" (Jahresbericht d. DMV Bd. 60 (1957) S. 4 9 ??? 7 7, reprinted in https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_I... <https://ehres.pagesperso-orange.fr/C.E.WORKS_fichiers/Ehresmann_C.-Oeuvres_II_1.pdf> The first category you describe is generally called the category of diagrams into Sets, Diag(Sets); the second one which is isomorphic, is called the category of (morphisms between) discrete fibrations. The category Diag(Sets), and more generally the (2-)category Diag(H) for any category H, have been extensively studied by Ren?? Guitart in some 1970's papers, in particular in D??compositions et lax-compl??tions, (avec L. Van den Bril), CTGD XVIII,4, p. 333-407, 1977. http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf <http://archive.numdam.org/article/CTGDC_1977__18_4_333_0.pdf> In the last years, with Alexandre Popoff, C. Agon and M. Andreatta, we have studied and applied Diag(H) in papers on Math/Music theory, naming its objects "Poly-Klumpenhouwer-Nets" (or PK-Net) with values in H, for instance in "From Nets to PK-Nets: a categorical approach", /Perspective of new music /54-2, 2016, 5-68. For other. references, consult my personal site https://ehres.pagesperso-orange.fr/ <https://ehres.pagesperso-orange.fr/> Kind regards Andr??e [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (3)
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Andrée Ehresmann -
streicher＠mathematik.tu-darmstadt.de -
Uwe Egbert Wolter