Proof

By the characterizaton of full and faithful adjoint (∞,1)-functors the condition on $i_{!}$ is equivalent to $i^{*}{i}_{!}\simeq \mathrm{Id}$. Since $(i^{*}{i}_{!}⊣i^{*}{i}_{*})$ it follows by essential uniqueness of adjoint (∞,1)-functors that also $i^{*}{i}_{*}\simeq \mathrm{Id}$.

Proof

By essential uniqueness of th global section geometric morphism: In both cases the direct image functor has as left adjoint that preserves the terminal object. Therefore

Analogously in the second case.

Proof

We present the (∞,1)-sheaf (∞,1)-category ${\mathrm{Sh}}_{(\mathrm{\infty },1)}(C_{\mathrm{th}})$ by the projective model structure on simplicial presheaves left Bousfield localized at the covering sieve inclusions

(as discussed at models for (∞,1)-sheaf (∞,1)-toposes).

Consider the right Kan extension ${\mathrm{Ran}}_i:[C^{\mathrm{op}},\mathrm{sSet}]\to [C_{\mathrm{th}}^{\mathrm{op}},\mathrm{sSet}]$ of simplicial presheaves along the functor $i$. On an object $K\times D\in C_{\mathrm{th}}$ it is given by the end-expression

where in the last step we use the Yoneda reduction-form of the Yoneda lemma.

This shows that the right adjoint to $(-)\circ i$ is itself given by precomposition with a functor, and hence has itself a further right adjoint, which gives us a total of four adjoint functors

From this are directly induced the corresponding simplicial Quillen adjunctions on the global projective and injective model structure on simplicial presheaves

Observe that ${\mathrm{Lan}}_i$, being a left Kan extension, sends representables to representables: we have

and by Yoneda reduction (more explicitly: observing that this is equivalently the formula for left Kan extension of the non-corepresentable $C_{\mathrm{th}}(K\times D,i(-)):C\to \mathrm{sSet}$ along the identity functor) this is

By the discussion at simplicial Quillen adjunction for the above Quillen adjunctions to descend to the Cech-local model structure on simplicial presheaves it suffices that the right adjoints preserve locally fibrant objects.

We first check that $(-)\circ i$ sends locally fibrant objects to locally fibrant objects.

To that end, let $\{U_i\to U\}$ be a covering family in $C$. Write ${\int }^{[k]\in \Delta }\Delta [k]\cdot {\coprod }_{i_0,\cdots ,i_k}(j(U_{i_0}){\times }_{j(U)}j(U_{i_1}){\times }_{j(U)}\cdots {\times }_{j(U)}j(U_k))$ for its Cech nerve, where $j$ denotes the Yoneda embedding. Recall by the definition of the ∞-cohesive site $C$ that all the fiber products of representable presheaves here are again themselves representable, hence $\cdots ={\int }^{[k]\in \Delta }\Delta [k]\cdot {\coprod }_{i_0,\cdots ,i_k}(j(U_{i_0}{\times }_U{U}_{i_1}{\times }_U\cdots {\times }_U{U}_k))$. This means that the left adjoint ${\mathrm{Lan}}_i$ preserves not only the coend and tensoring, but by the remark in the previous paragraph and the assumption that $i$ preserves pullbacks along covers we have that

By the assumption that $i$ preserves covers, this is the Cech nerve of a covering family in $C_{\mathrm{th}}$. Therefore for $F\in [C_{\mathrm{th}}^{\mathrm{op}},\mathrm{sSet}{]}_{\mathrm{proj},\mathrm{loc}}$ fibrant we have for all coverings $\{U_i\to U\}$ in $C$ that the descent morphism

is a weak equivalence, hence that $i^{*}F$ is locally fibrant.

To see that $(-)\circ p$ preserves locally fibrant objects, we apply the analogous reasoning after observing that its left adjoint $(-)\circ i$ preserves all limits and colimits of simplicial presheaves (as these are computed objectwise) and by observing that for $\{U_i\stackrel{p_i}{\to }U\}$ a covering family in $C_{\mathrm{th}}$ we have that its image under $(-)\circ i$ is its image under $p$, by the Yoneda lemma:

and using that $p$ preserves covers by assumption.

Therefore $(-)\circ i$ is a left and right local Quillen functor with left local Quillen adjoint ${\mathrm{Lan}}_i$ and right local Quillen adjoint $(-)\circ p$.

It follows that $i^{*}:{\mathrm{Sh}}_{(\mathrm{\infty },1)}(C_{\mathrm{th}})\to {\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ is given by the left derived functor of restriction along $i$, and $i_{*}:{\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\to {\mathrm{Sh}}_{(\mathrm{\infty },1)}(C_{\mathrm{th}})$ is given by the right derived functor of restriction along $p$.

Finally to see that also ${\mathrm{Ran}}_p$ preserves locally fibrant objects by the same reasoning as above, notice that for every covering family $\{U_i\to U\}$ in $C$ and every morphism $K\to p^{*}U$ in $C_{\mathrm{th}}$ we may find a covering $\{K_j\to K\}$ of $K$ such that we find commuting diagrams on the left of

because by adjunction these correspond to commuting diagrams as indicated on the right, which exist by definition of coverage on $C$ and lift through $p$ by assumption on $C_{\mathrm{th}}$.

