nLab
(infinity,1)-category of cartesian sections

Idea

Let $p:E\to X$ be an (∞,1)-functor of (∞,1)-categories. A cartesian section of $p$ is a section $\sigma :X\to E$ that sends all 1-morphisms in $X$ to Cartesian morphisms in $E$.

Definition

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Remarks

If $p:E\to X$ is a Cartesian fibration classified by an (∞,1)-functor $F:X\to (\mathrm{\infty },1){\mathrm{Cat}}^{\mathrm{op}}$ then ${\Gamma }_X^{\mathrm{cart}}(E)$ is equivalent to the limit of $F$

$${\Gamma }_X^{\mathrm{cart}}(E)\simeq \lim F.$$\Gamma_X^{cart}(E) \simeq lim F \,.

See the discussion at limit in a quasi-category for details.

Refernces

In corollary 3.3.3.2 of

• Jacob Lurie, Higher Topos Theory

the collection of cartesian sections of $p:E\to X$ appears as ${\mathrm{Maps}}_X^{♭}(X^{\#},E^{\mathrm{cart}})$.

Here

• the simplicial set ${\mathrm{Maps}}_X^{♯}(\cdots )$ is the simplicial set underlyying the internal hom of marked simplicial sets over $X$ (beginning of section 3.1.3);

• $X^{\#}$ is the simplicial set $X$ with all cells marked (beginning of section 3.1)

• and $E^{\mathrm{cart}}$ is $E$ with precisely all Cartesian morphisms marked (def. 3.1.1.9).