nLab differential cohesive (infinity,1)-topos

This is a subsection of the entry cohesive (∞,1)-topos. See there for background and context.

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Idea

A cohesive (∞,1)-topos is a context of ∞-groupoids that are equipped with a geometric notion of cohesion on their collections of objects and k-morphisms, for instance topological cohesion or smooth cohesion.

While the axioms of cohesion do imply the intrinsic existence of exponentiated infinitesimal spaces, they do not admit access to an explicit synthetic notion of infinitesimal extension.

Here we consider one extra axiom on a cohesive (∞,1)-topos that does imply a good intrinsic notion of synthetic differential extension, compatible with the given notion of cohesion. We speak of differential cohesion.

In a cohesive $(\infty,1)$-topos with differential cohesion there are for instance good intrinsic notions of formal smoothness and of de Rham spaces of objects.

Differential cohesion

We discuss extra structure on a cohesive (∞,1)-topos that encodes a refinement of the corresponding notion of cohesion to infinitesimal cohesion . More precisely, we consider inclusions $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ of cohesive

$(\infty,1)$-toposes that exhibit the objects of $\mathbf{H}_{th}$ as infinitesimal cohesive neighbourhoods of objects in $\mathbf{H}$.

Definition

Definition

Given a cohesive $(\infty,1)$-topos $\mathbf{H}$ we say that an infinitesimal cohesive neighbourhood of $\mathbf{H}$ is another cohesive $(\infty,1)$-topos $\mathbf{H}_{th}$ equipped with an adjoint quadruple of adjoint (∞,1)-functors of the form

$$(i_! \dashv i^* \dashv i_* \dashv i^!) : \mathbf{H} \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\hookrightarrow}}{\underset{i^!}{\leftarrow}}}} \mathbf{H}_{th}$$

where $i_!$ is a full and faithful and preserves finite products.

Conversely we will say that data as in def. equips the cohesive $\infty$-topos $\mathbf{H}$ with differential cohesion.

Remark

This definition is an abstraction of similar situations considered in (SimpsonTeleman) and in Kontsevich-Rosenberg. See also the section Infinitesimal thickening at Q-category.

Remark

This implies that also $i_*$ is a full and faithful (∞,1)-functor.

Proof

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

Remark

This definition captures the characterization of an infinitesimal object as having a single global point surrounded by an infinitesimal neighbourhood: as we shall see in more detail below, the (∞,1)-functor $i^*$ may be thought of as contracting away any infinitesimal extension of an object. Thus $X$ being an infinitesimal object amounts to $i^* X \simeq *$, and the (∞,1)-adjunction $(i_! \dashv i^*)$ then indeed guarantees that $X$ has only a single global point, since

$$\begin{aligned} \mathbf{H}_{th}(*, X) & \simeq \mathbf{H}_{th}(i_! *, X) \\ & \simeq \mathbf{H}(*, i^* X) \\ & \simeq \mathbf{H}(*, *) \\ & \simeq * \end{aligned} \,.$$

Proposition

The inclusion into the infinitesimal neighbourhood is necessarily a morphism of (∞,1)-toposes over ∞Grpd.

$$\array{ \mathbf{H} && \stackrel{(i^* \dashv i_*)}{\to} && \mathbf{H}_{th} \\ & {}_{\mathllap{\Gamma}}\searrow && \swarrow_{\mathrlap{\Gamma}} \\ && \infty Grpd }$$

as is the induced geometric morphism $(i_* \dashv i^!) : \mathbf{H}_{th} \to \mathbf{H}$

$$\array{ \mathbf{H}_{th} && \stackrel{(i_* \dashv i^!)}{\to} && \mathbf{H} \\ & {}_{\mathllap{\Gamma}}\searrow && \swarrow_{\mathrlap{\Gamma}} \\ && \infty Grpd } \,.$$

Moreover $i_*$ is necessarily a full and faithful (∞,1)-functor.

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

$$\begin{aligned} \Gamma_{\mathbf{H}_{th}}( i_* X ) & \simeq \mathbf{H}_{th}(*, i_* X) \\ & \simeq \mathbf{H}(i^* *, X) \\ & \simeq \mathbf{H}(*, X) \\ & \simeq \Gamma_{\mathbf{H}}(X) \end{aligned} \,.$$

Analogously in the second case.

We shall write

$$(\Pi_{inf} \dashv Disc_{inf} \dashv \Gamma_{inf}) := (i^* \dashv i_* \dashv i^!)$$

so that the global section geometric moprhism of $\mathbf{H}_{th}$ factors as

$$(\Pi_{\mathbf{H}_{th}} \dashv Disc_{\mathbf{H}_{th}} \dashv \Gamma_{\mathbf{H}_{th}}) : \mathbf{H}_{th} \stackrel{\overset{\Pi_{inf}}{\longrightarrow}}{\stackrel{\overset{Disc_{inf}}{\leftarrow}}{\underset{\Gamma_{inf}}{\longrightarrow}}} \mathbf{H} \stackrel{\overset{\Pi_{\mathbf{H}}}{\longrightarrow}}{\stackrel{\overset{Disc_{\mathbf{H}}}{\longleftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\longrightarrow}}} \infty Grpd \,.$$

We also consider the (∞,1)-monads/comonads induced from these reflections:

1. the reduction modality $\Re \coloneqq i_! i^\ast$ ;

2. the infinitesimal shape modality $\Im \coloneqq i_\ast i^\ast$;

3. the infinitesimal flat modality ${\&} \coloneqq i_* i^!$.

The above says that these interact with the modalities of the ambient cohesion, i.e.

1. the shape modality $ʃ$;

2. the flat modality $\flat$;

3. the sharp modality $\sharp$

as follows:

$$\array{ && && \Re \\ && && \bot \\ && ʃ & \subset & \Im \\ && \bot && \bot \\ \emptyset &\subset& \flat & \subset & {\&} \\ \bot & & \bot && \\ \ast & \subset& \sharp }$$

Here the inclusion sign $\subset$ is to mean that the modal types of the modality on the left are included in the modal types of the modality on the right.

For more details on this, see at geometry of physics – categories and toposes the section Elastic toposes.

Let for the remainder of this section an infinitesimal neighbourhood $\mathbf{H} \hookrightarrow \mathbf{H}_{th}$ be fixed.

Remark

More generally we may ask for a sequence of differential inclusions of $\infty$-toposes as above, reflecting ever higher orders of infinitesimals, hence notably a progression

$$ʃ \lt \Im = \Im_{(\infty)} \lt \cdots \lt \Im_{(3)} \lt \Im_{(2)} \lt \Im_{(1)} \lt id$$

of infinitesimal shape modalities of various order, yielding a further factorization of the shape unit as

$$X \to \Im_{(1)}X \to \Im_{(2)}X \to \Im_{(3)} X \to \cdots \to \Im X \to ʃ X \,.$$

Properties

From $\infty$-sheaves over infinitesimal neighbourhood sites

We give a presentation of classes of infinitesimal neighbourhoods by simplicial presheaves on suitable sites.

Definition

Let $C$ be an ∞-cohesive site. We say a site $C_{th}$

• equipped with a coreflective embedding

  $$(i \dashv p) : C \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} C_{th}$$

• such that

  • $i$ preserves finite products

  • $i$ preserves pullbacks along morphisms in covering families;

  • both $i$ and $p$ send covering families to covering families;

  • for all $\mathbf{U}$ in $C_{th}$ and covering families $\{U_i \to p(\mathbf{U})\}$ there is a lift through $p$ to a covering family $\{\mathbf{U}_i \to \mathbf{U}\}$

is an infinitesimal neighbourhood site of $C$.

Proposition

Let $C$ be an ∞-cohesive site and $(i \dashv p) : C \stackrel{\overset{i}{\hookrightarrow}}{\underset{p}{\leftarrow}} C_{th}$ an infinitesimal neighbourhood site.

