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function algebras on infinity-stacks

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(,1)(\infty,1)-Topos Theory

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structures in a cohesive (∞,1)-topos

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∞-Lie theory

Background

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Examples

\infty-Lie groupoids

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\infty-Lie algebroids

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Contents

Abstract

For TT 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 TT-algebras and on simplicial presheaves over duals of TT-algebras. We find mild general conditions under which this descends to the local model structure that models ∞-stacks over duals of TT-algebras. In these cases the left adjoint of the Quillen adjunction is given by sending \infty-stacks to their cosimplicial TT-algebras of functions with values in the canonical TT-line object, and the adjunction models small objects relative to a choice of a small full subcategory TCTAlg opT\subset C \subset T Alg^{op} of the localization

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

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

For the special case where TT 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 TT is the theory of smooth algebras (C 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 H\mathbf{H} may be understood as ∞-Lie groupoids, the objects in L\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 TT-algebras

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

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

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

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

Examples.

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

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

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

A morphism of Lawvere theories T 1T 2T_1 \to T_2 is again a product-preserving functor.

Definition

An abelian Lawvere theory TT is a morphism of Lawvere theories ab T:𝒜𝒷Tab_T : \mathcal{Ab} \to T.

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

ab T *:TAlgAb. 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:TAlgSetU_T : T Alg \to Set has a left adjoint, the free TT-algebra functor F T:SetTAlgF_T : Set \to T Alg. By the Yoneda lemma this sends the nn-element set (n)(n) to F T(n):kT(n,k)F_T(n) : k \mapsto T(n,k).

More generally, for any ATAlgA \in T Alg the copresheaf

TAlg(F T(),A):TSet T Alg(F_T(-), A) : T \to Set

is isomorphic to AA.

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

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

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

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

Model structure on cosimplicial TT-algebras

Theorem

There is a cofibrantly generated model structure on cosimplicial abelian groups Ab proj ΔAb^\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 ΔAb^{\Delta}, this is a simplicial model category.

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

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

induces a transferred model structure TAlg proj ΔT Alg^{\Delta}_{proj} on the category of cosimplicial TT-algebras, whose weak equivalences and fibrations are those morphisms that under (ab *) Δ(ab^*)^\Delta become weak equivalences and fibrations, respectively, in Ab proj ΔAb^\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 + (Ab)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 ΔCh + (Ab)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 ΔAb^\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 1T 2f : T_1 \to T_2 of such induces a pair of adjoint functors

(f *f *):T 2Algf *f *T 1Alg (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 TT is an abelian Lawvere theory we are given a morphism ab:AbTab : Ab \to T from the theory of abelian groups, this means that we have an adjunction

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

and hence also an adjunction

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

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

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

  • has a fibrant replacement functor;

  • has functorial path space objects for fibrant objects;

and for the simplicial enrichment that

  • (ab *) Δ(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( S nA n)= S nU T(A n)=((U T(A)) S) n. 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:AA Δ[1]. (-)^I : A \mapsto A^{\Delta[1]} \,.

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

AA IA×A 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 ()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 TT-algebras

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

We are interested here in that case that

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

is a small full subcategory of the opposite category of TT-algebras, for TT 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 Δ op(TAlg Δ) op. C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op} \,.

The prolonged Yoneda embedding

Write

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

for the ordinary Yoneda embedding and

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

for its degreewise simplicial prolongation

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

For BTAlgB \in T Alg and (TAlg Δ) s(T Alg^\Delta)_s denoting the simplicially enriched category of TT-algebras, we have a natural identification

j(A)(TAlg Δ) s op(,A). j(A) \simeq (T Alg^\Delta)^{op}_s(-, A) \,.
Proof

Using end/coend-calculus for handling the canonical enrichment of TAlg ΔT Alg^\Delta, we have for BTAlg op(TAlgΔ) opB \in T Alg^{op} \hookrightarrow (T Alg \Delta)^{op} and A(TAlg Δ) opA \in (T Alg^\Delta)^{op} natural isomorphisms

(TAlg Δ) s op(B,A) n (TAlg Δ) op(BΔ n,A) kΔTAlg(A k, Δ(k,n)B) kΔTAlg(Δ(k,n)A k,B) TAlg( kΔΔ(k,n)A k,B) TAlg(A n,B), \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)

kΔΔ(k,n)A klim (Δ/nΔATAlg)A n. \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(*)F_T(*) for the free TT-algebra on a single generator.

Definition

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

Observation

As a presheaf, the line object RR sends a TT-algebra BTAlgB \in T Alg to its underlying set U T(B)U_T(B)

R:BTAlg(F T(*),B)Set(*,U T(B))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 RR produces TT-algebras of functions . This is what the following definition deals with.

