A $\mathbf{\Pi}$-factorization system is a reflective factorization system?. The discussion given here is a terminological variant thereof.
We discuss orthogonal factorization systems in a cohesive $(\infty,1)$-topos that characterize or are characterized by the reflective subcategory of dicrete objects, with reflector $\mathbf{\Pi} : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{Disc}{\hookrightarrow} \mathbf{H}$.
Let $f : X \to Y$ be a morphism in $\mathbf{H}$, write $c_{\mathbf{\Pi}} f \to Y$ for the (∞,1)-pullback in
where the bottom morphism is the $(\Pi \dashv Disc)$-unit.
(1) We say that $c_{\mathbf{\Pi}} f$ (respectively $c_{\mathbf{\Pi}} f\to Y$) is the $\mathbf{\Pi}$-closure of $f$.
(2) $f$ is called $\mathbf{\Pi}$-closed if $X \simeq c_{\mathbf{\Pi}} f$.
(3) $f$ is called a $\mathbf{\Pi}$-equivalence if $\mathbf{\Pi} f$ is an equivalence.
The $\mathbf{\Pi}$-closure of $f$ is $\mathbf{\Pi}$-closed.
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 $\mathbf{\Pi}C_\mathbf{\Pi} f\simeq C_\mathbf{\Pi} f$.
If $\mathbf{H}$ has an ∞-cohesive site of definition, then every morphism $f : X \to Y$ in $\mathbf{H}$ factors as
such that $X \to c_{\mathbf{\Pi}} f$ is a $\mathbf{\Pi}$-equivalence and $C_\mathbf{\Pi}\to Y$ is $\mathbf{\Pi}$-closed.
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.
The pair of classes
is an orthogonal factorization system in $\mathbf{H}$.
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
For $f : X \to Y$ a $\mathbf{\Pi}$-closed morphism and $y : * \to Y$ a global element, the homotopy fiber $X_y := y^* X$ is a discrete object.
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.
There is also an internal formulation of cohesion. The following is a translation of the previous section in this language:
(1) The following statements are equivalent:
$(E,M)$ is a reflective factorization system in $H$.
There is a reflective subcategory $C\hookrightarrow H$ with reflector $\sharp$, $E$ is the class of morphisms whose $\sharp$-image is invertible in $C$, and $C=M/1$.
$(E,M)$ is a factorization system and $E$ satisfies 2-out -of-3.
$(E,M)$ is a factorization system and $M$ is the class of fibrant morphisms $P\to A$ which as dependent types $x:A\dashv P(x): Type$ satisfy $forall\, x \,in Rsc(P(x))$.
For every $H$-morphism $f:A\to B$ satisfying: $\sharp A$ and $\sharp B$ are contractible, also for all $b$ we have $\sharp \, hFiber(f,b)$ is contractible.
(2) The following statements are equivalent:
$(E,M)$ is a factorization system in $H$.
The class $(E,M)^\times:=\{M/x|x\in H,\M/x\hookrightarrow H/x\,is.refl,\,refl.fact\}$ is pullback-stable where $refl.fact$ means that each reflection is defined by $(E,M)$-factorization.
$(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, and for every $x$ the class of objects of $C.x$ is closed under composition.
The class of types $B$ satisfying $in Rsc (B)$ is closed under dependent sums.
(3) The following statements are equivalent:
$(C.x\subseteq H/x)_{x\in H}$ is a pullback-stable system of reflective subcategories of slices of $H$, for every $x$ the class of objects of $C.x$ is closed under composition, and all reflectors commute with pullbacks.
The (by (2)) to $(C.x\subseteq H/x)_{x\in H}$ corresponding factorization system $(E,M)$ is pullback stable.
A morphism $f \colon X \to Y$ in $\mathbf{H}_{th}$ is called $\mathbf{\Pi}_{inf}$-closed if its $\mathbf{\Pi}_{inf}$-unit naturality square
is an (∞,1)-pullback.
A morphism $f$ in $\mathbf{H}$ satisfies
is a pullback, iff $i_! f$ is $\mathbf{\Pi}_inf$-closed.
Such an $f$ is also called formally étale morphism.
This is established by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that
We say an object $X \in \mathbf{H}_{th}$ is formally smooth if the constant infinitesimal path inclusion, $X \to \mathbf{\Pi}_{inf}(X)$, def. \ref{Infinitesimal Paths And Reduction?}, is an effective epimorphism.
An object $X \in \mathbf{H}_{th}$ is formally smooth according to def. \ref{Formal Smoothness?} precisely if the canonical morphism
(induced from the adjoint quadruple $(i_! \dashv i^* \dashv i_* \dashv i^!)$, see there) is an effective epimorphism.
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.
For $f : X \to Y$ a morphism in $\mathbf{H}$, we say that
$f$ is a formally smooth morphism if the canonical morphism
is an effective epimorphism.
$f$ is a formally unramified morphism if this is a (-1)-truncated morphism. More generally, $f$ is an order-$k$ formally unramified morphisms for $(-2) \leq k \leq \infty$ if this is a k-truncated morphism.
$f$ is a formally étale morphism if this morphism is an equivalence, hence if
is an (∞,1)-pullback square.
