nLab function algebras on infinity-stacks

Abstract

For $T$ any abelian Lawvere theory, here we discuss – in a variation of the theme of Isbell conjugation, in generalization of (Toën) and following (Stel) – a simplicial Quillen adjunction between model category structures on cosimplicial $T$-algebras and on simplicial presheaves over duals of $T$-algebras. We find mild general conditions under which this descends to the local model structure that models ∞-stacks over duals of $T$-algebras. In these cases the left adjoint of the Quillen adjunction is given by sending $\infty$-stacks to their cosimplicial $T$-algebras of functions with values in the canonical $T$-line object, and the adjunction models small objects relative to a choice of a small full subcategory $T\subset C \subset T Alg^{op}$ of the localization

$$\mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C )$$

of the $(\infty,1)$-topos of $(\infty,1)$-sheaves over duals of $T$-algebras at those morphisms that induce isomorphisms in cohomology with coefficients the canonical $T$-line object.

For the special case where $T$ is the theory of ordinary commutative algebras this reproduces the situation of (Toën) and many statements are straightforward generalizations from that situation. For the case that $T$ is the theory of smooth algebras ($C^\infty$-rings) we obtain a refinement of this to the context of synthetic differential geometry. In these cases, in as far as objects in $\mathbf{H}$ may be understood as ∞-Lie groupoids, the objects in $\mathbf{L}$ may be understood as ∞-Lie algebroids.

As an application, we show how Anders Kock‘s simplicial model for synthetic combinatorial differential forms finds a natural interpretation as the differentiable $\infty$-stack of infinitesimal paths of a manifold. This construction is an $\infty$-categorical and synthetic differential resolution of the de Rham space functor introduced by Grothendieck for the cohomological description of flat connections. We observe that also the construction of the $\infty$-stack of modules lifts to the synthetic differential setup and thus obtain a notion of synthetic $\infty$-vector bundles with flat connection.

Models for $\infty$-stacks and their function algebras

Cosimplicial $T$-algebras

A good general notion of function algebras on generalized spaces are $T$-algebras, for $T$ a Lawvere theory. A good general notion of function algebras on internal ∞-groupoids in such spaces are cosimplicial $T$-algebras.

We recall some basics and then discuss a model category structure on cosimplicial $T$-algebras for the cases that $T$ contains the theory of abelian groups.

$T$-Algebras

A Lawvere theory may be thought of as a generalization of the theory of ordinary associative algebras.

A Lawvere theory is encoded in its syntactic category $T$, which by definition is a category with finite products such that every object is (isomorphic to) a finite cartesian power $x^k$ of a fixed object $x \in T$. We are to think of the hom-set $T(k,1)$ as the set of $k$-ary operations of the algebras defined by the theory. A $T$-algebra is accordingly a product-preserving functor $A : T \to Set$. Its image $U_T(A) \coloneqq A(1) \in Set$ is the underlying set, and its value $A(f) : U_T(A)^k \to U_T(A)$ on an element $f \in T(k,1)$ is the $k$-ary operation $f$ as implemented by $A$.

The category of $T$-algebras is the full subcategory $T Alg \subset [T, Set]$ of the category of presheaves on $T^{op}$ on these product-preserving functors.

Examples.

• The category $T = \mathcal{Ab}$ of free finitely generated abelian groups is the syntactic category of the Lawvere theory whose algebras are abelian groups.

• For $k$ a field, the category $T = k$ of free finitely generated $k$-algebras is the Lawvere theory whose algebras are $k$-associative algebras;

• The category $T =$ CartSp is the syntactic category whose algebras are smooth algebras.

A morphism of Lawvere theories $T_1 \to T_2$ is again a product-preserving functor.

Definition

An abelian Lawvere theory $T$ is a morphism of Lawvere theories $ab_T : \mathcal{Ab} \to T$.

For $T$ abelian, $T$-algebras have an underlying abelian group, given by the functor

$$ab_T^* : T Alg \to Ab \,.$$

This functor is a right adjoint.

For example associative algebras and smooth algebras are algebras over an abelian Lawvere theory, and their underlying abelian groups are the evident ones.

Similarly, the forgetful functor $U_T : T Alg \to Set$ has a left adjoint, the free $T$-algebra functor $F_T : Set \to T Alg$. By the Yoneda lemma this sends the $n$-element set $(n)$ to $F_T(n) : k \mapsto T(n,k)$.

More generally, for any $A \in T Alg$ the copresheaf

$$T Alg(F_T(-), A) : T \to Set$$

is isomorphic to $A$.

The free $T$-algebra $F_T(1)$ on a single generator may be thought of as the $T$-algebra of functions on the $T$-line. For instance

• for $T = k$ we have that $F_T(1) = k[X]$ is the free $k$-algebra on a single generator $X$;

• for $T = CartSp$ we have that $F_T(1) = C^\infty(\mathbb{R})$.

We say more on the canonical $T$-line object below in The Line object

Model structure on cosimplicial $T$-algebras

Theorem

There is a cofibrantly generated model structure on cosimplicial abelian groups $Ab^\Delta_{proj}$ whose weak equivalences are the morphisms that induce quasi-isomorphism under passage to normalized cochain complexes and fibrations are the degreewise surjections.

With respect to the canonical sSet-enrichment of the category of cosimplicial objects $Ab^{\Delta}$, this is a simplicial model category.

For $ab_T : \mathcal{Ab} \to T$ any abelian Lawvere theory, the adjunction

$$((ab_*)^\Delta \dashv (ab^*)^\Delta ) : T Alg^{\Delta} \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab^\Delta$$

induces a transferred model structure $T Alg^{\Delta}_{proj}$ on the category of cosimplicial $T$-algebras, whose weak equivalences and fibrations are those morphisms that under $(ab^*)^\Delta$ become weak equivalences and fibrations, respectively, in $Ab^\Delta_{proj}$.

