nLab foliated category


Foliated categories (catégories feuilletées) were introduced by Jean Benabou in an ongoing work which started in 1984 and is yet not published. They are a weaker structure than a fibered category, but still allow one to test for various standard properties of functors fibrewise.

The definition goes as follows: a functor P:XBP\colon \mathbf{X} \to \mathbf{B} makes X\mathbf{X} a foliated category (over B\mathbf{B}) if every morphism ff in X\mathbf{X} factors as a PP-vertical morphism vv (i.e. P(v)P(v) is an identity morphism in B\mathbf{B}), followed by a PP-cartesian morphism kk, i.e. f=kvf = k\circ v, and, further, PP-cartesian morphisms are closed under composition. (Note that this is the weaker notion of cartesian morphism.)

A morphism between foliated categories XX\mathbf{X}'\to \mathbf{X} (over B\mathbf{B}) is a functor over B\mathbf{B} that sends cartesian morphisms to cartesian morphisms, and such that for every object XX' of X\mathbf{X}', and morphism f:YF(X)f\colon Y\to F(X') in X\mathbf{X}, there is a factorisation f=F(k)vf = F(k)\circ v, where vv is vertical in X\mathbf{X} and kk is cartesian in X\mathbf{X}'. Bénabou calls such functors cartesian.


  • Jean Bénabou, Foncteurs cartésiens et catégories feuilletées (9 novembre 2012, journée Guitart, Paris) YouTube slides

  • Jean Bénabou, Du vieux et du neuf sur la construction de Grothendieck, March 2019, YouTube (the material on foliated categories—called simply foliations—starts after timestamp 1:01:00).

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Last revised on August 20, 2022 at 14:42:53. See the history of this page for a list of all contributions to it.