# 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 $E$.

# Remarks

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

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

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

# References

In corollary 3.3.3.2 of

the collection of cartesian sections of $p : E \to X$ appears as $Maps_X^\flat(X^#, E^{cart})$.

Here

• the simplicial set $Maps_X^\flat(\cdots)$ is the simplicial set underlying 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 $E^{cart}$ is $E$ with precisely all Cartesian morphisms marked (def. 3.1.1.9).

Revised on October 4, 2015 11:28:49 by DG? (93.232.217.155)