Then the (∞,1)-category of (∞,1)-sheaves on $C_{th}$ is a cohesive $(\infty,1)$-topos and the restriction $i^*$ along $i$ exhibits it as an infinitesimal neighbourhood of the cohesive $(\infty,1)$-topos over $C$.

$$( i_! \dashv i^* \dashv i_* \dashv i^! ) : Sh_{(\infty,1)}(C) \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^*}{\leftarrow}}{\stackrel{\overset{i_*}{\to}}{\stackrel{i^!}{\leftarrow}}}} Sh_{(\infty,1)}(C^{th}) \,.$$

Moreover, $i_!$ restricts on representables to the (∞,1)-Yoneda embedding factoring through $i$:

$$\array{ C &\hookrightarrow& Sh_{(\infty,1)}(C) \\ \downarrow^{\mathrlap{i}} && \downarrow^{\mathrlap{i_!}} \\ C_{th} &\hookrightarrow& Sh_{(\infty,1)}(C_{th}) } \,.$$

Proof

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

$$Sh_{(\infty,1)}(C_{th}) \simeq ([C_{th}^{op}, sSet]_{loc})^\circ$$

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

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

$$\begin{aligned} \mathrm{Ran}_{i} F : \mathbf{K} & \mapsto \int_{U \in C} \mathrm{sSet}( C_{\mathrm{th}}(i(U), \mathbf{K}) , F(U)) \\ & \simeq \int_{U \in C} \mathrm{sSet}( C(U, p(\mathbf{K})) , F(U)) \\ & \simeq F(p(\mathbf{K})) \\ & =: (p^* F)(\mathbf{K}) \end{aligned} \,,$$

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

$$[C^{op}, sSet] \stackrel{\overset{Lan_i}{\longrightarrow}}{\stackrel{\overset{(-)\circ i}{\longleftarrow}}{\stackrel{\overset{(-)\circ p}{\longrightarrow}}{\underset{Ran_p}{\longleftarrow}}}} [C_{th}^{op}, sSet] \,.$$

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

$$(Lan_i \dashv (-) \circ i) : [C^{op}, sSet]_{proj} \stackrel{\overset{Lan_i}{\to}}{\underset{(-)\circ i}{\leftarrow}} [C_{th}^{op}, sSet]_{proj} \,;$$

$$((-)\circ i \dashv (-) \circ p) : [C^{op}, sSet]_{proj} \stackrel{\overset{(-)\circ i}{\longleftarrow}} {\underset{(-)\circ p}{\longrightarrow}} [C_{th}^{op}, sSet]_{proj} \,;$$

$$((-) \circ p \dashv Ran_p) : [C^{op}, sSet]_{inj} \stackrel{\overset{(-)\circ p}{\longrightarrow}}{\underset{Ran_p}{\longleftarrow}} [C_{th}^{op}, sSet]_{inj} \,.$$

Observe that $Lan_i$, being a left Kan extension, sends representables to representables: we have

$$Lan_i C(-,T) : \mathbf{K} \mapsto \int^{U \in C} C_{th}(\mathbf{K}, i(U)) \cdot C(U,T)$$

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

$$\cdots \simeq C_{th}(\mathbf{K}, i(T)) \,.$$

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 $Lan_i$ 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

$$\begin{aligned} Lan_i C(\{U_i \to U\}) & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} Lan_i (j(U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k)) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} j i (U_{i_0} \times_U U_{i_1} \times_U \cdots \times_U U_k) \\ & \simeq \int^{[k] \in \Delta} \Delta[k] \cdot \coprod_{i_0, \cdots, i_k} j (i(U_{i_0}) \times_{i(U)} i(U_{i_1}) \times_{i(U)} \cdots \times_{i(U)} i(U_k)) \end{aligned} \,.$$

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

$$(i^* F)(U) = F(i(U)) \stackrel{}{\to} [C_{th}^{op}, sSet](C(\{i(U_i)\}), F) = [C^{op}, sSet](C(\{U_i\}), i^* F)$$

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 $\{\mathbf{U}_i \stackrel{p_i}{\to} \mathbf{U}\}$ a covering family in $C_{th}$ we have that its image under $(-) \circ i$ is its image under $p$, by the Yoneda lemma:

$$\begin{aligned} [C^{op}, sSet](K, ((-)\circ i) (\mathbf{U})) & \simeq C_{th}(i(K), \mathbf{U}) \\ & \simeq C(K, p(\mathbf{U})) \end{aligned}$$

and using that $p$ preserves covers by assumption.

Therefore $(-) \circ i$ is a left and right local Quillen functor with left local Quillen adjoint $Lan_i$ and right local Quillen adjoint $(-)\circ p$.

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

Finally to see that also $Ran_p$ 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 $\mathbf{K} \to p^* U$ in $C_{th}$ we may find a covering $\{\mathbf{K}_j \to \mathbf{K}\}$ of $\mathbf{K}$ such that we find commuting diagrams on the left of

$$\array{ \mathbf{K}_j &\to& p^* U_{i(j)} \\ \downarrow && \downarrow \\ \mathbf{K} &\to& p^* U } \;\;\; \leftrightarrow \;\;\; \array{ p(\mathbf{K}_j) & =& i^*(\mathbf{K}_j) &\to& U_{i(j)} \\ \downarrow && \downarrow && \downarrow \\ p(\mathbf{K}) &= & i^*(\mathbf{K}) &\to& U } \,,$$

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 $C_{th}$.

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 properties discussed there) along a full and faithful functor $i$ satisfies $Lan_i \circ i \simeq id$. It remains to observe that since $(-)\circ i$ is not only right but also left Quillen by the above, we have that $i^* Lan_i$ applied to a cofibrant object is already the derived functor of the composite.

Remark

Conversely this implies that $Sh_{(\infty,1)}(C_{th})$ is an ∞-connected (∞,1)-topos over Smooth∞Grpd, exhibited by the triple of adjunctions

$$(i^* \dashv i_* \dashv i^!) : SynthDiff \infty Grpd \to Smooth \infty Grpd \,.$$

Relation to infinitesimal cohesion

We discuss how differential cohesion in the sense of def. relates to infinitesimal cohesion.

under construction

Definition

Given differential cohesion, def. ,

$$\array{ \Re &\dashv& \Im &\dashv& {\&} \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp }$$

define operations $ʃ^{rel}$ and $\flat^{rel}$ by

$$ʃ^{rel} X \coloneqq (ʃ X) \underset{\Re X}{\coprod} X$$

$$\flat^{rel} X \coloneqq (\flat X) \underset{\Im}{\times} X \,.$$

Hence $ʃ^{rel} X$ makes a homotopy pushout square

$$\array{ \Re X &\longrightarrow& X \\ \downarrow && \downarrow \\ ʃ X &\longrightarrow& ʃ^{rel} X }$$

and $\flat^{rel}$ makes a homotopy pullback square

$$\array{ \flat^{rel} X &\longrightarrow& X \\ \downarrow && \downarrow \\ \flat X &\longrightarrow& \Im X } \,.$$

We call $ʃ^{rel}$ the relative shape modality and $\flat^{rel}$ the relative flat modality.

Proposition

The relative shape and flat modalities of def.

1. form an adjoint pair $(ʃ^{rel} \dashv \flat^{rel})$;

2. whose (co-)modal types are precisely the properly infinitesimal types, hence those for which $\flat \to \Im$ is an equivalence;

3. $ʃ^{rel}$ preserves the terminal object.

It follows that when $\flat^{rel}$ has a further right adjoint $\sharp^{rel}$ with equivalent modal types containing the codiscrete types, then this defines a level

$$\array{ \flat^{rel} &\dashv& \sharp^{rel} \\ \vee && \vee \\ \flat &\dashv& \sharp \\ \vee && \vee \\ \emptyset &\dashv& \ast }$$

hence an intermediate subtopos $\infty Grpd \hookrightarrow \mathbf{H}_{infinitesimal}\hookrightarrow \mathbf{H}_{th}$ which is infinitesimally cohesive.

This happens notably for the model of formal smooth ∞-groupoids and all its variants such as formal complex analytic ∞-groupoids etc. But in this case $(\flat^{rel} \dashv \sharp^{rel})$ does not provide Aufhebung for $(\flat \dashv \sharp)$.

(…)

Proposition

The counit of the relative flat modality is a formally étale morphism.

Proof

From the fact that the infinitesimal shape modality is idempotent and preserves homotopy pullbacks.

Structures in a differential cohesive $(\infty,1)$-topos

We discuss structures that are canonically present in a cohesive $(\infty,1)$-topos equipped with differential cohesion. These structures parallel the structures in a general cohesive (∞,1)-topos.

