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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{function algebras on infinity-stacks} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - contents]] \hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry} [[!include higher geometry - contents]] \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - contents]] \hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory} [[!include infinity-Lie theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{abstract}{Abstract}\dotfill \pageref*{abstract} \linebreak \noindent\hyperlink{Models}{Models for $\infty$-stacks and their function algebras}\dotfill \pageref*{Models} \linebreak \noindent\hyperlink{cosimplicial_algebras}{Cosimplicial $T$-algebras}\dotfill \pageref*{cosimplicial_algebras} \linebreak \noindent\hyperlink{TAlgebras}{$T$-Algebras}\dotfill \pageref*{TAlgebras} \linebreak \noindent\hyperlink{ModelTAlg}{Model structure on cosimplicial $T$-algebras}\dotfill \pageref*{ModelTAlg} \linebreak \noindent\hyperlink{simplicial_presheaves_on_duals_of_algebras}{Simplicial presheaves on duals of $T$-algebras}\dotfill \pageref*{simplicial_presheaves_on_duals_of_algebras} \linebreak \noindent\hyperlink{ProlongedYoneda}{The prolonged Yoneda embedding}\dotfill \pageref*{ProlongedYoneda} \linebreak \noindent\hyperlink{Line}{The line object}\dotfill \pageref*{Line} \linebreak \noindent\hyperlink{ModelPresheaves}{Model structure on simplicial presheaves}\dotfill \pageref*{ModelPresheaves} \linebreak \noindent\hyperlink{YonedaQuillenAdjunction}{The Yoneda-Quillen-adjunction}\dotfill \pageref*{YonedaQuillenAdjunction} \linebreak \noindent\hyperlink{Intrinsic}{Localization of the $(\infty,1)$-topos at $R$-cohomology}\dotfill \pageref*{Intrinsic} \linebreak \noindent\hyperlink{cohomology}{$R$-Cohomology}\dotfill \pageref*{cohomology} \linebreak \noindent\hyperlink{local_objects}{$R$-Local objects}\dotfill \pageref*{local_objects} \linebreak \noindent\hyperlink{DerivedGeometry}{In derived geometry}\dotfill \pageref*{DerivedGeometry} \linebreak \noindent\hyperlink{over_ordinary_associative_algebras}{Over ordinary associative algebras}\dotfill \pageref*{over_ordinary_associative_algebras} \linebreak \noindent\hyperlink{Examples}{Examples and applications}\dotfill \pageref*{Examples} \linebreak \noindent\hyperlink{rational_homotopy_theory}{Rational homotopy theory}\dotfill \pageref*{rational_homotopy_theory} \linebreak \noindent\hyperlink{lie_theory_in_the_cahiers_topos}{$\infty$-Lie theory in the $\infty$-Cahiers topos}\dotfill \pageref*{lie_theory_in_the_cahiers_topos} \linebreak \noindent\hyperlink{lie_algebroids}{$\infty$-Lie algebroids}\dotfill \pageref*{lie_algebroids} \linebreak \noindent\hyperlink{the_infinitesimal_path_groupoid_of_a_manifold}{The infinitesimal path $\infty$-groupoid of a manifold}\dotfill \pageref*{the_infinitesimal_path_groupoid_of_a_manifold} \linebreak \noindent\hyperlink{the_tangent_category_of_smooth_algebras}{The tangent category of smooth algebras}\dotfill \pageref*{the_tangent_category_of_smooth_algebras} \linebreak \noindent\hyperlink{appendix}{Appendix}\dotfill \pageref*{appendix} \linebreak \noindent\hyperlink{Enrichment}{Enrichment of categories of simplicial objects}\dotfill \pageref*{Enrichment} \linebreak \noindent\hyperlink{model_structure_on_cosimplicial_abelian_groups}{Model structure on cosimplicial abelian groups}\dotfill \pageref*{model_structure_on_cosimplicial_abelian_groups} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{abstract}{}\subsection*{{Abstract}}\label{abstract} For $T$ any abelian [[Lawvere theory]], here we discuss -- in a variation of the theme of [[Isbell conjugation]], in generalization of (\hyperlink{Toen}{To\"e{}n}) and following (\hyperlink{Stel}{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 \begin{displaymath} \mathbf{L} \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(C ) \end{displaymath} 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 (\hyperlink{Toen}{To\"e{}n}) and many statements