Homotopy left-exactness of a left derived functor

Question

Let $$ F: \mathcal{C} \leftrightarrows \mathcal{D} :G $$ be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors $$ \mathbb{L}F: \mathrm{Ho}(\mathcal{C}) \leftrightarrows \mathrm{Ho}(\mathcal{D}) :\mathbb{R}G $$ It is then a well-known fact that $\mathbb{L}F$ preserves homotopy colimits. Suppose $F$ preserves finite limits. Under what conditions does $\mathbb{L}F$ preserves finite homotopy limits?

Motivation: in the case I am interested in $\mathcal{C}$ and $\mathcal{D}$ are categories of simplicial presheaves on some sites, endowed with either the local injective or local projective model structures. The functors $F$ and $G$ arise from a functor between the underlying sites. Homotopy left-exactness of $\mathbb{L}F$ is what I need to claim that $\mathbb{L}F \dashv \mathbb{R}G$ induces a geometric morphism of the $\infty$-topoi presented by $\mathcal{C}$ and $\mathcal{D}$.

Answer 1

I do not know the answer for a general Quillen adjunction, but I will attempt to give a complete answer in the case you're interested in, when the adjunction $(F,G)$ is of the form $(f_!,f^\ast)$ for $f\colon C\to D$ a continuous functor between sites:

Claim. The functor $\mathbb{L}f_!$ preserves finite homotopy limits iff $f$ is a morphism of site, i.e., iff $f$ induces a geometric morphism between $1$-topoi.

(So for instance if finite limits exist in $C$ and $D$ and $f$ preserves them, then $\mathbb{L}f_!$ preserves finite homotopy limits.)

The proof of the claim will use higher topos theory. The underlying $(\infty,1)$-adjunction to $(F,G)$ is the adjunction

$$ f_!: Shv_\infty^\wedge(C)\to Shv_\infty^\wedge(D): f^\ast$$

between the $(\infty,1)$-topoi of hypercomplete sheaves, so we must figure out when this $f_!$ preserves finite limits. First of all it is clear that if $f_!$ preserves finite limits, then $f$ is a morphism of sites: the restriction of $f_!$ to $0$-truncated objects still preserves finite limits.

The other implication is less trivial: assume that $f$ is a morphism of sites. Let $Shv_\infty^\sim(C)$ denote the $1$-localic $(\infty,1)$-topos whose $0$-truncated objects are sheaves of sets on $C$ (see HTT Definition 6.4.5.8): its hypercompletion is the topos $Shv_\infty^\wedge(C)$ we're interested in (concretely, $Shv_\infty^\sim(C)\simeq Shv_\infty(C')$ where $C'$ is a site with finite limits equivalent to $C$). By definition of $1$-localic topoi, the functor $f^\ast\colon Shv_\infty^\sim(D)\to Shv_\infty^\sim(C)$ has a left exact left adjoint since its restriction to $0$-truncated objects has (by assumption). The functor $f_!\colon Shv_\infty^\wedge(C)\to Shv_\infty^\wedge (D)$ is the composition of the functors

$$ Shv_\infty^\wedge(C) \hookrightarrow Shv_\infty^\sim(C) \stackrel{f_!}{\to} Shv_\infty^\sim(D) \to Shv_\infty^\wedge(D)$$

all of which preserve finite limits.

ETA:

This leaves open the question of whether $f_!\colon Shv_\infty(C)\to Shv_\infty(D)$ preserves finite limits for $f$ a morphism of site. It's true if $C$ and $D$ have finite limits, by the above. My hunch is that it's not true in general, i.e., that the appropriate higher notion of morphism of sites is stronger than the usual one.

In case $f:C\to D$ is cocontinuous (but not necessarily continuous), then the functor $f_\ast\colon PSh_\infty(C)\to PSh_\infty(D)$ preserves sheaves (this follows from the corresponding fact for sheaves of sets because sieves are $0$-truncated). The induced adjunction $(f^\ast,f_\ast)$ between sheaves is always a geometric morphism because $f^\ast$ is the composition of three functors that preserve finite limits, as above. In particular, $f_\ast$ preserves hypercomplete objects and you get an induced geometric morphism between hypercompletions.