Infinitesimal paths and de Rham spaces

In the presence of differential cohesion there is an infinitesimal analog of the geometric paths ∞-groupoids.

Infinitesimal path $\infty$-groupoid

Definition

Define the adjoint triple of adjoint (∞,1)-functors corresponding to the adjoint quadruple $(i_! \dashv i^* \dashv i_* \dashv i^!)$:

$$(\Re \dashv \Im \dashv \& ) : (i_! i^* \dashv i_* i^* \dashv i_* i^! ) : \mathbf{H}_{th} \to \mathbf{H}_{th} \,.$$

We say that

• $\Re$ is the reduction modality.

• $\Im$ is the infinitesimal shape modality.

• $\&$ is the infinitesimal flat modality.

An object in the full sub-$\infty$-category

• of $\Re$ we call a reduced object

• of $\Im$ we call a coreduced object.

For $X\in \mathbf{H}_{th}$ we say that

• $\Im(X)$ is the infinitesimal path ∞-groupoid of $X$;

  The $(i^* \dashv i_*)$-unit

  $$X \to \Im(X)$$

  we call the constant infinitesimal path inclusion.

• $\Re(X)$ is the reduced cohesive ∞-groupoid underlying $X$.

  The $(i_* \dashv i^*)$-counit

  $$\Re X \to X$$

  we call the inclusion of the reduced part of $X$.

Remark

In traditional contexts see (SimpsonTeleman, p. 7) the object $\Im(X)$ is called the de Rham space of $X$ or the de Rham stack of $X$ . Here we may tend to avoid this terminology, since by the discussion at cohesive (∞,1)-topos – de Rham cohomology we have a good notion of intrinsic de Rham cohomology in any cohesive (∞,1)-topos already without equipping it with differential cohesion. From this point of view the object $\Im(X)$ is not primarily characterized by the fact that (in some models, see below) it does co-represent de Rham cohomology – because the object $\mathbf{\Pi}_{dR}(X)$ from above does, too – but by the fact that it does so in an explicitly (synthetic) infinitesimal way.

Observation

There is a canonical natural transformation

$$\Im(X) \to \int(X)$$

that factors the finite path inclusion through the infinitesimal one

$$\array{ && \Im(X) \\ & \nearrow && \searrow \\ X &&\to&& \int(X) } \,.$$

Proof

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

$$X \stackrel{i_{inf}X}{\to} Disc_{inf}\Pi_{inf} X \stackrel{Disc_{inf}(i_{\mathbf{H}}\Pi_{inf}X)}{\to} Disc_{\mathbf{H}} Disc_{inf} \Pi_{\mathbf{H}} \Pi_{inf} X = Disc_{\mathbf{H}_{th}} \Pi_{\mathbf{H}_{th}} \,.$$

Jet $\infty$-bundles

Notice that for $f : X \to Y$ any morphism in any (∞,1)-topos $\mathbf{H}$, there is the corresponding base change geometric morphism between the over-(∞,1)-toposes

$$(f^* \dashv f_*) : \mathbf{H}/X \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathbf{H}/Y \,.$$

Definition

For any object $X \in \mathbf{H}$ write

$$Jet : \mathbf{H}/X \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\to}} \mathbf{H}/\Im(X)$$

for the base change geometric morphism induced by the constant infinitesimal path inclusion $i : X \to \Im(X)$, def. .

For $(E \to X) \in \mathbf{H}/X$ we call $Jet(E) \to \Im(X)$ as well as its pullback $i^* Jet(E) \to X$ (depending on context) the jet bundle of $E \to X$.

Formally smooth/étale/unramified morphisms

Definition

We say an object $X \in \mathbf{H}_{th}$ is formally smooth if the constant infinitesimal path inclusion, $X \to \Im(X)$, def. ,

is an effective epimorphism.

In this form this is the evident $(\infty,1)$-categorical analog of the conditions as they appear for instance in (SimpsonTeleman, page 7).

Remark

An object $X \in \mathbf{H}_{th}$ is formally smooth according to def. precisely if the canonical morphism

$$i_! X \to i_* X$$

(induced from the adjoint quadruple $(i_! \dashv i^* \dashv i_* \dashv i^!)$, see there) is an effective epimorphism.

Proof

The canonical morphism is the composite

$$(i_! \to i_*) := i_! \stackrel{\eta i_!}{\to} \Im i_! := i_* i^* i_! \stackrel{\simeq}{\to} i_* \,.$$

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.

Remark

In this form this characterization of formal smoothness is the evident generalization of the condition given in (Kontsevich-Rosenberg, section 4.1). See the section Formal smoothness at Q-category for more discussion. Notice that by this remark the notation there is related to the one used here by $u^* = i_!$, $u_* = i^*$ and $u^! = i_*$.

Therefore we have the following more general definition.

Definition

For $f : X \to Y$ a morphism in $\mathbf{H}$, we say that

1. $f$ is a formally smooth morphism if the canonical morphism

   $$i_! X \to i_! Y \prod_{i_* Y} i_* Y$$

   is an effective epimorphism.

2. $f$ 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.

3. $f$ is a formally étale morphism if this morphism is an equivalence, hence if

   $$\array{ i_! X &\stackrel{i_! f}{\to}& i_! Y \\ \downarrow && \downarrow \\ i_* X &\stackrel{i_* f}{\to}& i_* Y }$$

   is an (∞,1)-pullback square.

Remark

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 $\mathbf{H}_{th}$:

Definition

A morphism $f \colon X \to Y$ in $\mathbf{H}_{th}$ is formally étale if it is $\Im$-closed, hence if its $\Im$-unit naturality square

$$\array{ X &\to& \Im(X) \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im(f)}} \\ Y &\to& \Im(y) }$$

is an (∞,1)-pullback.

Remark

A morphism $f$ in $\mathbf{H}$ is formally etale in the sense of def. precisely if its image $i_!(f)$ in $\mathbf{H}_{th}$ is formally etale in the sense of def. .

Proof

This is again given by the fact that $\Im = i_* i^*$ by definition and that $i_!$ is fully faithful, so that

$$\array{ i_! X &\to& \Im(i_! X) \simeq i_* i^* i_! X &\stackrel{\simeq}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* i^* i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! Y &\to& \Im(i_! Y) \simeq i_* i^* i_! Y &\stackrel{\simeq}{\to}& i_* Y } \,.$$

Proposition

The collection of formally étale morphisms in $\mathbf{H}$, def. , is closed under the following operations.

1. Every equivalence is formally étale.

2. The composite of two formally étale morphisms is itself formally étale.

3. If

   $$\array{ && Y \\ & {}^{\mathllap{f}}\nearrow &\swArrow_{\simeq}& \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z }$$

   is a diagram such that $g$ and $h$ are formally étale, then also $f$ is formally étale.

4. Any retract of a formally étale morphisms is itself formally étale.

5. 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 .

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

$$\array{ i_! X &\stackrel{i_! f }{\to}& i_! Y &\stackrel{i_! g}{\to}& Z \\ \downarrow && \downarrow && \downarrow \\ i_* X &\stackrel{i_* f }{\to}& i_* Y &\stackrel{i_* g}{\to}& i_* Z } \,.$$

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

$$Id \simeq ((i_! g \to i_*g) \to (i_! f \to i_*f) \to (i_! g \to i_*g))$$

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

$$\array{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\to& Y }$$

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

$$\array{ i_! A \times_Y X &\to& i_! X &\stackrel{\phi_X}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_! f}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\to& i_! Y &\stackrel{\phi_Y}{\to}& i_* Y }$$

and the other two sides are the pasting composite

$$\array{ i_! A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& i_* A \times_Y A &\stackrel{}{\to}& i_* X \\ \downarrow^{\mathrlap{i_! p}} && \downarrow^{\mathrlap{i_* p}} && \downarrow^{\mathrlap{i_* f}} \\ i_! A &\stackrel{\phi_A}{\to}& i_* A &\to& i_* Y } \,.$$

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.

Remark

The properties listed in prop. 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

• cohesive (∞,1)-topos – structure ∞-sheaves.