are straightforward generalizations from that situation. For the case that $T$ is the theory of \emph{[[smooth algebra]]s} ($C^\infty$-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 \emph{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. \hypertarget{Models}{}\subsection*{{Models for $\infty$-stacks and their function algebras}}\label{Models} \hypertarget{cosimplicial_algebras}{}\subsubsection*{{Cosimplicial $T$-algebras}}\label{cosimplicial_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 ∞-groupoid]]s in such spaces are [[cosimplicial object|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 group]]s. \hypertarget{TAlgebras}{}\paragraph*{{$T$-Algebras}}\label{TAlgebras} A [[Lawvere theory]] may be thought of as a generalization of the theory of ordinary [[associative algebra]]s. A Lawvere theory is encoded in its [[syntactic category]] $T$, which by definition is a category with finite [[product]]s 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 \textbf{$T$-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. \textbf{Examples}. \begin{itemize}% \item The category $T = \mathcal{Ab}$ of free finitely generated abelian groups is the syntactic category of the Lawvere theory whose algebras are abelian groups. \item For $k$ a field, the category $T = k$ of free finitely generated $k$-algebras is the Lawvere theory whose algebras are $k$-[[associative algebra]]s; \item The category $T =$ [[CartSp]] is the syntactic category whose algebras are [[smooth algebra]]s. \end{itemize} A morphism of Lawvere theories $T_1 \to T_2$ is again a product-preserving functor. \begin{defn} \label{}\hypertarget{}{} An \textbf{abelian Lawvere theory} $T$ is a morphism of Lawvere theories $ab_T : \mathcal{Ab} \to T$. \end{defn} For $T$ abelian, $T$-algebras have an underlying [[abelian group]], given by the functor \begin{displaymath} ab_T^* : T Alg \to Ab \,. \end{displaymath} 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 \begin{displaymath} T Alg(F_T(-), A) : T \to Set \end{displaymath} 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 \textbf{$T$-line}. For instance \begin{itemize}% \item for $T = k$ we have that $F_T(1) = k[X]$ is the free $k$-algebra on a single generator $X$; \item for $T = CartSp$ we have that $F_T(1) = C^\infty(\mathbb{R})$. \end{itemize} We say more on the canonical $T$-[[line object]] below in \hyperlink{Line}{The Line object} \hypertarget{ModelTAlg}{}\paragraph*{{Model structure on cosimplicial $T$-algebras}}\label{ModelTAlg} \begin{theorem} \label{ModelTransferToTAlg}\hypertarget{ModelTransferToTAlg}{} There is a [[cofibrantly generated model category|cofibrantly generated]] [[model structure on cosimplicial abelian groups]] $Ab^\Delta_{proj}$ whose weak equivalences are the morphisms that induce [[quasi-isomorphism]] under passage to [[Dold-Kan correspondence|normalized cochain complexes]] and fibrations are the degreewise surjections. With respect to the \hyperlink{Enrichment}{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]] \begin{displaymath} ((ab_*)^\Delta \dashv (ab^*)^\Delta ) : T Alg^{\Delta} \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab^\Delta \end{displaymath} 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 \hyperlink{Enrichment}{standard sSet-enrichment}. \end{theorem} \begin{proof} The proof of the existence of the [[model structure on cochain complexes]] in non-negative degree -- $Ch^\bullet_+(Ab)$ -- whose fibrations are the \emph{degreewise surjections} (and weak equivalences the usual [[quasi-isomorphism]])s is spelled out . 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 [[Moore complex|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 . 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 functor]]s \begin{displaymath} (f_* \dashv f^*) : T_2 Alg \stackrel{\overset{f_*}{\leftarrow}}{\underset{f^*}{\to}} T_1 Alg \end{displaymath} between their categories of algebras: the . 