This, too, is a simplicial model category with respect to its standard sSet-enrichment.

Proof

The proof of the existence of the model structure on cochain complexes in non-negative degree – $Ch^\bullet_+(Ab)$ – whose fibrations are the degreewise surjections (and weak equivalences the usual quasi-isomorphism)s is spelled out here.

By the dual Dold-Kan correspondence $Ab^\Delta \simeq Ch^\bullet_+(Ab)$ this induces the model structure on cosimplicial abelian groups whose fibrations are the degreewise surjections (using that the normalized cochain complex functor sends surjections to surjections).

That with the standard structure of an sSet-enriched category on $Ab^\Delta$ this constitutes a simplicial model category-structure is proven here.

Now we use the basic fact of Lawvere theories that any morphism $f : T_1 \to T_2$ of such induces a pair of adjoint functors

$$(f_* \dashv f^*) : T_2 Alg \stackrel{\overset{f_*}{\leftarrow}}{\underset{f^*}{\to}} T_1 Alg$$

between their categories of algebras: the adjunction of relatively free algebras.

Since by assumption that our $T$ is an abelian Lawvere theory we are given a morphism $ab : Ab \to T$ from the theory of abelian groups, this means that we have an adjunction

$$(ab_* \dashv ab^*) : T Alg \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab$$

and hence also an adjunction

$$(ab_*^\Delta \dashv (ab^*)^\Delta) : T Alg^\Delta \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} Ab^\Delta \,.$$

We need to check that the right adjoint $(ab^*)^\Delta$ induces the transferred model structure on $T Alg^\Delta$ from the above model structure $Ab^\Delta_{proj}$.

By the facts recalled at transferred model structure, we need to check that $T Alg^\Delta_{proj}$

• has a fibrant replacement functor;

• has functorial path space objects for fibrant objects;

and for the simplicial enrichment that

• $(ab^*)^\Delta$ preserves the powering.

The first condition is trivial, since all objects are fibrant. The last condition is evidently satisfied, since

$$U_T(A^S)_n = U_T(\prod_{S_n} A_n) = \prod_{S_n} U_T(A_n) = ((U_T(A))^S)_n \,.$$

Using this, we claim that we can take the path space object functor to be given by powering with the simplicial interval

$$(-)^I : A \mapsto A^{\Delta[1]} \,.$$

This is because $\Delta[0] \coprod \Delta[0] \hookrightarrow \Delta[1] \to \Delta[0]$ factors the co-diagonal in $sSet_{Quillen}$ by a cofibration followed by a weak equivalence between cofibrant objects. Accordingly the induced

$$A \to A^I \to A \times A$$

factors the digonal, and the morphism on the left is a weak equivalence, since it is the image under the left Quillen functor $A^{(-)}$ of a weak equivalence between cofibrant objects (and by the factorization lemma such weak equivalences are preserved by left Quillen functors).

Simplicial presheaves on duals of $T$-algebras

A good notion of a generalized space modeled on objects in a category $C$ is a sheaf on $C$. A good notion of an ∞-groupoid in such generalized spaces is an (∞,1)-sheaf on $C$. Such objects are modeled by the model structure on simplicial presheaves on $C$.

We are interested here in that case that

$$T \subset C \hookrightarrow T Alg^{op}$$

is a small full subcategory of the opposite category of $T$-algebras, for $T$ an abelian Lawvere theory. In the remainder of this section we assume such a choice to be fixed. Below in the section on Examples and applications we discuss concrete choices of interest.

Notice that such a choice induces also a full subcategory of (co)simplicial objects

$$C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op} \,.$$

The prolonged Yoneda embedding

Write

$$j : T Alg^{op} \to [C^{op}, Set]$$

for the ordinary Yoneda embedding and

$$j : (T Alg^\Delta)^{op} \to [C^{op}, sSet]$$

for its degreewise simplicial prolongation

$$j(A) : (B \in T Alg) \mapsto ([n] \mapsto T Alg(A_n, B)) \,.$$

Lemma

For $B \in T Alg$ and $(T Alg^\Delta)_s$ denoting the simplicially enriched category of $T$-algebras, we have a natural identification

$$j(A) \simeq (T Alg^\Delta)^{op}_s(-, A) \,.$$

Proof

Using end/coend-calculus for handling the canonical enrichment of $T Alg^\Delta$, we have for $B \in T Alg^{op} \hookrightarrow (T Alg \Delta)^{op}$ and $A \in (T Alg^\Delta)^{op}$ natural isomorphisms

$$\begin{aligned} (T Alg^\Delta)^{op}_s(B, A)_n & \coloneqq (T Alg^\Delta)^{op}(B \cdot \Delta^n, A) \\ & \simeq \int_{k \in \Delta} T Alg(A_k, \prod_{\Delta(k,n)} B) \\ & \simeq \int_{k \in \Delta} T Alg(\Delta(k,n)\cdot A_k, B) \\ & \simeq T Alg( \int^{k \in \Delta} \Delta(k,n) \cdot A_k, B) \\ & \simeq T Alg(A_n , B) \,, \end{aligned}$$

where in the last step we used the isomorphism (described at coend)

$$\int^{k \in \Delta} \Delta(k,n) \cdot A_k \simeq \lim_\to( \Delta/n \to \Delta \stackrel{A}{\to} T Alg) \simeq A_n \,.$$

The line object

The adjunction that we shall be concerned with is essentially Isbell conjugation. We recall some basics of Function T-algebras on presheaves.

Recall from above that we write $F_T(*)$ for the free $T$-algebra on a single generator.