In order to interpret the notion of formal smoothness, we turn now to the discussion of infinitesimal reduction.

Proposition

The operation $\Re$ is an idempotent projection of $\mathbf{H}_{th}$ onto the image of $\mathbf{H}$

$$\Re \Re \simeq \Re \,.$$

Accordingly also

$$\Im \Im \simeq \Im$$

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

$$\begin{aligned} \Re \Re & \simeq i_! i^* i_! i^* \\ & \simeq i_! i^* \\ & \simeq \Re \end{aligned} \,.$$

Observation

For every $X \in \mathbf{H}_{th}$, we have that $\Im(X)$ is formally smooth according to def. .

Proof

By prop. we have that

$$\Im(X) \to \Im \Im X$$

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

Infinitesimal $\mathbb{A}^1$-homotopy

Definition

A set of objects $\{D_\alpha \in \mathbf{H}_{th}\}_\alpha$ is said to exhibit the differential structure or

exhibit the infinitesimal thickening if the localization

$$L_{\{D_\alpha\}_\alpha} \mathbf{H}_{th} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H}_{th}$$

of $\mathbf{H}_{th}$ at the morphisms of the form $D_\alpha \times X \to X$ is exhibited by the infinitesimal shape modality $\Im$.

Remark

This is the infinitesimal analog of the notion of objects exhibiting cohesion, see at structures in cohesion – A1-homotopy and the continuum.

For more see at Lie differentiation.

Structure sheaves

We discuss how in differential cohesion $\mathbf{H}_{th}$ every object $X$ canonically induces its étale (∞,1)-topos $Sh_{\mathbf{H}_{th}}(X)$.

For $X \in \mathbf{H}_{th}$ any object in a differential cohesive $\infty$-topos, we formulate

1. the (∞,1)-topos denoted $\mathcal{X}$ or $Sh_\infty(X)$ of (∞,1)-sheaves over $X$, or rather of formally étale maps into $X$;

2. the structure (∞,1)-sheaf $\mathcal{O}_{X}$ of $X$.

The resulting structure is essentially that discussed (Lurie, Structured Spaces) if we regard $\mathbf{H}_{th}$ equipped with its formally étale morphisms, def. , as a (large) geometry for structured (∞,1)-toposes.

One way to motivate this is to consider structure sheaves of flat differential forms. To that end, let $G \in Grp(\mathbf{H}_{th})$ a differential cohesive ∞-group with de Rham coefficient object $\flat_{dR}\mathbf{B}G$ and for $X \in \mathbf{H}_{th}$ any differential homotopy type, the product projection

$$X \times \flat_{dR} \mathbf{B}G \to X$$

regarded as an object of the slice (∞,1)-topos $(\mathbf{H}_{th})_{/X}$ almost qualifies as a “bundle of flat $\mathfrak{g}$-valued differential forms” over $X$: for $U \to X$ an cover (a 1-epimorphism) regarded in $(\mathbf{H}_{th})_{/X}$, a $U$-plot of this product projection is a $U$-plot of $X$ together with a flat $\mathfrak{g}$-valued de Rham cocycle on $X$.

This is indeed what the sections of a corresponding bundle of differential forms over $X$ are supposed to look like – but only if $U \to X$ is sufficiently spread out over $X$, hence sufficiently étale. Because, on the extreme, if $X$ is the point (the terminal object), then there should be no non-trivial section of differential forms relative to $U$ over $X$, but the above product projection instead reproduces all the sections of $\flat_{dR} \mathbf{B}G$.

In order to obtain the correct cotangent-like bundle from the product with the de Rham coefficient object, it needs to be restricted to plots out of suficiently étale maps into $X$. In order to correctly test differential form data, “suitable” here should be “formally”, namely infinitesimally. Hence the restriction should be along the full inclusion

$$(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$$

of the formally étale maps of def. into $X$. Since on formally étale covers the sections should be those given by $\flat_{dR}\mathbf{B}G$, one finds that the corresponding “cotangent bundle” must be the coreflection along this inclusion. The following proposition establishes that this coreflection indeed exists.

Definition

For $X \in \mathbf{H}_{th}$ any object, write

$$(\mathbf{H}_{th})^{fet}_{/X} \hookrightarrow (\mathbf{H}_{th})_{/X}$$

for the full sub-(∞,1)-category of the slice (∞,1)-topos over $X$ on those maps into $X$ which are formally étale, def. .

We also write $FEt_{\mathbf{X}}$ or $Sh_{\mathbf{H}}(X)$ for $(\mathbf{H}_{th})_{/X}^{fet}$.

Proposition

The inclusion $\iota$ of def. is both reflective as well as coreflective, hence it fits into an adjoint triple of the form

$$(\mathbf{H}_{th})_{/X}^{fet} \stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{\iota}{\hookrightarrow}}{\underset{Et}{\leftarrow}}} (\mathbf{H}_{th})_{/X} \,.$$

Proof

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

$$\array{ Y &\to& X \times_{\Im(Y)} \Im(Y) &\to& \Im(Y) \\ & \searrow & \downarrow^{} && \downarrow^{\mathrlap{\Im(f)}} \\ && X &\to& \Im(X) } \,.$$

Therefore $(\mathbf{H}_{th})_{/X}^{fet}$, 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 $I \to (\mathbf{H}_{th})_{/X}^{fet}$ 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 $(\mathbf{H}_{th})_{/X}^{fet}$ 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

$$\array{ \underset{\to_i}{\lim} Y_i &\to& \Im \underset{\to_i}{\lim} Y_i \\ \downarrow && \downarrow \\ X &\to& \Im(X) }$$

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

$$\array{ \underset{\to_i}{\lim} Y_i &\to& \underset{\to_i}{\lim} \Im Y_i \\ \downarrow && \downarrow \\ X &\to& \Im(X) } \,.$$

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 $\Im(Y_i) \to \Im(X)$, by assumption that we are taking an $(\infty,1)$-colimit over formally étale morphisms.

Example

For the case that $X \simeq \ast$ in prop. , then the proof there shows that the étalification operation over the point is just ${\&}$:

$${\&} \simeq Et_{/\ast} \,.$$

Indeed, for any $X$ then ${\&} X \to \ast$ is a formally étale morphism since

$$\array{ {\&} X &\longrightarrow& \Im{\&} X \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \Im \ast } \;\;\; \simeq \;\;\; \array{ {\&} X &\stackrel{\simeq}{\longrightarrow}& {\&} X \\ \downarrow && \downarrow \\ \ast &\stackrel{\simeq}{\longrightarrow}& \ast }$$

is a homotopy pullback.

Proposition

The $\infty$-category $(\mathbf{H}_{th})_{/X}^{fet}$ is an (∞,1)-topos and the canonical inclusion into $(\mathbf{H}_{th})_{/X}$ is a geometric embedding.

Proof

By prop. the inclusion $(\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X}$ is reflective with reflector given by the $(\Im-equivalences , \Im-closed)$ factorization system. Since $\Im$ is a right adjoint and hence in particular preserves (∞,1)-pullbacks, the $\Im$-equivalences are stable under pullbacks. By the discussion at stable factorization system this is the case precisely if the corresponding reflector preserves finite (∞,1)-limits. Hence the embedding is a geometric embedding which exhibits a sub-(∞,1)-topos inclusion.

Definition

For $X \in \mathbf{H}_{th}$ we speak of

$$\mathcal{X} \coloneqq Sh_{\mathbf{H}_{th}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{fet}$$

also as the (petit) (∞,1)-topos of $X$ or the étale (∞,1)-topos of $X$.

Write

$$\mathcal{O}_X \colon \mathbf{H}_{th} \stackrel{(-) \times X}{\to} (\mathbf{H}_{th})_{/X} \stackrel{Et}{\to} (\mathbf{H}_{th})_{/X}^{fet} = Sh_{\mathbf{H}_{th}}(X)$$

for the composite (∞,1)-functor that sends any $A \in \mathbf{H}_{th}$ to the etalification, prop. , of the projection $A \times X \to X$.

We call $\mathcal{O}_X$ the structure sheaf of $X$.

