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

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

References

• 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).

A short summary is in a message at Category List.