Re: Discrete fibrations vs. functors into Set

["Andrée Ehresmann" <ehres@u-picardie.fr>]

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_II_1.pdf
<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



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