Definition

We call $R \coloneqq j(F_T(*))$ the line object in $[C^{op}, sSet]$.

Observation

As a presheaf, the line object $R$ sends a $T$-algebra $B \in T Alg$ to its underlying set $U_T(B)$

$$R : B \mapsto T Alg(F_T(*), B) \simeq Set(*, U_T(B)) \simeq U_T(B) \,.$$

This characterization may look simpler, but does not capture the important fact that homming into $R$ produces $T$-algebras of functions . This is what the following definition deals with.

Definition

($T$-algebras of functions)

For $X \in [C^{op}, sSet]$, the cosimplicial set

$$U_T(\mathcal{O}(X)) \coloneqq [C^{op},sSet](X_\bullet, R) \in Set$$

we call the cosimplicial set of $R$-valued functions on $X$. This is naturally the cosimplical set underlying the cosimplicial $T$-algebra

$$\mathcal{O}(X) : (k \in T) \mapsto [C^{op},sSet](X_\bullet, j(F_T(k))) \,.$$

We call $\mathcal{O}(X) \in T Alg^{op}$ the $T$-algebra of functions on $X$. This extends to a functor

$$\mathcal{O} : [C^{op}, sSet] \to (T Alg^{\Delta})^{op} \,.$$

In the next section we see that $(\mathcal{O} \dashv j)$ forms a simplicial Quillen adjunction.

Model structure on simplicial presheaves

Write $[C^{op}, sSet]_{proj}$ for the global projective model structure on simplicial presheaves over $C$. With the simplicial enrichment $[C^{op}, sSet]_s$ this is naturally a simplicial model category.

Let $S \subset mor [C^{op}, sSet]$ be a class of split hypercovers.

Definition

Write $[C^{op}, sSet]_{proj,loc}$ for the left Bousfield localization $[C^{op}, sSet]_{proj}$ at this class.

By general results on left Bousfield localization, this exists always for $S$ a small set, notably for $f$ the set of Cech nerve projections $C(U) \to X$ for covers $\{U_i \to X\}$ of the Grothendieck topology on $C$. By general results on the local model structure on simplicial presheaves, the localization also exists for $S$ the class of all (split) hypercovers.

The Yoneda-Quillen-adjunction

We relate now the model structure on cosimplicial T-algebras with the model structure on simplicial presheaves over $C \subset T Alg^{op}$ using the function algebra functor $\mathcal{O}$ and the prolonged Yoneda embedding $j$.

Theorem

The functors $j$ and $\mathcal{O}$ constitute a simplicial Quillen adjunction

$$(\mathcal{O} \dashv j) \;\colon\; (TAlg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}} [C^{op}, sSet]_{proj} \,.$$

Proof

We first establish the adjunction itself: using end-calculus for expressing hom-sets in functor categories we have for $X \in [C^{op}, sSet]$ and $A \in T Alg^\Delta$ natural isomorphisms

$$\begin{aligned} (T Alg^\Delta)^{op}(\mathcal{O}(X), A) & \coloneqq T Alg^\Delta (A(-), [C^{op}, sSet](X, j(F_T(-)))) \\ & \simeq \int_{k \in T} \int_{[n] \in \Delta} Set(A_n(k), [C^{op}, Set](X_n, j( F_T(k)) )) \\ & \simeq \int_{k \in T} \int_{[n] \in \Delta} \int_{B \in C} Set(A_n(k), Set(X_n(B), T Alg(F_T(k), B))) \\ & \simeq \int_{k \in T} \int_{[n] \in \Delta} \int_{B \in C} Set(A_n(k), Set(X_n(B), B(k))) \\ & \simeq \int_{k \in T} \int_{[n] \in \Delta} Set(X_n(B), \int_{k \in T} Set(A_n(k), B(k)) ) \\ & \simeq \int_{k \in T} \int_{[n] \in \Delta} Set(X_n(B), T Alg(A_n, B)) \\ & \simeq [C^{op}, sSet](X, j(A)) \,. \end{aligned} \,,$$

where the crucial step is the isomorphism $B(-) \simeq T Alg(F_T(-), B)$ for the line object discussed above. This computation is just simplicial-degreewise the adjunction discussed at Isbell duality – Function T-algebras on presheaves.

That this lifts to an $sSet$-enriched adjunction follows with the prolonged Yoneda lemma $j(A) \simeq (T Alg^\Delta)^{op}_s(-,A)$ and the $sSet$-tensoring and cotensoring of $[C^{op}, sSet]_s$ and $(T Alg^\Delta)^{op}_s$:

$$\begin{aligned} (T Alg^\Delta)^{op}_s(\mathcal{O}(X), A)_n & \coloneqq (T Alg^\Delta)^{op}(\mathcal{O}(X), A^{\Delta^n}) \\ & \simeq [C^{op}, sSet](X, j(A^{\Delta_n})) \\ & \simeq [C^{op}, sSet](X, (T Alg^\Delta)^{op}_s(-,A^{\Delta^n})) \\ & \simeq \int_{B \in C} sSet(X(B), (T Alg^\Delta)^{op}_s(B,A^{\Delta^n}))) \\ & \simeq \int_{B \in C} sSet(X(B), (T Alg^\Delta)^{op}_s(B \cdot \Delta^n,A))) \\ & \simeq \int_{B \in C} sSet(X(B)\times \Delta^n , (T Alg^\Delta)^{op}_s(B,A))) \\ & \simeq [C^{op}, sSet](X \cdot \Delta^n, j(A)) \\ & =: [C^{op}, sSet]_n(X, j(A))_n \,. \end{aligned}$$

By the pushout-product axiom satisfied by the $sSet$-enriched model category $(T Alg^\Delta)_s$ and using that in $(T Alg^\Delta_{proj})^{op}$ every object $B$ is cofibrant, we have that for $f : A_1 \to A_2$ a fibration (acyclic fibration) in $(T Alg^\Delta_{proj})^{op}$ and for $B \in C \subset T Alg^{op}$ any object, the morphism $j(A_1 \to A_2)(B) = (T Alg^{\Delta})^{op}_s(B,f)$ is a fibration (acyclic fibration) in $sSet$. Therefore $j(f)$ is a fibration (acyclic fibration) in $[C^{op}, sSet]_{proj}$.