This implies that $\{p^{*}{U}_i\to p^{*}U\}$ is a generalized cover in the terminology at model structure on simplicial presheaves, which by the discussion there implies that the corresponding Cech nerve equivalent to the sieve inclusion is a weak equivalence.

This establishes the quadruple of adjoint (∞,1)-functors as claimed.

It remains to see that $i_{!}$ is full and faithful.

For that notice the general fact that left Kan extension (see the propeties discussed there) along a full and faithful functor $i$ satisfies ${\mathrm{Lan}}_i\circ i\simeq \mathrm{id}$. It remains to observe that since $(-)\circ i$ is not only right but also left Quillen by the above, we have that $i^{*}{\mathrm{Lan}}_i$ applied to a cofibrant object is already the derived functor of the composite.

Proof

This is the formula for the unit of the composite adjunction $H_{\mathrm{th}}\stackrel{\stackrel{{\Pi }_{\inf }}{\to }}{\underset{{\mathrm{Disc}}_{\inf }}{\leftarrow }}H\stackrel{\stackrel{\Pi }{\to }}{\underset{\mathrm{Disc}}{\leftarrow }}\mathrm{\infty }\mathrm{Grpd}$:

Proof

The canonical morphism is the composite

By the condition that $i_{!}$ is a full and faithful (∞,1)-functor the second morphism here in an equivalence, as indicated, and hence the component of the composite on $X$ being an effective epimorphism is equivalent to the component $i_{!}X\to \Pi i_{!}X$ being an effective epimorphism.

Proof

The first statement follows since $\mathrm{\infty }$-pullbacks are well defined up to quivalence.

The second two statements follow by the pasting law for (∞,1)-pullbacks: let $f:X\to Y$ and $g:Y\to Z$ be two morphisms and consider the pasting diagram

If $f$ and $g$ are formally étale then both small squares are pullback squares. Then the pasting law says that so is the outer rectangle and hence $g\circ f$ is formally étale. Similarly, if $g$ and $g\circ f$ are formally étale then the right square and the total reactangle are pullbacks, so the pasting law says that also the left square is a pullback and so also $f$ is formally étale.

For the fourth claim, let $\mathrm{Id}\simeq (g\to f\to g)$ be a retract in the arrow (∞,1)-category $H^I$. By applying the natural transformation $\varphi :i_{!}\to I_{*}$ we obtain a retract

in the category of squares $H^{\square }$. We claim that generally, if the middle piece in a retract in $H^{\square }$ is an (∞,1)-pullback square, then so is its retract sqare. This implies the fourth claim.

To see this, we use that

1. (∞,1)-limits are computed by homotopy limits in any presentable (∞,1)-category $C$ presenting $H$;

2. homotopy limits in $C$ may be computed by the left and right adjoints provided by the derivator $\mathrm{Ho}(C)$ associated to $C$.

From this the claim follows as described in detail at retract in the section retracts of diagrams .

For the last claim, consider an (∞,1)-pullback diagram

where $f$ is formally étale.

Applying the natural transformation $\varphi :i_{!}\to i_{*}$ to this yields a square of squares. Two sides of this are the pasting composite

and the other two sides are the pasting composite

Counting left to right and top to bottom, we have that

• the first square is a pullback by assumption that $i_{!}$ preserves the given pullback;

• the second square is a pullback, since $f$ is formally étale.

• the total top rectangle is therefore a pullback, by the pasting law;

• the fourth square is a pullback since $i_{*}$ is right adjoint and so also preserves pullbacks;

• also the total bottom rectangle is a pullback, since it is equal to the top total rectangle;

• therefore finally the third square is a pullback, by the other clause of the pasting law. Hence $p$ is formally étale.

Proof

By definition of infinitesimal neighbourhood we have that $i_{!}$ is a full and faithful (∞,1)-functor. It follows that $i^{*}{i}_{!}\simeq \mathrm{Id}$ and hence

Proof

By prop. 6 we have that

is an equivalence. As such it is in particular an effective epimorphism.

Proof

The first statement is implied by the fact that both $i_{!}$ as well as $i_{*}$ are full and faithful. This means that if ${\Pi }_{\inf }(X)\simeq *$ then already $i^{*}X={\Pi }_{\inf }(X)\simeq *$. Since ${\Pi }_{H_{\mathrm{th}}}\simeq {\Pi }_H{\Pi }_{\inf }$ and ${\Pi }_H$ preserves the terminal object by cohesiveness, this implies the first claim.

The second statement follows by

Proof

For $X$ this is tautological, for $\Pi (X)$ it follows from the idempotency of Red and the $(i^{*}⊣i_{*})$-zig-zag identity.

Proof

Since both $i^{*}$ and $i_{*}$ are right adjoint, the infinitesimal path ∞-groupoid functor commutes with delooping. Therefore

Proof

This follows with using the above observation from the full and faithfulness of $i_{!}$ and $i_{*}$.

The former implies that with ${\Pi }_{\inf }(X)\simeq *$ already $i^{*}X={\Pi }_{\inf }X=*$. Since ${\Pi }_{H_{\mathrm{th}}}\simeq {\Pi }_H{\Pi }_{\inf }$ and since ${\Pi }_H$ preserves the point by cohesiveness, this implies the first claim.

For the latter we compute