Proof

Apply $\mathbf{\Pi}$ to the defining Pullback square, the result is a pullback square since $\mathbf{\Pi}Y$ is discrete. $\mathbf{\Pi}$ is idempotent. Hence by the Pullback pasting lemma $\mathbf{\Pi}C_\mathbf{\Pi} f\simeq C_\mathbf{\Pi} f$.

Proof

The factorization is given by the naturality of $\mathbf{\Pi}$ and the universal property of the $(\infty,1)$-pullback in def. \ref{Pi Closure?}.

Then by prop. \ref{Pi Preserves Pullbacks Over Discretes?} the functor $\mathbf{\Pi}$ preserves the $(\infty,1)$-pullback over the discrete object $\mathbf{\Pi}Y$ and since $\mathbf{\Pi}(X \to \mathbf{\Pi}X)$ is an equivalence, it follows that $\mathbf{\Pi}(X \to c_{\mathbf{\Pi}f})$ is an equivalence.

Proof

This follows by the general reasoning discussed at reflective factorization system:

By prop. \ref{Factorization Pi Equivalence Pi Closed?} we have the required factorization. It remains to check the orthogonality.

So let

be a square diagram in $\mathbf{H}$ where the left morphism is a $\mathbf{\Pi}$-equivalence and the right morphism is $\mathbf{\Pi}$-closed. Then by assumption there is a pullback square on the right in

By naturality of the adjunction unit, the total rectangle is equivalent to

Here by assumption the middle morphism is an equivalence. Therefore there is an essentially unique lift in the square on the right and hence a lift in the total square. Again by the universality of the adjunction, any such lift factors through $\mathbf{\Pi} B$ and hence also this lift is essentially unique.

Finally by universality of the pullback, this induces an essentially unique lift $\sigma$ in

Proof

By the def. \ref{Pi Closure?} and the pasting law we have that $y^* X$ is equivalently the $\infty$-pullback in

Since the terminal object is discrete, and since the right adjoint $Disc$ preserves $\infty$-pullbacks, this exhibits $y^* X$ as the image under $Disc$ of an $\infty$-pullback of $\infty$-groupoids.

Proof

This is again established given by the fact that$\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that

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 $i_! X \to \mathbf{\Pi} i_! X$ being an effective epimorphism.

Proof

This is again given by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that

Proof

The first statement follows since $\infty$-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 $Id \simeq (g \to f \to g)$ be a retract in the arrow (∞,1)-category $\mathbf{H}^I$. By applying the natural transformation $\phi : i_! \to I_*$ we obtain a retract

in the category of squares $\mathbf{H}^{\Box}$. We claim that generally, if the middle piece in a retract in $\mathbf{H}^\Box$ 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 $\mathbf{H}$;

2. homotopy limits in $C$ may be computed by the left and right adjoints provided by the derivator $Ho(C)$ 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 $\phi : i_! \to i_*$ 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 $i^* i_! \simeq Id$ and hence

Proof

By prop. \ref{Red Is Idempotent?} we have that

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

Proof

By the general discussion at reflective factorization system, the reflection is given by sending a morphism $f \colon Y \to X$ to $X \times_{\mathbf{\Pi}_{inf}(X)} \mathbf{\Pi}_{inf}(Y) \to Y$ and the reflection unit is the left horizontal morphism in

Therefore $(\mathbf{H}_{th})_{/X}^{et}$, being a reflective subcategory of a locally presentable (∞,1)-category, is (as discussed there) itself locally presentable. Hence by the adjoint (∞,1)-functor theorem it is now sufficient to show that the inclusion preserves all small (∞,1)-colimits in order to conclude that it also has a right adjoint (∞,1)-functor.

So consider any diagram (∞,1)-functor $I \to (\mathbf{H}_{th})_{/X}^{et}$ out of a small (∞,1)-category. Since the inclusion of $(\mathbf{H}_{th})_{/X}^{fet}$ is full, it is sufficient to show that the $(\infty,1)$-colimit over this diagram taken in $(\mathbf{H}_{th})_{/X}$ lands again in $(\mathbf{H}_{th})_{/X}^{et}$ in order to have that $(\infty,1)$-colimits are preserved by the inclusion. Moreover, colimits in a slice of $\mathbf{H}_{th}$ are computed in $\mathbf{H}_{th}$ itself (this is discussed at slice category - Colimits).

Therefore we are reduced to showing that the square

is an (∞,1)-pullback square. But since $\mathbf{\Pi}_{inf}$ is a left adjoint it commutes with the $(\infty,1)$-colimit on objects and hence this diagram is equivalent to

This diagram is now indeed an (∞,1)-pullback by the fact that we have universal colimits in the (∞,1)-topos $\mathbf{H}_{th}$, hence that on the left the component $Y_i$ for each $i \in I$ is the (∞,1)-pullback of $\mathbf{\Pi}_{inf}(Y_i) \to \mathbf{\Pi}_{inf}(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.