This establishes that $j$ is a right Quillen functor and completes the proof.

The following theorems say that the obstructions to making this Quillen adjunction descent to local model structures on simplicial presheaves are mild.

Proposition

Let $J$ be a subcanonical coverage on $C \subset (TAlg^\Delta)^{op}$, $X \in Ob(C)$ and $f : Y \to j(X)$ a split hypercover with respect to $J$.

Then for $i \neq 1$ we have that $f$ induces an isomorphism in $R$-cohomology in degree $i$: $H^i(\mathcal{O}(f)) : H^i(\mathcal{O}(X)) \stackrel{\simeq}{\to} H^i(\mathcal{O}(Y))$ .

Proof

Regard $f$ as a simplicial object in the overcategory

$$Sh(C)/X \simeq Sh(C/X) \,.$$

Write

$$\bar f \in Ab(Sh(C)/X)^{\Delta^{op}}$$

for the degreewise free abelian group object of that, a simplicial object in the category of abelian group objects in the sheaf topos over $C$. The chain homology of the corresponding normalized chain complex vanishes in positive degree (as discussed here):

$$H_{n \geq 1}(\bar f) = 0 \,.$$

Let now by the Freyd-Mitchell embedding theorem

$$i : Ab(Sh(C/Y)) \hookrightarrow R Mod$$

be a full and faithful functor from the abelian category of abelian group object into the category of $R$-module over some ring $R$.

Write $K \in R Mod$ for the canonical $T$-line object regarded first as the abelian group object $U_T(-) \times X \in Ab(Sh(C/X))$ and then injected with $i$ into $R Mod$.

Using this, the cochain cohomology $H^i(\mathcal{O}(Y_\bullet))$ that we are after is equivalently the cohomology of

$$\begin{aligned} \mathcal{O}(Y) & \simeq Sh(C)^\Delta(Y_\bullet, U_T(-)) \\ & \simeq Sh(C)/X(f_\bullet , U_T(-) \times X) \\ & \simeq Ab(Sh(C)/X)( \bar f_\bullet, U_T(-)\times X) \\ & \simeq R Mod( i(\bar f_\bullet), i(U_T(-) \times X) ) \\ & \simeq R Mod( i(\bar f_\bullet), K ) \end{aligned} \,.$$

To compute this, we use the universal coefficient theorem, which says that we have an exact sequence

$$0 \to Ext^1(H_{n-1}(i(\bar f_\bullet), K)) \to H^n(R Mod(i(\bar f_\bullet), K)) \to Ab(H_n(i(\bar f_\bullet)), C) \to 0 \,.$$

By the above fact that the homology $H_n(i(\bar f))$ vanishes in positive degree, this gives finally that

$$H^n(\mathcal{O}(Y))$$

vanishes in degree $n \geq 2$. That it also vanishes in degree 0 is seen to be equivalent to the sheaf condition on $X$, which is true by the assumption that we are working with a subcanonical coverage.

Theorem

(passage to local model structure)

If for all split (hyper-)covers $f \in S$ we have that $H^1(\mathcal{O}(f))$ is an isomorphisms then $(\mathcal{O} \dashv j)$ is a simplicial Quillen adjunction to the local model structure on simplicial presheaves.

$$(\mathcal{O} \dashv j) : (TAlg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}} [C^{op}, sSet]_{proj, loc} \,.$$

Proof

By the previous proposition we have that under the given assumptions every (hyper-)cover $f : Y\to X$ in $[C^{op}, sSet]$ is taken by $\mathcal{O}$ to a weak equivalence.

Using this we can follow the remainder of the argument of Toën, prop. 2.2.2:

Since the model structure on simplicial presheaves is a left proper model category and since left Bousfield localization preserves left properness, we have that $[C^{op}, sSet]_{proj,loc}$ is left proper. Since moreover left Bousfield localization does not change the class of cofibrations, we know that $\mathcal{O}$ still preserves cofibrations.

Then by the recognition theorem for simplicial Quillen adjunction it is sufficient to check that $j$ sends fibrant objects $A \in (T Alg^\Delta_{proj})^{op}$ to local objects with respect to the morphisms $f$.

Since by definition of hypercovers, their domain and codomain is cofibrant (codomain because it is a representable, domain by assumption that it is a degreewise coproduct of representables with disjoint degeneracies, see the discussion of cofibrancy in the projective structure at model structure on simplicial presheaves), this means that it is sufficient to check that for all $f$ and fibrant $c$ we have that $[C^{op}, sSet]_s(f, j(c))$ is a weak equivalence. But by the adjunction $(\mathcal{O} \dashv j)$ this is isomorphically $(T Alg^\Delta)^{op}(\mathcal{O}(f), c)$.

Now by the above propositions and assumptions, we have that $\mathcal{O}(f)$ is a weak equivalence. Since all objects in $(T Alg^\Delta_{proj})^{op}$ are cofibrant, it is a weak equivalence between cofibrant objects. With the factorization lemma it follows that in an enriched model category the enriched hom of a weak equivalence into a fibrant object is a weak equivalence.