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 group]]s, this means that we have an [[adjunction]] \begin{displaymath} (ab_* \dashv ab^*) : T Alg \stackrel{\overset{ab_*}{\leftarrow}}{\underset{ab^*}{\to}} Ab \end{displaymath} and hence also an adjunction \begin{displaymath} (ab_*^\Delta \dashv (ab^*)^\Delta) : T Alg^\Delta \stackrel{\overset{}{\leftarrow}}{\underset{}{\to}} Ab^\Delta \,. \end{displaymath} 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 $T Alg^\Delta_{proj}$ \begin{itemize}% \item has a fibrant replacement functor; \item has functorial [[path space object]]s for fibrant objects; \end{itemize} and for the simplicial enrichment that \begin{itemize}% \item $(ab^*)^\Delta$ preserves the [[power]]ing. \end{itemize} The first condition is trivial, since all objects are fibrant. The last condition is evidently satisfied, since \begin{displaymath} U_T(A^S)_n = U_T(\prod_{S_n} A_n) = \prod_{S_n} U_T(A_n) = ((U_T(A))^S)_n \,. \end{displaymath} Using this, we claim that we can take the path space object functor to be given by [[power]]ing with the simplicial [[interval]] \begin{displaymath} (-)^I : A \mapsto A^{\Delta[1]} \,. \end{displaymath} 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 \begin{displaymath} A \to A^I \to A \times A \end{displaymath} 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). \end{proof} \hypertarget{simplicial_presheaves_on_duals_of_algebras}{}\subsubsection*{{Simplicial presheaves on duals of $T$-algebras}}\label{simplicial_presheaves_on_duals_of_algebras} A good notion of a generalized [[space]] modeled on objects in a category $C$ is a [[nLab:sheaf]] on $C$. A good notion of an [[nLab:∞-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 \begin{displaymath} T \subset C \hookrightarrow T Alg^{op} \end{displaymath} is a [[small category|small]] [[full subcategory]] of the [[nLab: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 \hyperlink{Examples}{Examples and applications} we discuss concrete choices of interest. Notice that such a choice induces also a full subcategory of (co)simplicial objects \begin{displaymath} C^{\Delta^{op}} \hookrightarrow (T Alg^\Delta)^{op} \,. \end{displaymath} \hypertarget{ProlongedYoneda}{}\paragraph*{{The prolonged Yoneda embedding}}\label{ProlongedYoneda} Write \begin{displaymath} j : T Alg^{op} \to [C^{op}, Set] \end{displaymath} for the ordinary [[nLab:Yoneda embedding]] and \begin{displaymath} j : (T Alg^\Delta)^{op} \to [C^{op}, sSet] \end{displaymath} for its degreewise simplicial prolongation \begin{displaymath} j(A) : (B \in T Alg) \mapsto ([n] \mapsto T Alg(A_n, B)) \,. \end{displaymath} \begin{lemma} \label{}\hypertarget{}{} For $B \in T Alg$ and $(T Alg^\Delta)_s$ denoting the \hyperlink{Enrichment}{simplicially enriched} category of $T$-algebras, we have a natural identification \begin{displaymath} j(A) \simeq (T Alg^\Delta)^{op}_s(-, A) \,. \end{displaymath} \end{lemma} \begin{proof} Using [[nLab:end]]/[[nLab:coend]]-calculus for handling the \hyperlink{Enrichment}{canonical enrichment} of $T Alg^\Delta$, we have for $B \in T Alg^{op} \hookrightarrow (T Alg \Delta)^{op}$ and $A \in (T Alg^\Delta)^{op}$ [[nLab:natural transformation|natural]] isomorphisms \begin{displaymath} \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} \end{displaymath} where in the last step we used the isomorphism (described at [[coend]]) \begin{displaymath} \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 \,. \end{displaymath} \end{proof} \hypertarget{Line}{}\paragraph*{{The line object}}\label{Line} The adjunction that we shall be concerned with is essentially [[Isbell conjugation]]. We recall some basics of . Recall from \hyperlink{TAlgebras}{above} that we write $F_T(*)$ for the free $T$-algebra on a single generator. \begin{defn} \label{}\hypertarget{}{} We call $R \coloneqq j(F_T(*))$ the \textbf{line object} in $[C^{op}, sSet]$. \end{defn} \begin{lemma} \label{}\hypertarget{}{} As a presheaf, the line object $R$ sends a $T$-algebra $B \in T Alg$ to its underlying set $U_T(B)$ \begin{displaymath} R : B \mapsto T Alg(F_T(*), B) \simeq Set(*, U_T(B)) \simeq U_T(B) \,. \end{displaymath} \end{lemma} This