Remark

For $X, A \in \mathbf{H}_{th}$ and for $U \to X$ a formally étale morphism in $\mathbf{H}_{th}$ (hence like an open subset of $X$), we have that

$$\begin{aligned} \mathcal{O}_{X}(A)(U) & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , \mathcal{O}_{X}(A) ) \\ & \coloneqq Sh_{\mathbf{H}_{th}}(X)( U , Et(X \times A) ) \\ & \simeq (\mathbf{H}_{th})_{/X}(U, X \times A) \\ & \simeq \mathbf{H}_{th}(U,A) \\ & \simeq A(U) \end{aligned} \,,$$

where we used the ∞-adjunction $(\iota \dashv Et)$ of prop. and the (∞,1)-Yoneda lemma.

This means that $\mathcal{O}_{X}(A)$ behaves as the sheaf of $A$-valued functions over $X$.

Proposition

The functor $\mathcal{O}_X$ of def. is indeed an $\mathbf{H}_{th}$-structure sheaf in the sense of structured (∞,1)-toposes, for $\mathbf{H}_{th}$ regarded as a (large) geometry (for structured (∞,1)-toposes) with the formally étale morphisms being the “admissible morphisms”.

This is the analog of (Lurie, Structured Spaces, prop. 2.2.11).

Proof

We need to check that $\mathcal{O}_{X}$ preserves finite (∞,1)-limits and formally étale covers (where covers here in the canonical topology on the given toposes are 1-epimorphisms). The first statement follows since $\mathcal{O}_{X}$ is right adjoint to the forgetful functor

$$Sh_{\mathbf{H}}(X) \simeq (\mathbf{H}_{th})_{/X}^{fet} \hookrightarrow (\mathbf{H}_{th})_{/X} \stackrel{\underset{X}{\sum}}{\to} \mathbf{H}_{th}$$

For the second statement, let $p \colon \widehat Y \longrightarrow Y$ be any 1-epimorphism which is also a formally étale. We need to show that also $Et(X \times p)$ is a 1-epimorphism. By the discussion at effective epimorphism in an (∞,1)-category for this it is sufficient that the 0-truncation $\tau_0 Et(X \times p)$ is an epimorphism in the underlying sheaf topos, hence that every generalized element of $Et(X \times Y)$ has a lift to $Et(X \times \widehat{Y})$ after refinement along a cover.

By the fact that $Et$ is right adjoint to the inclusion, by construction, this means that it is sufficient to show that given a diagram in $\mathbf{H}_{th}$ of the form

$$\array{ && && X \times \widehat{Y} \\ && && \downarrow \\ U && \longrightarrow && X \times Y \\ & \searrow && \swarrow \\ && X }$$

with $U \to X$ formally étale, this can be completed to a diagram of the form

$$\array{ \widehat{U} && \longrightarrow && X \times \widehat{Y} \\ \downarrow && && \downarrow \\ U && \longrightarrow && X \times Y \\ & \searrow && \swarrow \\ && X }$$

with $\widehat{U} \to U$ a formally étale 1-epimorphism. But since both 1-epimorphisms as well as formally étale morphisms are stable under (∞,1)-pullback we can take $\widehat U \coloneqq \widehat{Y} \times_Y U$.

Remark

In the case that $\mathbf{H}_{th}$ happens to have an (∞,1)-site of definition whose covers are (generated) from formally étale morphisms (a small geometry (for structured (∞,1)-toposes)), then $\mathbf{H}_{th}$ is the classifying (∞,1)-topos for structure sheaves and $\mathcal{O}_X$ in prop. may be regarded as the inverse image of the classifying geometric morphism

$$Sh_{\mathbf{H}_{th}}(X) \stackrel{\overset{\mathcal{O}_X}{\leftarrow}}{\underset{}{\longrightarrow}} \mathbf{H}_{th} \,.$$

Example

Let $G \in Grp(\mathbf{H}_{th})$ be an ∞-group and write $\flat_{dR} \mathbf{B}G \in \mathbf{H}_{th}$ for the corresponding de Rham coefficient object. Then

$$\mathcal{O}_X(\flat_{dR}\mathbf{B}G) \in Sh_{\mathbf{H}}(X)$$

we may call the $G$-valued flat cotangent sheaf of $X$.

Remark

For $U \in \mathbf{H}_{th}$ a test object (say an object in a (∞,1)-site of definition, under the Yoneda embedding) a formally étale morphism $U \to X$ 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

$$\array{ U &&\to&& \mathcal{O}_X(\flat_{dR}\mathbf{B}G) \\ & \searrow && \swarrow \\ && X } \,.$$

By the fact that $Et(-)$ is right adjoint, such diagrams are in bijection to diagrams

$$\array{ U &&\to&& X \times \flat_{dR} \mathbf{B}G \\ & \searrow && \swarrow \\ && X }$$

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 sections of the $G$-valued flat cotangent sheaf $\mathcal{O}_X(\flat_{dR}\mathbf{B}G)$ 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 étale 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 $X$.

Proposition

For $X \in \mathbf{H}_{th}$ an object in a differentially cohesive $\infty$-topos, then its petit structured $\infty$-topos $Sh_{\mathbf{H}_{th}}(X)$, according to def. , is locally ∞-connected.

Proof

We need to check that the composite

$$\infty Grpd \stackrel{Disc}{\longrightarrow} \mathbf{H}_{th} \stackrel{(-) \times X}{\longrightarrow} (\mathbf{H}_{th})_{/X} \stackrel{L}{\longrightarrow} Sh_{\mathbf{H}}(X)$$

preserves (∞,1)-limits, so that it has a further left adjoint. Here $L$ is the reflector from prop. . Inspection shows that this composite sends an object $A \in \infty Grpd$ to $\Im(Disc(A)) \times X \to X$:

$$\array{ \Im(Disc(A)) \times X &\longrightarrow& \Im(Disc(A) \times X) & \simeq \Im(Disc(A)) \times \Im(X) \\ \downarrow &{}^{(pb)}& \downarrow \\ X &\longrightarrow& \Im(X) } \,.$$

By the discussion at slice (∞,1)-category – Limits and colimits an (∞,1)-limit in the slice $(\mathbf{H}_{th})_{/X}$ is computed as an (∞,1)-limit in $\mathbf{H}$ of the diagram with the slice cocone adjoined. By right adjointness of the inclusion $Sh_{\mathbf{H}}(X) \hookrightarrow (\mathbf{H}_{th})_{/X}$ the same is then true for $Sh_{\mathbf{H}}(X) \coloneqq (\mathbf{H}_{th})_{/X}^{et}$.

Now for $A \colon J \to \infty Grpd$ a diagram, it is taken to the diagram $j \mapsto \Im(Disc(A_j)) \times X \to X$ in $Sh_{\mathbf{H}}(X)$ and so its $\infty$-limit is computed in $\mathbf{H}$ over the diagram locally of the form

$$\array{ X \times \Im(Disc(A_{j})) &&\longrightarrow&& X \times \Im(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X } \simeq \array{ X \times \Im(Disc(A_{j})) &&\longrightarrow&& X \times \Im(Disc(A_{j'})) \\ & \searrow && \swarrow \\ && X \times \ast } \,.$$

Since $\infty$-limits commute with each other this limit is the product of

1. $\underset{\leftarrow}{\lim}_j \Im(Disc(A_j))$

2. $\underset{\leftarrow}{\lim}_{J \star \Delta^0} X$ (over the co-coned diagram constant on $X$).

For the first of these, since the infinitesimal shape modality $\Im$ is in particular a right adjoint (with left adjoint the reduction modality), and since $Disc$ is also right adjoint by cohesion, we have a natural equivalence

$$\underset{\leftarrow}{\lim}_j \Im(Disc(A_j)) \simeq \Im(Disc(\underset{\leftarrow}{\lim}_j(A_j))) \,.$$

For the second, the $\infty$-limit over an $\infty$-category $J \star \Delta^0$ of a functor constant on $X$ is

$$\begin{aligned} \underset{\leftarrow}{\lim}_{J \star \Delta^0} X & \simeq \underset{\leftarrow}{\lim}_{J \star \Delta^0} [\ast, X] \\ & \simeq [\underset{\rightarrow}{\lim}_{J \star \Delta^0} \ast, X] \\ & \simeq [{\vert {J \star \Delta^0}\vert}, X] \\ & \simeq [\ast, X] \simeq X \end{aligned} \,,$$

where the last line follows since ${J \star \Delta^0}$ has a terminal object and hence contractible geometric realization.