The following proposition asserts that the Quillen adjunction that we have established is very special, in that it is the model-category theoretic analog of a reflective subcategory. Below in the section Localization of the (∞,1)-topos at R-cohomology we see that this indeed presents such a reflective inclusion in (∞,1)-category theory.

Proposition

When restricted along $C^{\Delta^{op}} \subset (T Alg^\Delta)^{op}$ the functor $j$ is homotopy full and faithful in that for all $A \in C^{\Delta^{op}}$ we have that the canonical morphism

$$A \to \mathbb{L}\mathcal{O} \; \mathbb{R}j \; A$$

into the image of the derived functors of $j$ and $\mathcal{O}$ is an isomorphism in the homotopy category $Ho(T Alg^\Delta_{proj})$.

Proof

With the above results, this follows verbatim as the proof of the analogous (Toën, corollary 2.2.3).

Localization of the $(\infty,1)$-topos at $R$-cohomology

We consider now the cohomology localization of $Sh_{(\infty,1)}(C)$ at the canonical line object.

In this section we discuss that in terms of the (∞,1)-category theory that is presented by the model category theoretic structures above, these serve to establish the following intrinsic statement.

Theorem

The Quillen adjunction $(\mathcal{O} \dashv j)$ is a presentation that models $C$-small objects (…) in the reflective sub-(∞,1)-category

$$\mathbf{L}_C \stackrel{\stackrel{\mathcal{O}}{\leftarrow}}{\hookrightarrow} \mathbf{H} \coloneqq Sh_{(\infty,1)}(C)$$

of the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(C)$, where $\mathbb{L}_C$ is the localization at those morphisms that induce isomorphisms in intrinsic $R$-cohomology, for $R$ the canonical T-line object.

We obtain a proof of this after the following discussions.

Remark

The resulting localization modality $Spec \mathcal{O}$ we might call the affine modality. It is similar to exhibiting $C$ as a total category.

$R$-Cohomology

Since $T$ is assumed to be an abelian Lawvere theory, the T-line object $R \in [C^{op}, sSet]$ canonically has the structure of an abelian group object in $[C^{op}, sSet]$. As such it presents a 0-truncated ∞-group in the $Sh_{(\infty,1)}(C)$, and so we may consider its Eilenberg-MacLane objects $\mathbf{B}^n R$ for $n \in \mathbb{N}$.

The following proposition provides a model for these Eilenberg-MacLane objects.

Write $\Xi : Ch^\bullet_+ \to Ab^\Delta$ for the dual Dold-Kan correspondence map. Notice that the free $\mathcal{Ab}$-algebra is $F_{Ab}(*) = \mathbb{Z}$, the free abelian group on a single generator, the integers. Write $F_{Ab}(*)[n] = \mathbb{Z}[n]$ for the cochain complex concentrated in degree $n$ on $F_{Ab}(*)$. For $ab_* : Ab \to T Alg$ the left adjoint to the underlying abelian group functor $ab^* : T Alg \to Ab$ we have then that $ab_* \Xi (F_{Ab}(*)[n])$ is the cosimplicial $T$-algebra which in degree $k$ is a product of copies of the free $T$-algebra corresponding to the product of copies $\mathbb{Z}$ in $\Xi \mathbb{Z}[n]$.

Proposition

For $n \in \mathbb{N}$ the object $\mathbf{B}^n R \in Sh_{(\infty,1)}(C)$ is presented in $[C^{op}, sSet]_{proj,loc}$ by

$$\mathbf{B}^n R_{chn} \coloneqq j(ab_* \Xi(F_{Ab}(*)[n]) \,.$$

Every (∞,1)-topos such as $\mathbf{H} = Sh_{(\infty,1)}(C)$ comes with its intrinsic notion of abelian cohomology: for $X \in \mathbf{H}$ any object and for $A \in \mathbf{H}$ a ∞-group object with arbitrary deloopings $\mathbf{B}^n A$, the $n$th cohomology group of $X$ with coefficients in $A$ is

$$H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X,\mathbf{B}^n A) \,.$$

In terms of the model category presentation by $[C^{op}, sSet]_{proj,loc}$ and writing $X \in [C^{op}, sSet]$ for a representative of $X \in \mathbf{H}$ this is the hom-set in the homotopy category

$$\cdots \simeq Ho_{[C^{op}, sSet]_{proj,loc}}(X, \mathbf{B}^n A_{chn}) \,.$$

Proposition

For $X \in [C^{op}, sSet]$ representing an object $X \in \mathbf{H}$, the intrinsic $R$-cohomology of $X$ coincides with the cochain cohomology of its cosimplicial function algebra $\mathbb{L}\mathcal{O}(X) \in T Alg^\Delta$:

$$H^n(X,R) \simeq H^n( \mathbb{L} \mathcal{O}(X)) \,.$$

Proof

Notice that $ab_* \Xi(\mathbb{Z}[n])$, being the image of a cofibrant object in $Ab^\Delta$, is cofibrant in $T Alg^\Delta_{proj}$, hence fibrant in $(T Alg^\Delta_{proj})^{op}$.

Using this, we compute as follows

$$\begin{aligned} H(X,\mathbf{B}^n R) & = Ho_{[C^{op}, sSet]_{proj}}(X, j(ab_* \Xi(\mathbb{Z}[n])) \\ & \simeq Ho_{(T Alg^\Delta_{proj})^{op}}(\mathbb{L}\mathcal{O}(X), ab_* \Xi(\mathbb{Z}[n]) \\ & \simeq Ho_{(T Alg^\Delta_{proj})}( ab_* \Xi(\mathbb{Z}[n]), \mathbb{L}\mathcal{O}(X) ) \\ & \simeq Ho_{(Ab^\Delta_{proj})}( \Xi(\mathbb{Z}[n]), ab^* \mathbb{L}\mathcal{O}(X) ) \\ & \simeq Ho_{Ch^\bullet}( \mathbb{Z}[n], N^\bullet ab^* \mathbb{L}\mathcal{O}(X) ) \\ & \simeq H^n(\mathbb{L}\mathcal{O}(X)) \end{aligned}$$

This is essentially the argument of (Toën, corollary 2.2.6).