Definition

(TT-algebras of functions)

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

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

we call the cosimplicial set of RR-valued functions on XX. This is naturally the cosimplical set underlying the cosimplicial TT-algebra

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

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

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

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

Model structure on simplicial presheaves

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

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

Definition

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

By general results on left Bousfield localization, this exists always for SS a small set, notably for ff the set of Cech nerve projections C(U)XC(U) \to X for covers {U iX}\{U_i \to X\} of the Grothendieck topology on CC. By general results on the local model structure on simplicial presheaves, the localization also exists for SS 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 CTAlg opC \subset T Alg^{op} using the function algebra functor 𝒪\mathcal{O} and the prolonged Yoneda embedding jj.

Theorem

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

(𝒪j):(TAlg proj Δ) opj𝒪[C op,sSet] proj. (\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[C op,sSet]X \in [C^{op}, sSet] and ATAlg ΔA \in T Alg^\Delta natural isomorphisms

(TAlg Δ) op(𝒪(X),A) TAlg Δ(A(),[C op,sSet](X,j(F T()))) kT [n]ΔSet(A n(k),[C op,Set](X n,j(F T(k)))) kT [n]Δ BCSet(A n(k),Set(X n(B),TAlg(F T(k),B))) kT [n]Δ BCSet(A n(k),Set(X n(B),B(k))) kT [n]ΔSet(X n(B), kTSet(A n(k),B(k))) kT [n]ΔSet(X n(B),TAlg(A n,B)) [C op,sSet](X,j(A))., \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()TAlg(F T(),B)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 sSetsSet-enriched adjunction follows with the prolonged Yoneda lemma j(A)(TAlg Δ) s op(,A)j(A) \simeq (T Alg^\Delta)^{op}_s(-,A) and the sSetsSet-tensoring and cotensoring of [C op,sSet] s[C^{op}, sSet]_s and (TAlg Δ) s op(T Alg^\Delta)^{op}_s:

(TAlg Δ) s op(𝒪(X),A) n (TAlg Δ) op(𝒪(X),A Δ n) [C op,sSet](X,j(A Δ n)) [C op,sSet](X,(TAlg Δ) s op(,A Δ n)) BCsSet(X(B),(TAlg Δ) s op(B,A Δ n))) BCsSet(X(B),(TAlg Δ) s op(BΔ n,A))) BCsSet(X(B)×Δ n,(TAlg Δ) s op(B,A))) [C op,sSet](XΔ n,j(A)) =:[C op,sSet] n(X,j(A)) n. \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 sSetsSet-enriched model category (TAlg Δ) s(T Alg^\Delta)_s and using that in (TAlg proj Δ) op(T Alg^\Delta_{proj})^{op} every object BB is cofibrant, we have that for f:A 1A 2f : A_1 \to A_2 a fibration (acyclic fibration) in (TAlg proj Δ) op(T Alg^\Delta_{proj})^{op} and for BCTAlg opB \in C \subset T Alg^{op} any object, the morphism j(A 1A 2)(B)=(TAlg Δ) s op(B,f)j(A_1 \to A_2)(B) = (T Alg^{\Delta})^{op}_s(B,f) is a fibration (acyclic fibration) in sSetsSet. Therefore j(f)j(f) is a fibration (acyclic fibration) in [C op,sSet] proj[C^{op}, sSet]_{proj}.

This establishes that jj 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 JJ be a subcanonical coverage on C(TAlg Δ) opC \subset (TAlg^\Delta)^{op}, XOb(C)X \in Ob(C) and f:Yj(X)f : Y \to j(X) a split hypercover with respect to JJ.

Then for i1i \neq 1 we have that ff induces an isomorphism in RR-cohomology in degree ii: H i(𝒪(f)):H i(𝒪(X))H i(𝒪(Y))H^i(\mathcal{O}(f)) : H^i(\mathcal{O}(X)) \stackrel{\simeq}{\to} H^i(\mathcal{O}(Y)) .

Proof

Regard ff as a simplicial object in the overcategory

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

Write

f¯Ab(Sh(C)/X) Δ op \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 CC. The chain homology of the corresponding normalized chain complex vanishes in positive degree (as discussed here):

H n1(f¯)=0. H_{n \geq 1}(\bar f) = 0 \,.

Let now by the Freyd-Mitchell embedding theorem

i:Ab(Sh(C/Y))RMod 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 RR-module over some ring RR.

Write KRModK \in R Mod for the canonical TT-line object regarded first as the abelian group object U T()×XAb(Sh(C/X))U_T(-) \times X \in Ab(Sh(C/X)) and then injected with ii into RModR Mod.