In conclusion this shows that $\infty$-limits are preserved by $L \circ (-)\times X\circ Disc$.

Proposition

Let $f \;\colon\; Y \longrightarrow X$ be a formally étale morphism in a differentially cohesive $\infty$-topos $\mathbf{H}$. Then pullback $f^\ast$ is the inverse image of an étale morphism of structured (∞,1)-toposes between the corresponding étale toposes, def. , hence there is an étale geometric morphism

$$Sh_{\mathbf{H}}(Y) \simeq Sh_{\mathbf{H}}(X)/_{f} \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}} Sh_{\mathbf{H}}(X)$$

and an equivalence of structure sheaves

$$\mathcal{O}_Y \simeq f^\ast \mathcal{O}_X \,.$$

Proof

Since the inclusion of the point into the interval is an op-final (∞,1)-functor we have (by this proposition) an equivalence of over-(∞,1)-categories

$$\mathbf{H}/_{Y} \simeq \mathbf{H}/_{f} \simeq (\mathbf{H}/_{X})/_f \,.$$

Since $f$ is formally étale by assumption and since formally étale morphisms are closed under composition, this restricts to an equivalence $Sh_{\mathbf{H}}(Y) \simeq (Sh_{\mathbf{H}}(X))/_f$.

For the equivalence of structure sheaves it is sufficient to show for each coefficient $A \in \mathbf{H}_{th}$ an equivalence

$$\mathcal{O}_Y(A) \simeq (f^\ast \mathcal{O}_X(A))$$

in $Sh_{\mathbf{H}}(Y)$. But by definition () $\mathcal{O}_Y(A) \coloneqq Et(A \times Y)$ and similarly for $\mathcal{O}_X$ and since $Et$ is right adjoint to the inclusion $Sh_{\mathbf{H}}(Y) \hookrightarrow \mathbf{H}_{Y}$ we have

$$f^\ast \mathcal{O}_X(A) = f^\ast Et(A \times X) \simeq Et(f^\ast(A \times X)) \simeq Et(A \times Y) = \mathcal{O}_Y(A) \,.$$

Sheaves of (quasi-coherent) modules

We discuss the abstract formulation of sheaves of modules and of quasicoherent sheaves on petit $\infty$-toposes in differential cohesion.

under construction – check

Definition

For $X \in \mathbf{H}_{th}$ an object and $(Sh_{\mathbf{H}}(X), \mathcal{O}_X)$ its $\mathbf{H}_{th}$-structured (infinity,1)-topos, according to def. , consider the composite functor

$$\mathcal{O}_X^{\mathbf{H}} \;\colon\; \mathbf{H} \stackrel{i_!}{\hookrightarrow} \mathbf{H}_{th} \stackrel{Et X^\ast}{\longrightarrow} Sh_{\mathbf{H}}(X) \,,$$

then an $\mathcal{O}_X$-module $\mathcal{F}$ on $X$ is an extension of $\mathcal{O}_X^{\mathbf{H}}$ by a limit-preserving functor through $i_!$

$$\array{ \mathbf{H} &\stackrel{\mathcal{O}_X^{\mathbf{H}}}{\longrightarrow}& Sh_{\mathbf{H}}(X) \\ {}^{\mathllap{i_!}}\downarrow & \nearrow_{\mathrlap{\mathcal{F}}} \\ \mathbf{H}_{th} } \,.$$

Example

In particular $\mathcal{O}_X$ is canonically a module over itself by setting $\mathcal{F} = \mathcal{O}_X$.

This is a slight abstraction of the definition in (Lurie QC, section 2.3). See at quasicoherent sheaf – In higher geometry – As extension of the structure sheaf.

Liouville-Poincaré cocycle

Definition

For $X,A \in \mathbf{H}_{th}$ and with $\mathcal{O}_X(A) \in Sh_{\mathbf{H}}(X)$ as in def. , write

$$\theta_X(A) \;\colon\; \underset{X}{\sum} \iota \mathcal{O}_X(A) \to A$$

for the morphism in $\mathbf{H}$ which is the $(\underset{X}{\sum} \dashv X^*)$-adjunct $\underset{X}{\sum}\iota Et(X^* A) \to A$ of the counit $\iota Et(X^* A) \to X^* A$ of the $(\iota \dashv Et)$-coreflection of def. .

This $\theta_X(A)$ we call the Liouville-Poincaré $A$-cocycle on $\underset{X}{\sum} \iota \mathcal{O}_X(A)$.

Example

Consider the model of differential cohesion given by $\mathbf{H}_{th} =$ SynthDiff∞Grpd. Write $\Omega^1 \in \mathbf{H }\stackrel{i_!}{\hookrightarrow} \mathbf{H}_{th}$ for the abstract sheaf of differential 1-forms.

Then for $X \in SmthMfd \hookrightarrow \mathbf{H}$ a smooth manifold, we have that

$$\underset{X}{\sum} \iota \mathcal{O}_X(\Omega^1) \to X$$

is the cotangent bundle

$$T^* X \to X$$

of the manifold: because for $i_U \colon U \to X$ an open subset of the manifold regarded as an object of $Sh_{\mathbf{H}}(X)$, a section $\iota(\sigma_U)$ of $T^* X|_U \to U$ is equivalently a map $\sigma \colon i_U \to \mathcal{O}_X(\Omega^1)$ in $Sh_{\mathbf{H}_{th}}(X)$, which by the $(\iota \dashv Et)$-adjunction is a map $\iota(i_U) \to X \times \Omega^1$ in $(\mathbf{H}_{th})_{/X}$ which finally is equivalently a map $U \to \Omega^1$ in $\mathbf{H}_{th}$ hence an element in $\Omega^1(U)$.

So the Liouville-Poincaré $\Omega^1$-cocycle according to is a differential 1-form

$$\theta \;\colon\; \underset{X}{\sum}\iota \mathcal{O}_X(\Omega^1) \to \Omega^1$$

on the total space of the cotangent bundle. For

$$\sigma \;\colon\; X \to \mathcal{O}_X(\Omega^1) \;\;\; \in Sh_{\mathbf{H}}(X)$$

a section of the cotangent bundle, the pullback form $\sigma^* \theta$ on $X$ is the composite

$$\underset{X}{\sum}\iota X \stackrel{\underset{X}{\sum}\iota(\sigma)}{\to} \underset{X}{\sum}\iota \mathcal{O}_X(\Omega^1) \stackrel{\theta}{\to} \Omega^1 \,,$$

hence the adjunct

$$\iota X \stackrel{\iota(\sigma)}{\to} \iota \mathcal{O}_X(\Omega^1) \stackrel{}{\to} X^* \Omega^1 \,,$$

hence by definition

$$\iota(X) \stackrel{\iota(\sigma)}{\to} \iota(Et(X \times \Omega^1)) \stackrel{}{\to} X^*\Omega^1 \,,$$

hence the adjunct

$$X \stackrel{\sigma}{\to} Et(X \times \Omega^1) \stackrel{id}{\to} Et(X \times \Omega^1)$$

hence the original $\sigma$. This is the defining property which identifies $\that$ as the traditional Liouville-Poincaré 1-form.

Manifolds and étale groupoids

An ordinary topological/Lie étale groupoid is one whose source/target map is an étale map. We consider now a notion that can be formulated in the presence of infinitesimal cohesion which generalizes this.

Definition

A groupoid object $\mathcal{G}_\bullet$ is an étale ∞-groupoid if the equivalent (via the higher Giraud theorem) effective epimorphism (the atlas) $\mathcal{G}_0 \longrightarrow \mathcal{G}$ is a formally étale morphism.

Let now $V$ be any object

(For instance let $\mathbb{A}^1 \in \mathbf{H}$ be a canonical line object that exhibits the cohesion of $\mathbf{H}$ in the sense of structures in a differential infinity-topos – A1 homotopy / The continuum and take $V = \coprod_n \mathbb{A}^n$.)

Definition

A $V$-manifold is an object $X$ such that there exists a $V$-cover $U$, namely a correspondence from $V$ to $X$

$$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$$

such that both morphisms are formally étale morphisms and such that $U \to X$ is in addition an effective epimorphism.

