nLab
Cartesian fibration

Idea

A Cartesian fibration of simplicial sets is a morphism between simplicial sets that generalizes the notion of Grothendieck fibration from categories to quasi-categories.

This means that precisely if an ∞-functor $p:C\to D$ if a Cartesian fibration is it possible to interpret its value over any morphism $f:d_1\to d_2$ in $D$ as an ∞-functor $p^{-1}(f):p^{-1}(d_2)\to p^{-1}(d_1)$ between the fibers $p^{-1}(d_2)$ and $p^{-1}(d_1)$ over its source and target objects.

So a Cartesian fibration $p:C\to D$ determines and is determined by an ∞-functor $S_p:D\to (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}$ from $D$ into the opposite category of the (∞,1)-category of (∞,1)-categories.

This correspondence is mediated by the universal fibration of (∞,1)-categories which is a Cartesian fibration $p_{\mathrm{univ}}:Z\to (\mathrm{\infty }\mathrm{,1})\mathrm{Cat}$ such that for every Cartesian fibration $p:C\to D$ there is a functor $S_p:D\to (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}$ such that $p$ is the ($(\mathrm{\infty }\mathrm{,1})$-categorical) pullback of $p_{\mathrm{univ}}$

$$\begin{array}{ccc}C& \to & Z\\ {\downarrow }^p& & {\downarrow }^{p_{\mathrm{univ}}}\\ D& \stackrel{S_p}{\to }& (\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}^{\mathrm{op}}\end{array}.$$\array{ C &\to& Z \\ \downarrow^p && \downarrow^{p_{univ}} \\ D &\stackrel{S_p}{\to}& (\infty,1)Cat^{op} } \,.

As described at generalized universal bundle, this pullback construction is the $(\mathrm{\infty }\mathrm{,1})$-categorical analog of the Grothendieck construction for ordinary categories.

Definition

A Cartesian fibration is an inner fibration of simplicial sets $p:X\to S$ such that for every edge $f:X\to Y$ of $S$ and every lift $\tilde{y}$ of $y$ (that is, $p(\tilde{y})=y$), there is a Cartesian edge $\tilde{f}:\tilde{x}\to \tilde{y}$ in $X$ lifting $f$.

Properties

Proposition

Basic properties:

• Every isomorphism of simplicial sets is a Cartesian fibration.

• Cartesian fibrations are stable under base change in SSet.

• The composite of two Cartesian fibrations is again a Cartesian fibration.

Proof

This is proposition 2.4.2.3 in HTT.

Proposition (over $\mathrm{\infty }$-groupoids)

Let $p:X\to S$ be a morphism of simplicial sets where $S$ is a Kan complex. Then the following are equivalent:

1. $p$ is a Cartesian fibration

2. $p$ os a coCartesian fibrationn

3. $p$ is a categorical fibration

Proof

This is proposition 3.3.1.8 in HTT.

References

Definition 2.4.2.1 in

• Jacob Lurie, Higher Topos Theory