characterization may look simpler, but does not capture the important fact that homming into $R$ produces \emph{$T$-algebras of functions} . This is what the following definition deals with. \begin{defn} \label{}\hypertarget{}{} \textbf{($T$-algebras of functions)} For $X \in [C^{op}, sSet]$, the [[nLab:cosimplicial set]] \begin{displaymath} U_T(\mathcal{O}(X)) \coloneqq [C^{op},sSet](X_\bullet, R) \in Set \end{displaymath} we call the cosimplicial set of $R$-valued \emph{functions} on $X$. This is naturally the cosimplical set underlying the cosimplicial $T$-algebra \begin{displaymath} \mathcal{O}(X) : (k \in T) \mapsto [C^{op},sSet](X_\bullet, j(F_T(k))) \,. \end{displaymath} We call $\mathcal{O}(X) \in T Alg^{op}$ the \textbf{$T$-algebra of functions} on $X$. This extends to a functor \begin{displaymath} \mathcal{O} : [C^{op}, sSet] \to (T Alg^{\Delta})^{op} \,. \end{displaymath} \end{defn} In the next section we see that $(\mathcal{O} \dashv j)$ forms a [[simplicial Quillen adjunction]]. \hypertarget{ModelPresheaves}{}\paragraph*{{Model structure on simplicial presheaves}}\label{ModelPresheaves} 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 hypercover]]s. \begin{defn} \label{}\hypertarget{}{} Write $[C^{op}, sSet]_{proj,loc}$ for the [[left Bousfield localization]] $[C^{op}, sSet]_{proj}$ at this class. \end{defn} 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 [[cover]]s $\{U_i \to X\}$ 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. \hypertarget{YonedaQuillenAdjunction}{}\subsubsection*{{The Yoneda-Quillen-adjunction}}\label{YonedaQuillenAdjunction} We relate now the \hyperlink{ModelTAlg}{model structure on cosimplicial T-algebras} with the \hyperlink{ModelPresheaves}{model structure on simplicial presheaves} over $C \subset T Alg^{op}$ using the \hyperlink{Line}{function algebra functor} $\mathcal{O}$ and the \hyperlink{ProlongedYoneda}{prolonged Yoneda embedding} $j$. \begin{theorem} \label{}\hypertarget{}{} The functors $j$ and $\mathcal{O}$ constitute a [[simplicial Quillen adjunction]] \begin{displaymath} (\mathcal{O} \dashv j) \;\colon\; (TAlg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}} [C^{op}, sSet]_{proj} \,. \end{displaymath} \end{theorem} \begin{proof} We first establish the [[nLab:adjunction]] itself: using [[nLab:end]]-calculus for expressing [[nLab:hom-set]]s in [[nLab:functor category|functor categories]] we have for $X \in [C^{op}, sSet]$ and $A \in T Alg^\Delta$ [[nLab:natural transformation|natural]] isomorphisms \begin{displaymath} \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} \,, \end{displaymath} where the crucial step is the isomorphism $B(-) \simeq T Alg(F_T(-), B)$ for the line object discussed \hyperlink{Line}{above}. This computation is just simplicial-degreewise the adjunction discussed at . That this lifts to an $sSet$-enriched adjunction follows with the \hyperlink{ProlongedYoneda}{prolonged Yoneda lemma} $j(A) \simeq (T Alg^\Delta)^{op}_s(-,A)$ and the $sSet$-[[nLab:tensoring]] and [[nLab:cotensoring]] of $[C^{op}, sSet]_s$ and $(T Alg^\Delta)^{op}_s$: \begin{displaymath} \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} \end{displaymath} By the pushout-product axiom satisfied by the $sSet$-[[nLab: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. \end{proof} The following theorems say that the obstructions to making this Quillen adjunction descent to [[nLab:local model structures on simplicial presheaves]] are mild. \begin{prop} \label{}\hypertarget{}{} 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 $i$: $H^i(\mathcal{O}(f)) : H^i(\mathcal{O}(X)) \stackrel{\simeq}{\to} H^i(\mathcal{O}(Y))$ . \end{prop} \begin{proof} Regard $f$ as a [[simplicial object]] in the [[overcategory]] \begin{displaymath} Sh(C)/X \simeq Sh(C/X) \,. \end{displaymath} Write \begin{displaymath} \bar f \in Ab(Sh(C)/X)^{\Delta^{op}} \end{displaymath} 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 ): \begin{displaymath} H_{n \geq 1}(\bar f) = 0 \,. \end{displaymath} Let now by the [[Freyd-Mitchell embedding theorem]] \begin{displaymath} i : Ab(Sh(C/Y)) \hookrightarrow R Mod \end{displaymath} 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 $T$-\hyperlink{Line}{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{displaymath} \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} \,. \end{displaymath} To compute this, we use the [[universal coefficient theorem]], which says that we have an [[exact sequence]] \begin{displaymath} 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 \,. \end{displaymath} By the above fact that the homology $H_n(i(\bar f))$ vanishes in positive degree, this gives finally that \begin{displaymath} H^n(\mathcal{O}(Y)) \end{displaymath} 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]]. \end{proof} \begin{theorem} \label{}\hypertarget{}{} \textbf{(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]]. \begin{displaymath} (\mathcal{O} \dashv j) : (TAlg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\hookrightarrow}} [C^{op}, sSet]_{proj, loc} \,. \end{displaymath} \end{theorem} \begin{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 \hyperlink{Toen}{To\"e{}n, 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 [[proper model category|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 it is sufficient to check that $j$ sends fibrant objects $A \in (T Alg^\Delta_{proj})^{op}$ to [[local object]]s 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. \end{proof} 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 \hyperlink{Intrinsic}{Localization of the (∞,1)-topos at R-cohomology} we see that this indeed presents such a reflective inclusion in [[(∞,1)-category theory]]. \begin{theorem} \label{}\hypertarget{}{} When restricted along $C^{\Delta^{op}} \subset (T Alg^\Delta)^{op}$ the functor $j$ is \emph{homotopy full and faithful} in that for all $A \in C^{\Delta^{op}}$ we have that the canonical morphism \begin{displaymath} A \to \mathbb{L}\mathcal{O} \; \mathbb{R}j \; A \end{displaymath} into the image of the [[derived functor]]s of $j$ and $\mathcal{O}$ is an [[isomorphism]] in the [[homotopy category]] $Ho(T Alg^\Delta_{proj})$. \end{theorem} \begin{proof} With the above results, this follows verbatim as the proof of the analogous (\hyperlink{Toen}{To\"e{}n, corollary 2.2.3}). \end{proof} \hypertarget{Intrinsic}{}\subsection*{{Localization of the $(\infty,1)$-topos at $R$-cohomology}}\label{Intrinsic} 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 [[presentable (∞,1)-category|presented]] by the model category theoretic structures \hyperlink{Models}{above}, these serve to establish the following intrinsic statement. \begin{theorem} \label{}\hypertarget{}{} The Quillen adjunction $(\mathcal{O} \dashv j)$ is a [[nLab:presentable (∞,1)-category|presentation]] that models $C$-small objects (\ldots{}) in the [[reflective sub-(∞,1)-category]] \begin{displaymath} \mathbf{L}_C \stackrel{\stackrel{\mathcal{O}}{\leftarrow}}{\hookrightarrow} \mathbf{H} \coloneqq Sh_{(\infty,1)}(C) \end{displaymath} of the [[nLab:(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(C)$, where $\mathbb{L}_C$ is the [[nLab:localization of an (∞,1)-category|localization]] at those morphisms that induce isomorphisms in intrinsic $R$-[[cohomology]], for $R$ the \hyperlink{Line}{canonical T-line object}. \end{theorem} We obtain a proof of this after the following discussions. \begin{remark} \label{}\hypertarget{}{} The resulting [[localization]] [[modality]] $Spec \mathcal{O}$ we might call the \emph{[[affine modality]]}. It is similar to exhibiting $C$ as a [[total category]]. \end{remark} \hypertarget{cohomology}{}\subsubsection*{{$R$-Cohomology}}\label{cohomology} Since $T$ is assumed to be an abelian Lawvere theory, the \hyperlink{Line}{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 object]]s $\mathbf{B}^n R$ 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 $\Xi \mathbb{Z}[n]$. \begin{prop} \label{}\hypertarget{}{} 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 \begin{displaymath} \mathbf{B}^n