See also at smooth manifold – general abstract geometric formulation

Frame bundles

We discuss how each manifold $X$ in differential cohesion as in def. is associated a canonical frame bundle classified by a morphism $X \to \mathbf{B}GL(V)$.

Definition

For $X$ any object in differential cohesion, its infinitesimal disk bundle $T_{inf} X \to X$ is the homotopy pullback

$$\array{ T_{inf} X &\stackrel{ev}{\longrightarrow}& X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& \Im X }$$

of the unit of its infinitesimal shape modality along itself.

More generally, given a filtration of differential cohesion by orders of infinitesimals, remark , then the order-$k$ infinitesimal disk bundle is the homotopy pullback in

$$\array{ T_{inf} X &\stackrel{ev}{\longrightarrow}& \Im_{(k)}X \\ \downarrow^{\mathrlap{p}} && \downarrow \\ X &\longrightarrow& \Im X } \,.$$

Remark

The Atiyah groupoid of $T_{inf} X$ is the jet groupoid of $X$

Remark

With respect to the base change geometric morphism

$$\mathbf{H}_{/X} \stackrel{\overset{p_!}{\longrightarrow}}{\stackrel{\overset{p^\ast}{\longleftarrow}}{\underset{p_\ast}{\longrightarrow}}} \mathbf{H}_{/\Im X}$$

then then infinitesimal disk bundle of $X$ is

$$T_{inf} X \simeq p^\ast p_! X \,,$$

where on the right $X$ is regarded as sitting by the identity morphism over itself.

Written in this form it follows from the adjoint triple above that bundle morphisms

$$\array{ T_{inf}X && \longrightarrow && E \\ & \searrow && \swarrow \\ && X }$$

are equivalently sections of $p^\ast p_\ast E$. But such bundle morphisms are equivalently jets of $E$ and hence $p^\ast p_\ast E$ is the jet bundle of $E$. See there for more.

Lemma

If $\iota \colon U \to X$ is a formally étale morphism, def. , then

$$\iota^\ast T_{inf} X \simeq T_{inf}U \,.$$

Proof

By the definition of formal étaleness and using the pasting law we have an equivalence of pasting diagrams of homotopy pullbacks of the following form:

$$\array{ \iota^\ast T_{inf} X &\longrightarrow& T_{inf} X &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& X &\longrightarrow& \Im X } \;\;\;\; \simeq \;\;\;\; \array{ T_{inf} U &\longrightarrow& U &\longrightarrow& X \\ \downarrow && \downarrow && \downarrow \\ U &\longrightarrow& \Im U &\longrightarrow& \Im X }$$

Definition

For $V$ an object, a framing on $V$ is a trivialization of its infinitesimal disk bundle, def. , i.e. an object $\mathbb{D}^V$ – the typical infinitesimal disk or formal disk – and a (chosen) equivalence

$$\array{ T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n \\ & \searrow && \swarrow_{\mathrlap{p_1}} \\ && V } \,.$$

Definition

For $V$ a framed object, def. , we write

$$GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V)$$

for the automorphism ∞-group of its typical infinitesimal disk/formal disk.

Remark

When the infinitesimal shape modality exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order infinitesimal neighbourhood of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the general linear group. When $\mathbb{D}^n$ is instead a higher order or even the whole formal neighbourhood, then $GL(n)$ is rather a jet group. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation “$GL(V)$” here, consistent with the fact that we have no index on the infinitesimal shape modality. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

This class of examples of framings is important:

Proposition

Every differentially cohesive ∞-group $G$ is canonically framed (def. ) such that the horizontal map in def. is given by the left action of $G$ on its infinitesimal disk at the neutral element:

$$ev \colon T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G \,.$$

Proof

By the discussion at Mayer-Vietoris sequence in the section Over an ∞-group and using that the infinitesimal shape modality preserves group structure, the defining homotopy pullback of $T_{inf} G$ is equivalent to the pasting of pullback diagrams of the form

$$\array{ T_{inf} G &\stackrel{}{\longrightarrow}& \mathbb{D}^G_e &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow && \downarrow \\ G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G &\stackrel{}{\longrightarrow}& \Im G } \,,$$

where the right square is the defining pullback for the infinitesimal disk $\mathbb{D}^G$. For the left square we find by this proposition that $T_{inf} G \simeq G\times \mathbb{D}^G$ and that the top horizontal morphism is as claimed.

By lemma it follows that:

Proposition

For $V$ a framed object, def. , let $X$ be a $V$-manifold, def. . Then the infinitesimal disk bundle, def. , of $X$ canonically trivializes over any $V$-cover $V \leftarrow U \rightarrow X$ , i.e. there is a homotopy pullback of the form

$$\array{ U \times \mathbb{D}^V &\longrightarrow& T_{inf} X \\ \downarrow && \downarrow \\ U &\longrightarrow& X } \,.$$

This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-fiber ∞-bundle.

Remark

By this discussion this fiber fiber ∞-bundle is the associated ∞-bundle of an essentially uniquely determined $\mathbf{Aut}(\mathbb{D}^V)$-principal ∞-bundle.

Definition

Given a $V$-manifold $X$, def. , for framed $V$, def. , then its frame bundle $Fr(X)$ is the $GL(V)$-principal ∞-bundle given by prop. via remark .

Remark

As in remark , this really axiomatizes in general higher order frame bundles with the order implicit in the nature of the infinitesimal shape modality.

Remark

By prop. the construction of frame bundles in def. is functorial in formally étale maps between $V$-manifolds.

$G$-Structures

We discuss the formalization of G-structures and integrability of G-structures in differential cohesion

Let $V$ be framed, def. , let $G$ be an ∞-group and $G \to GL(V)$ a homomorphism to the general linear group of $V$, def. , hence

$$G\mathbf{Struc}\colon \mathbf{B}G \longrightarrow \mathbf{B}GL(V)$$

a morphism between the deloopings.

Definition

For $X$ a $V$-manifold, def. , a G-structure on $X$ is a lift of the structure group of its frame bundle, def. , to $G$, hence a diagram

$$\array{ X && \longrightarrow&& \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) }$$

hence a morphism

$$\mathbf{c} \colon \tau_X \longrightarrow G\mathbf{Struc}$$

in the slice (∞,1)-topos.

In fact $G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(n)}$ is the moduli ∞-stack of such $G$-structures.

The double slice $(\mathbf{H}_{/\mathbf{B}GL(n)})_{/G\mathbf{Struc}}$ is the (∞,1)-category of such $G$-structures.

Example

If $V$ is framed, def. , then it carries the trivial $G$-structure, which we denote by

$$\mathbf{c}_0 \colon \tau_{V} \longrightarrow G\mathbf{Struc} \,.$$

Definition

For $V$ framed, def. , and $X$ a $V$-manifold, def. , then a $G$-structure $\mathbf{c}$ on $X$, def. , is integrable (or locally flat) if there exists a $V$-cover

$$\array{ && U \\ & \swarrow && \searrow \\ V && && X }$$

such that the correspondence of frame bundles induced via remark

$$\array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && && \tau_X }$$

(a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$) extends to a sliced correspondence between $\mathbf{c}$ and the trivial $G$-structure $\mathbf{c}_0$ on $V$, example , hence to a diagram in $\mathbf{H}_{/\mathbf{B}GL(V)}$ of the form

$$\array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} }$$

On the other hand, $\mathbf{c}$ is called infinitesimally integrable (or torsion-free) if such an extension exists (only) after restriction to all infinitesimal disks in $X$ and $U$, hence after composition with the counit

$$\flat^{rel} U \longrightarrow U$$

of the relative flat modality, def. (using that by prop. this is also formally étale and hence induces map of frame bundles):

$$\array{ && \tau_{\flat^{rel} U} \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} } \,.$$

Remark

As before, if the given reduction modality encodes order-$k$ infinitesimals, then the infinitesimal integrability in def. is order-$k$ integrability. For $k = 1$ this is torsion-freeness.

Flat $\infty$-connections and infinitesimal local systems

We discuss the intrinsic flat cohomology in an infinitesimal neighbourhood.