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

𝒪(Y) Sh(C) Δ(Y ,U T()) Sh(C)/X(f ,U T()×X) Ab(Sh(C)/X)(f¯ ,U T()×X) RMod(i(f¯ ),i(U T()×X)) RMod(i(f¯ ),K). \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

0Ext 1(H n1(i(f¯ ),K))H n(RMod(i(f¯ ),K))Ab(H n(i(f¯ )),C)0. 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(f¯))H_n(i(\bar f)) vanishes in positive degree, this gives finally that

H n(𝒪(Y)) H^n(\mathcal{O}(Y))

vanishes in degree n2n \geq 2. That it also vanishes in degree 0 is seen to be equivalent to the sheaf condition on XX, 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 fSf \in S we have that H 1(𝒪(f))H^1(\mathcal{O}(f)) is an isomorphisms then (𝒪j)(\mathcal{O} \dashv j) is a simplicial Quillen adjunction to the local model structure on simplicial presheaves.

(𝒪j):(TAlg proj Δ) opj𝒪[C op,sSet] proj,loc. (\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:YXf : Y\to X in [C op,sSet][C^{op}, sSet] is taken by 𝒪\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[C^{op}, sSet]_{proj,loc} is left proper. Since moreover left Bousfield localization does not change the class of cofibrations, we know that 𝒪\mathcal{O} still preserves cofibrations.

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

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 ff and fibrant cc we have that [C op,sSet] s(f,j(c))[C^{op}, sSet]_s(f, j(c)) is a weak equivalence. But by the adjunction (𝒪j)(\mathcal{O} \dashv j) this is isomorphically (TAlg Δ) op(𝒪(f),c)(T Alg^\Delta)^{op}(\mathcal{O}(f), c).

Now by the above propositions and assumptions, we have that 𝒪(f)\mathcal{O}(f) is a weak equivalence. Since all objects in (TAlg proj Δ) op(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 Δ op(TAlg Δ) opC^{\Delta^{op}} \subset (T Alg^\Delta)^{op} the functor jj is homotopy full and faithful in that for all AC Δ opA \in C^{\Delta^{op}} we have that the canonical morphism

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

into the image of the derived functors of jj and 𝒪\mathcal{O} is an isomorphism in the homotopy category Ho(TAlg proj Δ)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 (,1)(\infty,1)-topos at RR-cohomology

We consider now the cohomology localization of Sh (,1)(C)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 (𝒪j)(\mathcal{O} \dashv j) is a presentation that models CC-small objects (…) in the reflective sub-(∞,1)-category

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

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

We obtain a proof of this after the following discussions.

Remark

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

RR-Cohomology

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

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

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

Proposition

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

B nR chnj(ab *Ξ(F Ab(*)[n]). \mathbf{B}^n R_{chn} \coloneqq j(ab_* \Xi(F_{Ab}(*)[n]) \,.

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

H n(X,A)π 0H(X,B nA). 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[C^{op}, sSet]_{proj,loc} and writing X[C op,sSet]X \in [C^{op}, sSet] for a representative of XHX \in \mathbf{H} this is the hom-set in the homotopy category

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

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

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

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

Using this, we compute as follows

H(X,B nR) =Ho [C op,sSet] proj(X,j(ab *Ξ([n])) Ho (TAlg proj Δ) op(𝕃𝒪(X),ab *Ξ([n]) Ho (TAlg proj Δ)(ab *Ξ([n]),𝕃𝒪(X)) Ho (Ab proj Δ)(Ξ([n]),ab *𝕃𝒪(X)) Ho Ch ([n],N ab *𝕃𝒪(X)) H n(𝕃𝒪(X)) \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).

RR-Local objects

Definition

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

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

By the above proposition this is equivalent to saying that the derived functor 𝕃𝒪\mathbb{L}\mathcal{O} takes ff to a weak equivalence.

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

H(f,K):H(Y,K)H(X,K) \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 Hom [C op,sSet] proj,loc(f,K)\mathbb{R}Hom_{[C^{op}, sSet]_{proj,loc}}(f,K) (of Kan complexes).

Proposition

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

C Δ op(TAlg Δ) opj[C op,sSet] proj,cov. 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 (𝒪j)(\mathcal{O} \dashv j)-adjunction that we have established above, as well as the fact that TAlg proj ΔT Alg^{\Delta}_{proj} is a cofibrantly generated model category for TT the theory of ordinary commutative algebras. But by our result on the model structure on TAlg we have that for general TT 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 TAlg proj ΔT 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-(,1)(\infty,1)-category of ((TAlg proj Δ) op) ((T Alg^\Delta_{proj})^{op})^\circ on the objects in C Δ op(TAlg Δ) opC^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op} is a reflective sub-(,1)(\infty,1)-category

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

and that the left adjoint to the embedding inverts precisely the RR-equivalences. Hence L\mathbf{L} is the full sub-(,1)(\infty,1)-category of H\mathbf{H} on RR-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 TT-algebras to an (∞,1)-site inside the opposite of an (∞,1)-category of ∞-algebras over an (∞,1)-algebraic theory.