R_{chn} \coloneqq j(ab_* \Xi(F_{Ab}(*)[n]) \,. \end{displaymath} \end{prop} Every [[(∞,1)-topos]] such as $\mathbf{H} = Sh_{(\infty,1)}(C)$ comes with its [[cohomology|intrinsic notion of abelian cohomology]]: for $X \in \mathbf{H}$ any object and for $A \in \mathbf{H}$ a [[∞-group]] object with arbitrary [[delooping]]s $\mathbf{B}^n A$, the $n$th cohomology group of $X$ with coefficients in $A$ is \begin{displaymath} H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X,\mathbf{B}^n A) \,. \end{displaymath} 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]] \begin{displaymath} \cdots \simeq Ho_{[C^{op}, sSet]_{proj,loc}}(X, \mathbf{B}^n A_{chn}) \,. \end{displaymath} \begin{prop} \label{RCohomologyByO}\hypertarget{RCohomologyByO}{} 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 $\mathbb{L}\mathcal{O}(X) \in T Alg^\Delta$: \begin{displaymath} H^n(X,R) \simeq H^n( \mathbb{L} \mathcal{O}(X)) \,. \end{displaymath} \end{prop} \begin{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{displaymath} \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} \end{displaymath} \end{proof} This is essentially the argument of (\hyperlink{Toen}{To\"e{}n, corollary 2.2.6}). \hypertarget{local_objects}{}\subsubsection*{{$R$-Local objects}}\label{local_objects} \begin{defn} \label{}\hypertarget{}{} We say a morphism $f : X \to Y$ in $[C^{op}, sSet]$ is an \textbf{$R$-equivalence} if it induces isomorphisms in $R$-cohomology. \begin{displaymath} H^i(f,R) : H^i(Y,R) \stackrel{\simeq}{\to} H^i(X,R) \,. \end{displaymath} By prop. \ref{RCohomologyByO} 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 \textbf{$R$-[[local object]]} if for all $R$-equivalences $f$ we have that \begin{displaymath} \mathbf{H}(f,K) : \mathbf{H}(Y,K) \to \mathbf{H}(X,K) \end{displaymath} 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 complex]]es). \end{defn} \begin{prop} \label{}\hypertarget{}{} 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 $C^{\Delta^{op}}$ \begin{displaymath} C^{\Delta^{op}} \hookrightarrow (T Alg^{\Delta})^{op} \stackrel{\mathbb{R}j}{\to} [C^{op}, sSet]_{proj,cov} \,. \end{displaymath} \end{prop} \begin{proof} We may explicitly see this by observing that the proof of (\hyperlink{Toen}{To\"e{}n, 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 \hyperlink{ModelTransferToTAlg}{result on the model structure on TAlg} (theorem \ref{ModelTransferToTAlg}) 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 \begin{displaymath} \mathbf{L} \stackrel{\overset{\mathbb{L} \mathcal{O}}{\longleftarrow}}{\underset{\mathbb{R} i}{\hookrightarrow}} \mathbf{H} \coloneqq Sh_{(\infty,1)}(C) \end{displaymath} 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. \end{proof} \hypertarget{DerivedGeometry}{}\subsection*{{In derived geometry}}\label{DerivedGeometry} 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]]. \hypertarget{over_ordinary_associative_algebras}{}\subsubsection*{{Over ordinary associative algebras}}\label{over_ordinary_associative_algebras} Let $k$ be a [[field]] of characteristic $0$. \begin{defn} \label{}\hypertarget{}{} Write $(cdgAlg_k^{op})^\circ$ for the [[(∞,1)-category]] that is [[presentable (∞,1)-category|presented]] by the [[model structure on dg-algebras|model structure on unbounded commutative cochain dg-algebras]] over $k$. Write \begin{displaymath} i : (cdgAlg_k^{op})^\circ_- \subset (cdgAlg_k^{op})^\circ \end{displaymath} for the full [[sub-(∞,1)-category]] on cochain dg-algebras concentrated in non-positive degree. \end{defn} Let $C \subset (cdgAlg_k^{op})^\circ_-$ be a [[small (∞,1)-category|small]] full sub-$(\infty,1)$-category equipped with the structure of a [[subcanonical coverage|subcanonical]] [[(∞,1)-site]]. Set \begin{displaymath} \mathbf{H} \coloneqq Sh_{(\infty,1)}(C) \,. \end{displaymath} \begin{defn} \label{}\hypertarget{}{} Write \begin{displaymath} \mathcal{O} : \mathbf{H} \to ((cdgAlg_k^{op})^\circ \end{displaymath} for the $(\infty,1)$-[[Yoneda extension]] of the inclusion $C \hookrightarrow ((cdgAlg_k^{op})^\circ_, \hookrightarrow ((cdgAlg_k^{op})^\circ$. \end{defn} \begin{remark} \label{}\hypertarget{}{} By the [[(∞,1)-co-Yoneda lemma]] we may express any $X \in Sh_{(\infty,1)}(C)$ by an [[(∞,1)-colimit]] over representables \begin{displaymath} X \simeq {\lim_\to}_i U_i \;\; \in Sh(C) \,. \end{displaymath} The functor $\mathcal{O}$ simply evaluates this colimit in $((cdgAlg_k^{op})^\circ$, which is the [[(∞,1)-limit]] in the [[opposite (∞,1)-category]] \begin{displaymath} \mathcal{O}X \simeq {\lim_\leftarrow}_i \mathcal{O}(U_i) \;\; \in (cdgAlg_k)^\circ \,, \end{displaymath} where we write $\mathcal{O}(U_i)$ simply for the object $U_i$ regarded in the opposite category. \end{remark} \begin{lemma} \label{}\hypertarget{}{} 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|right]] [[adjoint (∞,1)-functor]]. \begin{displaymath} j : ((cdgAlg_k^{op})^\circ \to Sh_{(\infty,1)}(C) \,. \end{displaymath} This is given by \begin{displaymath} Spec(A) : U \mapsto ((cdgAlg_k^{op})^\circ(U,A) \,. \end{displaymath} \end{lemma} \begin{proof} This follows by the general yoga of [[Kan extension]]s. Explcitly, we check the hom-equivalence \begin{displaymath} \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} \,. \end{displaymath} \end{proof} This is considered in (\hyperlink{BenZviNadler}{Ben-Zvi/Nadler, prop. 3.1}). \begin{lemma} \label{}\hypertarget{}{} The above \hyperlink{YonedaQuillenAdjunction}{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 \begin{displaymath} T Alg_k^{op} \hookrightarrow (T Alg_k^{\Delta})^{op} \end{displaymath} \end{lemma} \begin{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 model structure|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. \end{proof} \hypertarget{Examples}{}\subsection*{{Examples and applications}}\label{Examples} \begin{prop} \label{}\hypertarget{}{} The conditons of the above theorem are satisfied for instance for \begin{itemize}% \item $T$ the theory of ordinary commutative algebras over a field $k$ and $J$ the [[fpqc topology]]. In this case the adjunction is that considered in (\hyperlink{Toen}{To\"e{}n}). \item $T$ the theory of [[nLab:smooth algebra]]s and $C \hookrightarrow T Alg^{op}$ the [[site]] of the [[Cahiers topos]]. This is what we discuss in more detail below. \end{itemize} \end{prop} \hypertarget{rational_homotopy_theory}{}\subsubsection*{{Rational homotopy theory}}\label{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]]. \hypertarget{lie_theory_in_the_cahiers_topos}{}\subsubsection*{{$\infty$-Lie theory in the $\infty$-Cahiers topos}}\label{lie_theory_in_the_cahiers_topos} In this section we study the general theory for the case that \begin{itemize}% \item $T \coloneqq$ [[CartSp]] is the ([[syntactic category]] of the) Lawvere theory of [[smooth algebra]]s. \end{itemize} 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 topos]]es that are [[Models for Smooth Infinitesimal Analysis|well adapted models]] for [[synthetic differential geometry]]. We consider here the choice \begin{itemize}% \item $C \subset Smooth Alg^{op}$ is the [[site]] for the [[Cahiers topos]]. \end{itemize} \begin{defn} \label{}\hypertarget{}{} 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 \textbf{$(\infty,1)$-Cahiers-topos} is the [[(∞,1)-category of (∞,1)-sheaves]] on [[ThCartSp]] with respect to the good open cover coverage. \end{defn} \begin{remark} \label{}\hypertarget{}{} The good open cover [[coverage]] generates the [[Grothendieck topology]] of all [[open cover]]s 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 , the analogous statement holds true for the [[(∞,1)-topos]]es 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 nerve]]s of all good open cover. Notice that the construction of [[good open cover]]s (see there) on [[paracompact space]]s (such as [[Cartesian space]]s) by geodescally convex regions shows that we may always find a good open cover all whose finite non-empty intersections are [[diffeomorphism|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 \emph{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 