$R$-Local objects

Definition

We say a morphism $f : X \to Y$ in $[C^{op}, sSet]$ is an $R$-equivalence if it induces isomorphisms in $R$-cohomology.

$$H^i(f,R) : H^i(Y,R) \stackrel{\simeq}{\to} H^i(X,R) \,.$$

By prop. this is equivalent to saying that the derived functor $\mathbb{L}\mathcal{O}$ takes $f$ to a weak equivalence.

We say an object $K \in [C^{op}, sSet]_{proj,loc}$ is an $R$-local object if for all $R$-equivalences $f$ we have that

$$\mathbf{H}(f,K) : \mathbf{H}(Y,K) \to \mathbf{H}(X,K)$$

is an equivalence, equivalently if the derived hom-space functor produces a weak equivalence $\mathbb{R}Hom_{[C^{op}, sSet]_{proj,loc}}(f,K)$ (of Kan complexes).

Proposition

The $R$-local objects of $[C^{op}, sSet]_{proj,loc}$ that are equivalent to those in the image of $C^{\Delta^{op}} \hookrightarrow [C^{op}, sSet]$ span precisely the homotopy-essential image of the restriction of $\mathbb{R}j$ to $C^{\Delta^{op}}$

$$C^{\Delta^{op}} \hookrightarrow (T Alg^{\Delta})^{op} \stackrel{\mathbb{R}j}{\to} [C^{op}, sSet]_{proj,cov} \,.$$

Proof

We may explicitly see this by observing that the proof of (Toën, theorem 2.2.9) goes through verbatim: it only uses the general properties of the $(\mathcal{O} \dashv j)$-adjunction that we have established above, as well as the fact that $T Alg^{\Delta}_{proj}$ is a cofibrantly generated model category for $T$ the theory of ordinary commutative algebras. But by our result on the model structure on TAlg (theorem ) we have that for general $T$ this is the transferred model structure of the model structure on cosimplicial abelian groups, which is cofibrantly generated. Hence by the general properties of transferred model structures, also $T Alg^\Delta_{proj}$ is.

But more abstractly, we can also simply use the general theory of reflective sub-(∞,1)-categories and their characterization as the reflective localization of an (∞,1)-category at a set of weak equivalences: from the above we know that on the full sub-$(\infty,1)$-category of $((T Alg^\Delta_{proj})^{op})^\circ$ on the objects in $C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op}$ is a reflective sub-$(\infty,1)$-category

$$\mathbf{L} \stackrel{\overset{\mathbb{L} \mathcal{O}}{\longleftarrow}}{\underset{\mathbb{R} i}{\hookrightarrow}} \mathbf{H} \coloneqq Sh_{(\infty,1)}(C)$$

and that the left adjoint to the embedding inverts precisely the $R$-equivalences. Hence $\mathbf{L}$ is the full sub-$(\infty,1)$-category of $\mathbf{H}$ on $R$-local objects.

In derived geometry

We now discuss function algebras on $\infty$-stacks more generally in the context of derived geometry, meaning that we we pass in the above from sites inside the opposite of a 1-category of $T$-algebras to an (∞,1)-site inside the opposite of an (∞,1)-category of ∞-algebras over an (∞,1)-algebraic theory.

Over ordinary associative algebras

Let $k$ be a field of characteristic $0$.

Definition

Write $(cdgAlg_k^{op})^\circ$ for the (∞,1)-category that is presented by the model structure on unbounded commutative cochain dg-algebras over $k$.

Write

$$i : (cdgAlg_k^{op})^\circ_- \subset (cdgAlg_k^{op})^\circ$$

for the full sub-(∞,1)-category on cochain dg-algebras concentrated in non-positive degree.

Let $C \subset (cdgAlg_k^{op})^\circ_-$ be a small full sub-$(\infty,1)$-category equipped with the structure of a subcanonical (∞,1)-site.

Set

$$\mathbf{H} \coloneqq Sh_{(\infty,1)}(C) \,.$$

Definition

Write

$$\mathcal{O} : \mathbf{H} \to (cdgAlg_k^{op})^\circ$$

for the $(\infty,1)$-Yoneda extension of the inclusion $C \hookrightarrow (cdgAlg_k^{op})^\circ_- \hookrightarrow (cdgAlg_k^{op})^\circ$.

Remark

By the (∞,1)-co-Yoneda lemma? we may express any $X \in Sh_{(\infty,1)}(C)$ by an (∞,1)-colimit over representables

$$X \simeq {\lim_\to}_i U_i \;\; \in Sh(C) \,.$$

The functor $\mathcal{O}$ simply evaluates this colimit in $(cdgAlg_k^{op})^\circ$, which is the (∞,1)-limit in the opposite (∞,1)-category

$$\mathcal{O}X \simeq {\lim_\leftarrow}_i \mathcal{O}(U_i) \;\; \in (cdgAlg_k)^\circ \,,$$

where we write $\mathcal{O}(U_i)$ simply for the object $U_i$ regarded in the opposite category.

