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I know from classical category theory that if $C$ is a small category and $X$ is a presheaf, that there is a canonical equivalence $$Set^{C^{op}}/X \simeq Set^{\left(C/X\right)^{op}},$$ where $C/X$ is the category of elements of $X$ (i.e. the Grothendieck construction of $X$). Moreover, if $C$ carries a Grothendieck topology, this statement is true for sheaves, where $C/X$ inherits a canonical topology from $C$.

Can I make a similar statement if I go to infinity sheaves, and if so, does anyone have a reference? Thanks. If someone knows a way of proving this model theoretically, that would also be nice.

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I should be careful about the wording "canonical topology", I don't mean anything about representables. – David Carchedi Mar 19 '11 at 2:25
up vote 2 down vote accepted

This is Corollary in HTT. Somehow I overlooked this.

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