Over ordinary associative algebras

Let kk be a field of characteristic 00.

Definition

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

Write

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

Set

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

Write

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

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

Remark

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

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

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

𝒪Xlim i𝒪(U i)(cdgAlg k) , \mathcal{O}X \simeq {\lim_\leftarrow}_i \mathcal{O}(U_i) \;\; \in (cdgAlg_k)^\circ \,,

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

Observation

By construction 𝒪\mathcal{O} is a colimit-preserving (,1)(\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) Sh (,1)(C). j : ((cdgAlg_k^{op})^\circ \to Sh_{(\infty,1)}(C) \,.

This is given by

Spec(A):U((cdgAlg k op) (U,A). 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

Sh C(X,SpecA) H(lim iU i,SpecA) lim iH(U i,SpecA) lim iC(U i,A) lim i(cdgAlg k) (A,𝒪(U i)) (cdgAlg k) (A,lim i𝒪(U i)) (cdgAlg k op) (lim i𝒪(U i),A). \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 TT the theory of commutative kk-algebras is compatible with this in that it also does model the (,1)(\infty,1)-Yoneda extension of the inclusion

TAlg k op(TAlg k Δ) op 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[C op,sSet] proj,locX \in [C^{op}, sSet]_{proj,loc} has a cofibrant resolution of the form [k]ΔΔ[k] i nU i n\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] i nU i n[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 Δ\mathbf{\Delta} are Reedy cofibrant cosimplicial simplicial sets, this is moreover equivalent to [k]Δ[k] i nU i n\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} U_{i_n} and this is still cofibrant. Now the left Quillen functor 𝒪\mathcal{O} takes this to [k]Δ[k] i n𝒪(U i n)\int^{[k]} \mathbf{\Delta}[k] \cdot \coprod_{i_n} \mathcal{O}(U_{i_n}). Since every object in (TAlg Δ) op(T Alg^{\Delta})^{op} is cofibrant, this coend is still a homotopy colimit.

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

  • TT the theory of ordinary commutative algebras over a field kk and JJ the fpqc topology.

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

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

Rational homotopy theory

TT the Lawvere theory of \mathbb{Q}-algebras. Then (𝒪j)(\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

Write SmoothAlgTAlgSmooth Alg \coloneqq T Alg for the category of smooth algebras. Sheaf toposes on sub-sites CSmoothAlg opC \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

Definition

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

The (,1)(\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 ThCartSpThCartSp with covering families coming from all open covers of Cartesian spaces is equivalent to the sheaf topos on ThCartSpThCartSp 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 (,1)(ThCartSp goodopen)Sh_{(\infty,1)}(ThCartSp_{good-open}) by the left Bousfield localization of [ThCartSp op,sSet] proj[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 iX}\{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)[ThCartSp op,sSet] projC(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[ThCartSp^{op}, sSet]_{proj,cov} for the left Bousfield localization of the global projective model structure [ThCartSp op] proj[ThCartSp^{op}]_{proj} at the Cech nerves C(U)X×SC(U) \to X\times S of good open covers {U i×SX×S}\{U_i \times S \to X \times S\} in ThCartSp.

We have that

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

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

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

    is a weak equivalence (of Kan complexes).

Lemma

The Cech nerves projeections p U:C(U)X×Sp_U : C(U) \to X \times S induce isomorphisms on the cohomology of their cosimplicial function algebras: H p(𝒪(p U))H^p(\mathcal{O}(p_U)) is an isomorphism, for all pp \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

(SmoothAlg proj Δ) opj𝒪[ThCartSp op,sSet] proj,cov. (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 (,1)(\infty,1)-Cahiers topos we call synthetic differential ∞-Lie groupoids.

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

LH=Sh (,1)(ThCartSp) \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 LH\mathbf{L} \hookrightarrow \mathbf{H} we may compute ∞-Lie algebra cohomology in H\mathbf{H}.

(…)

The infinitesimal path \infty-groupoid of a manifold

(…)

For UCartSpU \in CartSp let

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

be the simplicial object of infinitesimal simplices in UU.

Definition

We call

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

the infinitesimal path \infty-Lie groupoid of UU.

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 C^\infty-rings.

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

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

This serves the purpose of presenting the \infty-stack of \infty-vector bundles on TAlg opT 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(TAlg Δ) opA \in (T Alg^{\Delta})^{op} and SsSetS \in sSet we have that the tensoring is given by

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

with the product taken in TAlgT Alg.

Model structure on cosimplicial abelian groups

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

References

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

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

The generalization to arbitrary abelian TT-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 , master thesis (2010) (web)

  • Herman Stel, Cosimplicial C 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 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 C^\infty-rings with a functor 𝒪\mathcal{O} taking them to cosimplicial-simplicial C 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

Revised on August 3, 2014 22:47:59 by David Corfield (46.208.114.209)