An order-(-2) formally unramified morphism is equivalently a formally étale morphism.
Only for 0-truncated $X$ does formal smoothness together with formal unramifiedness imply formal étaleness.
Even more generally we can formulate formal smoothness in $\mathbf{H}_{th}$:
(—)
(—)
A morphism $f$ in $\mathbf{H}$ satisfies
is a pullback, iff $i_! f$ is $\mathbf{\Pi}_inf$-closed.
Such an $f$ is also called formally étale morphism.
This is again given by the fact that $\mathbf{\Pi}_{inf} = i_* i^*$ by definition and that $i_!$ is fully faithful, so that
The collection of formally étale morphisms in $\mathbf{H}$, def. \ref{Formal Relative Smoothness By Canonical Morphism?}, is closed under the following operations.
Every equivalence is formally étale.
The composite of two formally étale morphisms is itself formally étale.
If
is a diagram such that $g$ and $h$ are formally étale, then also $f$ is formally étale.
Any retract of a formally étale morphisms is itself formally étale.
The (∞,1)-pullback of a formally étale morphisms is formally étale if the pullback is preserved by $i_!$.
The statements about closure under composition and pullback appears as(KontsevichRosenberg, prop. 5.4, prop. 5.6). Notice that the extra assumption that $i_!$ preserves the pullback is implicit in their setup, by remark \ref{Relation To RK?}.
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)-limits are computed by homotopy limits in any presentable (∞,1)-category $C$ presenting $\mathbf{H}$;
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.
The properties listed in prop. \ref{Properties Of Formally Etale Morphisms?} correspond to the axioms on the open maps (“admissible maps”) in a geometry (for structured (∞,1)-toposes) (Lurie, def. 1.2.1). This means that a notion of formally étale morphisms induces a notion of locally algebra-ed (∞,1)toposes/structured (∞,1)-toposes in a cohesive context. This is discuss in
In order to interpret the notion of formal smoothness, we turn now to the discussion of infinitesimal reduction.
The operation $\mathbf{Red}$ is an idempotent projection of $\mathbf{H}_{th}$ onto the image of $\mathbf{H}$
Accordingly also
and
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
For every $X \in \mathbf{H}_{th}$, we have that $\mathbf{\Pi}_{inf}(X)$ is formally smooth according to def. \ref{Formal Smoothness?}.
By prop. \ref{Red Is Idempotent?} we have that
is an equivalence. As such it is in particular an effective epimorphism.
For $X \in \mathbf{H}_{th}$ any object, write
for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on those maps into $X$ which are formally étale, def. \ref{Formally Etale In HTh?}.
The inclusion of def. \ref{Etale Slice?} is both reflective as well as coreflective, hence it fits into an adjoint triple of the form
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.
Using this étalification, we can now turn de Rham coefficient objects into genuine cotangent bundle.
Let $G \in Grp(\mathbf{H}_{th})$ be an ∞-group and let write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object.
For $X \in \mathbf{H}_{th}$ any object, consider the projection $X \times \flat_{dR}\mathbf{B}G \to X$ as an object in the slice (∞,1)-topos $(\mathbf{H}_{th})_{/X}$. Then write
for its étalifiation, the coreflection by prop. \ref{Etalification Is Coreflection?}. The sections of this object we call the flat sections of the $G$-valued cotangent bundle of $X$.
For $U \in \mathbf{H}_{th}$ a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formall étale morphsim $U \to X$ is like an open map/open embedding. Regarded as an object in $(\mathbf{H}_{th})_{/X}^{fet}$ we may consider the sections over $U$ of the cotangent bundle as defined above, which in $\mathbf{H}_{th}$ are diagrams
By the fact that $Et(-)$ is right adjoint, such diagrams are in bijection to diagrams
where we are now simply including on the left the formally étale map $(U \to X)$ along $(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$.
In other words, the (flat) sections of the $G$-valued cotangent bundle $Et(X \times \flat_{dR}\mathbf{B}G) \to X$ are just the sections of $X \times \flat_{dR}\mathbf{B}G \to X$ itself, only that the domain of the section is constrained to be a formally é patch of $X$.
But then by the very nature of $\flat_{dR}\mathbf{B}G$ it follows that the flat sections of the $G$-valued cotangent bundle of $X$ are indeed nothing but the flat $G$-valued differential forms on $X$.
Let $G$ be an ∞-group and write $\mathbf{d} \colon G \to \flat_{dR}\mathbf{B}G$ for its Maurer-Cartan form.
For $S \colon X \to G$ a morphism in $\mathbf{H}_{th}$, hence $G$-valued function, its derivative is the composite
Since the identity on $X$ is formally étale, This we may regard as a section of the $G$-valued cotangent bundle, def. \ref{Cotangent Bundle?},
The critical locus $\{x \in X | \mathbf{d}S = 0\}$ of $S$ is the homotopy fiber of this section in $(\mathbf{H}_{th})_{/X}^{fet}$, hence the $(\infty,1)$-pullback
See at derived critical locus? for more discussion of this.
Last revised on December 2, 2012 at 18:27:35. See the history of this page for a list of all contributions to it.