nLab foliated category

Idea

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(not to be confused with the notion of foliations in differential geometry), were introduced by Jean Bénabou in unpublished work dating back to 1984.

They are a weaker structure than fibered categories, but still allow one to test for various standard properties of functors fibre-wise.

Definition

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

1. 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 weak $P$-cartesian morphism $k$, i.e. $f = k\circ v$,

2. The class of weak $P$-cartesian morphisms is closed under composition.

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.

In indexed form

While every functor $P: X \to B$ corresponds to a normal lax functor $dP : B^{\mathrm{op}} \to \mathbf{Prof}$, foliated categories can be characterized as those that factor through the inclusion $\mathbf{ParCat} \to \mathbf{Prof}$, where $\mathbf{ParCat}$ is the 2-category of partial functors.

Indeed, the only obstruction to full representability of the profunctors $dP(h)$, with $h:b \to b'$ in the base $B$, is the existence of all weak cartesian lifts: the domain of definition of the partial functor corresponding to $dP(h)$ will be spanned by those objects $p \in dP(b')$ that admit a weak cartesian lift (this corresponds to Axiom 1 above). Axiom 2 then assures the laxator is well-defined.

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).

A short summary is in this message on the Category Theory Mailing List.