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I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can't find it).

Classically, given two sites $C$ and $D$, a morphism of sites $f:C\to D$ induces a geometric morphism of topos $f_{*}:Shv(C)\to Shv(D)$. Does the statement still hold in the $\infty$ context?

What I'm really after is the following: Let $C$ be an $\infty$ site and let $c$ be an object of $C$. Then $C_{/c}$ inherits a topology from $C$ by restricting the covering sieves to objects in the overcategory. The forgetful functor $f:C_{/c}\to C$ is trivially a morphism of $\infty$ sites. Does $f$ induce a functor $$ f^*:Shv(C)\to Shv(C_{/c})$$ preserving finite $\infty$ limits?

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  • $\begingroup$ It might be valuable to record something in the nlab $\endgroup$
    – Bas Spitters
    Oct 29, 2015 at 14:23
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    $\begingroup$ The particular case you care about is covered by the fact that $Shv(C/c)\simeq Shv(C)/y(c),$ where $y$ is the Yoneda embedding. With this insight, the geometric morphism you seek IS in HTT; it's an etale geometric morphism. The fact I claimed is Proposition 2.2.1 here: arxiv.org/abs/1312.2204 $\endgroup$
    – David Carchedi
    Nov 7, 2015 at 7:34

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I never found any discussion about this in HTT either, but it turns out to work exactly the same way as in SGA 4.

That is, let $f^* : P(D) \to P(C)$ denote the restriction functor on presheaves, and $f_!$ its left adjoint (given by left Kan extension). Since $f$ preserves covering families, $f_!$ preserves local equivalences, so by adjunction $f^*$ preserves sheaves and induces a functor $$ f^*_s : Sh(D) \to Sh(C). $$ Define $$ f_!^s := a_D f_! i_C : Sh(C) \to Sh(D), $$ where $i_C$ is the inclusion $Sh(C) \hookrightarrow P(C)$ and $a_D : P(D) \to Sh(D)$ is the associated sheaf functor. It is straightforward to check that this is left adjoint to $f^*_s$. The fact that $f_!^s$ is left-exact follows from the fact that $i_C$ commutes with (small) limits, $f_!$ commutes with finite limits, and $a_D$ commutes with finite limits (I am assuming of course that $D$ is small).

The fact that $f_! : P(C) \to P(D)$ commutes with finite limits follows from the pointwise formula for the Kan extension $$ f_! F(d) = \varinjlim_{(c \in C, d \to f(c)) \in (d/f)} F(c) $$ where, since $f$ is left-exact, the comma category $(d/f)$ is filtered. (The commutativity of filtered colimits and finite limits in an $\infty$-topos is Example 7.3.4.7 in HTT.)

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  • $\begingroup$ ah, perfect. I thought the argument might go the same way as in the classical case but I wasn't completely sure. Thanks for the response. $\endgroup$
    – Daniel Grady
    Oct 28, 2015 at 11:05
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    $\begingroup$ Of course, as in SGA, we may replace the assumption that $f : C \to D$ preserves finite limits with the assumption that $f_! : \mathrm P (C) \to \mathrm P (D)$ does (or, better, that $a_D f_! : \mathrm P (C) \to \mathrm{Sh} (D)$ does) when $C$ does not have all finite limits. $\endgroup$
    – Zhen Lin
    Oct 28, 2015 at 11:51

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