Observation

By construction $\mathcal{O}$ is a colimit-preserving $(\infty,1)$-functor between locally presentable (∞,1)-categories. Accordingly, by the adjoint (∞,1)-functor theorem is has a right adjoint (∞,1)-functor.

$$j : (cdgAlg_k^{op})^\circ \to Sh_{(\infty,1)}(C) \,.$$

This is given by

$$Spec(A) : U \mapsto (cdgAlg_k^{op})^\circ(U,A) \,.$$

Proof

This follows by the general yoga of Kan extensions. Explcitly, we check the hom-equivalence

$$\begin{aligned} Sh_C(X, Spec A) & \simeq \mathbf{H}({\lim_{\to}}_i U_i, Spec A) \\ & \simeq {\lim_\leftarrow}_i \mathbf{H}(U_i, Spec A) \\ & \simeq {\lim_\leftarrow}_i C(U_i, A) \\ & \simeq {\lim_\leftarrow}_i (cdgAlg_k)^\circ(A, \mathcal{O}(U_i)) \\ & \simeq (cdgAlg_k)^\circ(A, {\lim_\to}_i \mathcal{O}(U_i)) \\ & \simeq (cdgAlg_k^{op})^\circ({\lim_\to}_i \mathcal{O}(U_i), A) \end{aligned} \,.$$

This is considered in (Ben-Zvi/Nadler, prop. 3.1).

Observation

The above Yoneda-Quillen adjunction for $T$ the theory of commutative $k$-algebras is compatible with this in that it also does model the $(\infty,1)$-Yoneda extension of the inclusion

$$T Alg_k^{op} \hookrightarrow (T Alg_k^{\Delta})^{op}$$

Proof

By the general discussion of cofibrant replacement in the projective model structure on simplicial presheaves we have that every $X \in [C^{op}, sSet]_{proj,loc}$ has a cofibrant resolution of the form $\int^{[k] \in \Delta} \mathbf{\Delta}[k] \cdot \coprod_{i_n} U_{i_n}$, where the integrand the integrand we have the fat simplex tensored degreewise with a coproduct of representables such that the degenerate cells split off as direct summands (a split hypercover). This makes $[n] \mapsto \coprod_{i_n} U_{i_n}$ Reedy cofibrant an therefore the whole coend is a model for its homotopy colimit.

Since both the simplex as well as the fat simplex $\mathbf{\Delta}$ are Reedy cofibrant cosimplicial simplicial sets, this is moreover equivalent to $\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} U_{i_n}$ and this is still cofibrant. Now the left Quillen functor $\mathcal{O}$ takes this to $\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} \mathcal{O}(U_{i_n})$. Since every object in $(T Alg^{\Delta})^{op}$ is cofibrant, this coend is still a homotopy colimit.

This shows that the derived functor of the left Quillen functor $\mathcal{O}$ sends the decomposition of any $\infty$-stack as the $(\infty,1)$-colimit over representable to the $(\infty,1)$-colimit of the images of these representables.

Examples and applications

Proposition (Examples)

The conditons of the above theorem are satisfied for instance for

• $T$ the theory of ordinary commutative algebras over a field $k$ and $J$ the fpqc topology.

  In this case the adjunction is that considered in (Toën).

• $T$ the theory of smooth algebras and $C \hookrightarrow T Alg^{op}$ the site of the Cahiers topos. This is what we discuss in more detail below.

Rational homotopy theory

$T$ the Lawvere theory of $\mathbb{Q}$-algebras. Then $(\mathcal{O} \dashv j)$ reproduces the setup discussed at rational homotopy theory in an (∞,1)-topos.

$\infty$-Lie theory in the $\infty$-Cahiers topos

In this section we study the general theory for the case that

• $T \coloneqq$ CartSp is the (syntactic category of the) Lawvere theory of smooth algebras.

Write $Smooth Alg \coloneqq T Alg$ for the category of smooth algebras. Sheaf toposes on sub-sites $C \subset Smooth Alg^{op}$ are well known to provide smooth toposes that are well adapted models for synthetic differential geometry.

We consider here the choice

• $C \subset Smooth Alg^{op}$ is the site for the Cahiers topos.

Definition

The Cahiers topos is the sheaf topos $Sh(ThCartSp)$ on the site ThCartSp $\subset CartSp Alg^{op}$ with coverage given by the families $\{U_i \times S \stackrel{(p,Id)}{\to} X \times S\}$, where $U \in$ CartSp, $S$ is an infinitesimal space (the dual of a Weil algebra) and where $\{U_i \to X\}$ is a good open cover in CartSp.

The $(\infty,1)$-Cahiers-topos is the (∞,1)-category of (∞,1)-sheaves on ThCartSp with respect to the good open cover coverage.

Remark

The good open cover coverage generates the Grothendieck topology of all open covers on CartSp. Therefore the sheaf toposes on $ThCartSp$ with covering families coming from all open covers of Cartesian spaces is equivalent to the sheaf topos on $ThCartSp$ with only good open covering.

By the discussioin at Cech localization of simplicial presheaves at a coverage, the analogous statement holds true for the (∞,1)-toposes over these sites.

Therefore we may model $Sh_{(\infty,1)}(ThCartSp_{good-open})$ by the left Bousfield localization of $[ThCartSp^{op}, sSet]_{proj}$ at the Cech nerves of all good open cover. Notice that the construction of good open covers (see there) on paracompact spaces (such as Cartesian spaces) by geodescally convex regions shows that we may always find a good open cover all whose finite non-empty intersections are diffeomorphic to an open ball, hence to a Cartesian space. We shall adopt for the present purposes therefore that a cover $\{U_i \to X\}$ is good if all finite intersections are isomorphic to Cartesian spaces.

The point is that with this definition, the Cech nerve $C(U) \in [ThCartSp^{op}, sSet]_{proj}$ is cofibrant, by the characterization of cofibrant objects in the projective model structure.

As a consequence of this, we have the following useful technical result.