Definition

For $X, A \in \mathbf{H}_{th}$ we say that

$$H_{infflat}(X,A) := \pi_0 \mathbf{H}(\Im(X), A) \simeq \pi_0 \mathbf{H}(X, \mathbf{\flat}_{inf}A)$$

(where $(\Im \dashv \mathbf{\flat}_{inf})$ is given by def. ) is the infinitesimal flat cohomology of $X$ with coefficient in $A$.

Note

In traditional contexts this is also called crystalline cohomology or just de Rham cohomology . Since we already have an intrinsic notion of de Rham cohomology in any cohesive (∞,1)-topos, which is similar to but may slightly differ from infinitesimal flat differential cohomology, we shall say synthetic de Rham cohomology for the notion of def. if we wish to honor traditional terminology. In this case we shall write

$$H_{dR,synth}(X,A) := \pi_0 \mathbf{H}_{th}(\Im(X), A) \,.$$

Note

By the above observation we have canonical morphisms

$$\mathbf{H}_{flat}(X,A) \to \mathbf{H}_{infflat}(X,A) \to \mathbf{H}(X,A)$$

The objects on the left are principal ∞-bundles equipped with flat ∞-connection . The first morphism forgets their higher parallel transport along finite volumes and just remembers the parallel transport along infinitesimal volumes. The last morphism finally forgets also this connection information.

Definition

For $A \in \mathbf{H}_{th}$ an abelian ∞-group object we say that the de Rham theorem for $A$-coefficients holds in $\mathbf{H}_{th}$ if for all $X \in \mathbf{H}_{th}$ the infinitesimal path inclusion

$$\Im(X) \to \mathbf{\Pi}(X)$$

is an equivalence in $A$-cohomology, hence if for all $n \in \mathbb{N}$ we have that

$$\pi_0 \mathbf{H}_{th}(\mathbf{\Pi}(X), \mathbf{B}^n A) \to \pi_0 \mathbf{H}_{th}(\Im(X), \mathbf{B}^n A)$$

is an isomorphism.

If we follow the notation of note and moreover write $\vert X \vert = \vert \Pi X \vert$ for the intrinsic geometric realization, then this becomes

$$H^{\bullet}_{dR, synth}(X,A) \simeq H^\bullet(|X|, A_{disc}) \,,$$

where on the right we have ordinary cohomology in Top (for instance realized as singular cohomology) with coefficients in the discrete group $A_{disc} := \Gamma A$ underlying the cohesive group $A$.

In certain contexts of infinitesimal neighbourhoods of cohesive $\infty$-toposes the de Rham theorem in this form has been considered in (SimpsonTeleman).

Formal cohesive $\infty$-groupoids

Recall that a groupoid object in an (infinity,1)-category is equivalently an 1-epimorphism $X \longrightarrow \mathcal{G}$, thought of as exhibiting an atlas $X$ for the groupoid $\mathcal{G}$.

Now an $\infty$-Lie algebroid is supposed to be an $\infty$-groupoid which is only infinitesimally extended over its base space $X$. Hence:

A groupoid object $p \colon X \longrightarrow \mathcal{G}$ is infinitesimal if under the reduction modality $\Re$ (equivalently under the infinitesimal shape modality $\Im$) the atlas becomes an equivalence: $\Re(p), \Im(p) \in Equiv$.

For example the tangent $\infty$-Lie algebroid $T X$ of any $X$ is the unit of the infinitesimal shape modality.

$$\eta^{\Im}_X \;\colon\; X \stackrel{}{\longrightarrow} \Im X \,.$$

It follows that every such $\infty$-Lie algebroid $X \to \mathcal{G}$ canonically maps to the tangent $\infty$-Lie algebroid of $X$ – the anchor map. The naturality square of the unit $\eta^{\Im}_{p}$ exhibits the morphism:

$$\array{ X & \stackrel{id}{\longrightarrow} & X \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{\eta^\Im_X}} \\ && \Im X \\ \downarrow && \downarrow^{\mathrlap{\Im p}}_\simeq \\ \mathcal{G} &\stackrel{\eta^{\Im}_{\mathcal{G}}}{\longrightarrow}& \Im \mathcal{G} }$$

Lie theory

(…)

The discussion at synthetic differential ∞-groupoid – Lie differentiation immediately generalizes to produce a concept of Lie differentiation in any differentially cohesive context. This Lie differentiation is just the flat modality of the differential cohesion but regarded as cohesive over its induced infinitesimal cohesion. As such, there is a left adjoint to Lie differentiation, given by the corresponding shape modality. However, the substance of Lie theory here will be to restrict this adjunction to geometric ∞-stacks. On the geometric $\infty$-stacks the Lie differentiation via passage to infinitesimal cohesion will yield actual $L_\infty$-algebras, but some structure is required to make the formal Lie integration of these lang indeed in geometric ∞-stacks.

(…)

Deformation theory

For $C^{op}$ any (∞,1)-site the construction of the tangent (∞,1)-category $T_{C} \to C$ provides a canonical infinitesimal thickening of $C$:

$$C \stackrel{\overset{cod}{\leftarrow}}{\stackrel{\overset{\Delta}{\to}}{\underset{dom}{\leftarrow}}} C^{\Delta[1]} \stackrel{\overset{L}{\to}}{\underset{\Omega^\infty}{\leftarrow}} \,,$$

where the $\infty$-functor pair on the right forms a $cod$-relative (∞,1)-adjunction. The composite $L \circ i$ is the cotangent complex functor for $C$ and $\Omega^\infty$ is fiberwise the canonical map out of the stabilization.

The image of $i$ is contained in that of $\Omega^\infty$. Therefore we may restrict the $(cod \dashv i)$-adjunction on the right to the full sub-(∞,1)-category $\tilde T_C$ of $C^{\Delta[1]}$ on thise objects in the image of $\Omega^\infty$. This yields an infinitesimal neighbourhood of (∞,1)-sites

$$C^{op} \stackrel{\overset{i}{\hookrightarrow}}{\underset{cod}{\leftarrow}} (\tilde T_C)^{op} \,.$$

(…)

Idelic structure

See at differential cohesion and idelic structure.

Examples

• formal smooth ∞-groupoid
• infinitesimally thickened Sierpinski topos

Related concepts

• infinitesimal cohesive (infinity,1)-topos

• differential homotopy type theory

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

  $(\esh \dashv \flat \dashv \sharp )$

  • discrete object, codiscrete object, concrete object

  • points-to-pieces transform

  • structures in cohesion

• dR-shape modality$\dashv$ dR-flat modality

  $\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

• classical modality

tangent cohesion

• differential cohomology diagram

differential cohesion

• (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)

  $(\Re \dashv \Im \dashv \&)$

  • reduced object, coreduced object, formally smooth object

  • formally étale map

  • structures in differential cohesion

graded differential cohesion

• fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality

  $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$

singular cohesion

$$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$$

References

The material discussed here corresponds to the most part to sections 3.5 and 3.10 of

• Urs Schreiber, differential cohomology in a cohesive topos

Exposition:

• Fosco Loregian, Cohesion in Rome, talk notes, Dec. 2019 (pdf, pdf)

For references on the general notion of cohesive (∞,1)-topos, see there.

The following literature is related to or subsumes by the discussion here.

Something analogous to the notion of ∞-connected site and the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos is the content of section 2.16. of

• Carlos Simpson, Constantin Teleman, Section 3 of: deRham theorem for $\infty$-stacks (1997) [pdf, pdf]

The infinitesimal path ∞-groupoid adjunction $(\Re \dashv \Im \dashv \&)$ is essentially discussed in section 3 there.

The characterization of infinitesimal extensions and formal smoothness by adjoint functors (in 1-category theory) is considered in

• Maxim Kontsevich, Alexander Rosenberg, Noncommutative spaces, preprint MPI-2004-35 (pdf, ps, dvi)

in the context of Q-categories .

The notion of forming petit $(\infty,1)$-toposes of étale objects over a given object appears in

• Jacob Lurie, Structured Spaces

• David Carchedi, The Meaning of Étale Stacks (arXiv:1212.2282)

Discussion for derived differential geometry:

• Luigi Alfonsi, Charles A. S. Young, Towards non-perturbative BV-theory via derived differential cohesive geometry [arXiv:2307.15106]