Foliated categories (French: catégories feuilletées), or simply foliations (not to be confused with the notion of foliations in differential geometry), were introduced by Jean Bénabou in unpublished work dating back to 1984.

A functor$P\colon \mathbf{X} \to \mathbf{B}$ makes its domaincategory$\mathbf{X}$ a foliated category (over $\mathbf{B}$) if the following conditions hold:

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. $f = k\circ v$,

If closure under composition is not required, we obtain the notion of prefoliated category.

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, Cartesian functors and foliated categories, talk at Oxford (1 May 2012) [YouTube]

Jean Bénabou, Foncteurs cartésiens et catégories feuilletées, talk at Journée Guitart, Paris (9 November 2012) [YouTube, slides, pdf]

Jean Bénabou, Du vieux et du neuf sur la construction de Grothendieck, talk at Paris-Diderot (March 2019) [YouTube]

(the material on foliated categories – called simply foliations — starts at 1:01:00).

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Last revised on December 28, 2023 at 11:49:33.
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