Definition/Observation

Write $[ThCartSp^{op}, sSet]_{proj,cov}$ for the left Bousfield localization of the global projective model structure $[ThCartSp^{op}]_{proj}$ at the Cech nerves $C(U) \to X\times S$ of good open covers $\{U_i \times S \to X \times S\}$ in ThCartSp.

We have that

• this presents the $(\infty,1)$-Cahiers topos $Sh_{(\infty,1)}(ThCartSp) \simeq ([ThCartSp^{op}, sSet]_{proj,cov})$;

• the fibrant objects of $[ThCartSp^{op}, sSet]_{proj,cov}$ are precisely those fibrant objects $A \in [ThCartSp^{op}, sSet]_{proj}$ such that for all goop open covers $\{ U_i \times S \to X \times S\}$ with Cech nerve $p_U : C(U) \to X \times S$ we have that

  $$[ThCartSp^{op}, sSet]( p_U , A )$$

  is a weak equivalence (of Kan complexes).

Lemma

The Cech nerves projections $p_U : C(U) \to X \times S$ induce isomorphisms on the cohomology of their cosimplicial function algebras: $H^p(\mathcal{O}(p_U))$ is an isomorphism, for all $p \in \mathbb{N}$.

Proof

This is a standard fact about Cech cohomology. An explicit way to see it is to choose a smooth partition of unity subordinate to the cover. See Coboundaries for Cech cocycles.

This means that the assumptions of the Theorem on passage to the local model structure are satisfied.

Corollary

We have a simplicial Quillen adjunction

$$(Smooth Alg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} [ThCartSp^{op}, sSet]_{proj,cov} \,.$$

$\infty$-Lie algebroids

Definition

The objects of the $(\infty,1)$-Cahiers topos we call synthetic differential ∞-Lie groupoids.

The objects of the reflective sub-$(\infty,1)$-category of $R$-local objects in the $(\infty,1)$-Cahiers topos

$$\mathbf{L} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(ThCartSp)$$

we call ∞-Lie algebroids.

A connected $\infty$-Lie algebroid we call an ∞-Lie algebra.

(…)

Passing along the embedding $\mathbf{L} \hookrightarrow \mathbf{H}$ we may compute ∞-Lie algebra cohomology in $\mathbf{H}$.

(…)

The infinitesimal path $\infty$-groupoid of a manifold

(…)

For $U \in CartSp$ let

$$U^{\Delta^\bullet_{inf}} \in C^{\Delta^{op}}$$

be the simplicial object of infinitesimal simplices in $U$.

Definition

We call

$$\mathbf{\Pi}_{inf}(U) \coloneqq \mathbb{R}j\; (U^{\Delta^\bullet_{inf}}) \in [C^{op}, sSet]$$

the infinitesimal path $\infty$-Lie groupoid of $U$.

Or the path $\infty$-Lie algebroid .

(…)

The tangent category of smooth algebras

(…)

The tangent category of the category of smooth algebras is the category of modules over $C^\infty$-rings.

Proposition This abstract definition of module over $C^\infty$-rings reproduces the definition given by Kock.

The tangent category of the category of simplicial $C^\infty$-rings is …

This serves the purpose of presenting the $\infty$-stack of $\infty$-vector bundles on $T Alg^{op}$.

(…)

Appendix

Enrichment of categories of simplicial objects

We make use of the canonical structure of an sSet-enriched category on any category of cosimplicial objects in a category with all limits and colimits (see there).

Example

For $A \in (T Alg^{\Delta})^{op}$ and $S \in sSet$ we have that the tensoring is given by

$$(A \cdot S)_n = \prod_{S_s} A \in T Alg \,,$$

with the product taken in $T Alg$.

Model structure on cosimplicial abelian groups

We use the model category structure on $Ab^\Delta$ whose fibratin are the degreewise surjections, and whose weak equivalences are the usual quasi-isomorphisms under the dual Dold-Kan correspondence $Ab^\Delta \simeq Ch^\bullet_+(Ab)$.

The model structure is described in detail at model structure on chain complexes - the projective structure.

The structure of a simplicial model category is described in detail at model structure on cosimplicial abelian groups.

Related concepts

• geometric stack, geometric ∞-stack

• Lie integration

References

The Quillen adjunction over abelian $T$-algebras that we consider generalizes that discussed in

• Bertrand Toën, Champs affines, Selecta Math. new series 12 (2006), no. 1, 39-135 (arXiv:math/0012219, doi:10.1007/s00029-006-0019-z)

over ordinary commutative $k$-algebras. See also rational homotopy theory in an (infinity,1)-topos.

The generalization to arbitrary abelian $T$-algebras and the application to synthetic differential geometry is the content of

• Herman Stel, $\infty$-Stacks and their function algebras – with applications to $\infty$-Lie theory, MSc thesis, Utrecht 2010 (dspace:1874/203187, web)

• Herman Stel, Cosimplicial $C^\infty$-rings and the de Rham complex of Euclidean space (arXiv:1310.7407)

on which this entry here is based.

The considerations in

• David Spivak, Derived smooth manifolds, Duke Math. J. Volume 153, Number 1 (2010), 55-128. (pdf)

on derived smooth manifolds may be considered as complementary to the approach taken here: there simplicial $C^\infty$-rings are considered, instead of cosimplicial ones. A fully comprehensive treatment of derived synthetic differential geometry would consider the combination of both aspects: simplicial presheaves on duals of simplicial $C^\infty$-rings with a functor $\mathcal{O}$ taking them to cosimplicial-simplicial $C^\infty$-rings.

For ordinary commutative algebras the generalizaton of Toen’s setup to geometry over duals of simplicial algebras is used for instance in

• David Ben-Zvi, David Nadler, Loop spaces and connections (arXiv:1002.3636)