in the projective model structure. As a consequence of this, we have the following useful technical result. \end{remark} \begin{lemma} \label{}\hypertarget{}{} Write $[ThCartSp^{op}, sSet]_{proj,cov}$ for the [[left Bousfield localization]] of the global projective model structure $[ThCartSp^{op}]_{proj}$ at the [[Cech nerve]]s $C(U) \to X\times S$ of [[good open cover]]s $\{U_i \times S \to X \times S\}$ in [[ThCartSp]]. We have that \begin{itemize}% \item this presents the $(\infty,1)$-Cahiers topos $Sh_{(\infty,1)}(ThCartSp) \simeq ([ThCartSp^{op}, sSet]_{proj,cov})$; \item 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 \begin{displaymath} [ThCartSp^{op}, sSet]( p_U , A ) \end{displaymath} is a weak equivalence (of [[Kan complex]]es). \end{itemize} \end{lemma} \begin{lemma} \label{}\hypertarget{}{} 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 $p \in \mathbb{N}$. \end{lemma} \begin{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 . \end{proof} This means that the assumptions of the \hyperlink{PassageToLocalTheorem}{Theorem on passage to the local model structure} are satisfied. \begin{cor} \label{}\hypertarget{}{} We have a [[simplicial Quillen adjunction]] \begin{displaymath} (Smooth Alg^\Delta_{proj})^{op} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} [ThCartSp^{op}, sSet]_{proj,cov} \,. \end{displaymath} \end{cor} \hypertarget{lie_algebroids}{}\paragraph*{{$\infty$-Lie algebroids}}\label{lie_algebroids} \begin{defn} \label{}\hypertarget{}{} The objects of the $(\infty,1)$-Cahiers topos we call \textbf{synthetic differential} [[∞-Lie groupoid]]s. The objects of the reflective sub-$(\infty,1)$-category of $R$-local objects in the $(\infty,1)$-Cahiers topos \begin{displaymath} \mathbf{L} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H} = Sh_{(\infty,1)}(ThCartSp) \end{displaymath} we call \textbf{[[∞-Lie algebroid]]}s. A [[connected]] $\infty$-Lie algebroid we call an \textbf{[[∞-Lie algebra]]}. \end{defn} (\ldots{}) Passing along the embedding $\mathbf{L} \hookrightarrow \mathbf{H}$ we may compute [[∞-Lie algebra cohomology]] in $\mathbf{H}$. (\ldots{}) \hypertarget{the_infinitesimal_path_groupoid_of_a_manifold}{}\paragraph*{{The infinitesimal path $\infty$-groupoid of a manifold}}\label{the_infinitesimal_path_groupoid_of_a_manifold} (\ldots{}) For $U \in CartSp$ let \begin{displaymath} U^{\Delta^\bullet_{inf}} \in C^{\Delta^{op}} \end{displaymath} be the simplicial object of in $U$. \textbf{Definition} We call \begin{displaymath} \mathbf{\Pi}_{inf}(U) \coloneqq \mathbb{R}j\; (U^{\Delta^\bullet_{inf}}) \in [C^{op}, sSet] \end{displaymath} the infinitesimal path $\infty$-Lie groupoid of $U$. Or the \textbf{path $\infty$-Lie algebroid} . (\ldots{}) \hypertarget{the_tangent_category_of_smooth_algebras}{}\paragraph*{{The tangent category of smooth algebras}}\label{the_tangent_category_of_smooth_algebras} (\ldots{}) The [[tangent category]] of the category of [[smooth algebra]]s is the category of modules over $C^\infty$-rings. \textbf{Proposition} This abstract definition of module over $C^\infty$-rings reproduces the definition given by Kock. The tangent category of the category of \emph{simplicial} $C^\infty$-rings is \ldots{} This serves the purpose of presenting the $\infty$-stack of $\infty$-vector bundles on $T Alg^{op}$. (\ldots{}) \hypertarget{appendix}{}\subsection*{{Appendix}}\label{appendix} \hypertarget{Enrichment}{}\subsubsection*{{Enrichment of categories of simplicial objects}}\label{Enrichment} 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). \begin{example} \label{}\hypertarget{}{} For $A \in (T Alg^{\Delta})^{op}$ and $S \in sSet$ we have that the tensoring is given by \begin{displaymath} (A \cdot S)_n = \prod_{S_s} A \in T Alg \,, \end{displaymath} with the [[nLab:product]] taken in $T Alg$. \end{example} \hypertarget{model_structure_on_cosimplicial_abelian_groups}{}\subsubsection*{{Model structure on cosimplicial abelian groups}}\label{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-isomorphism]]s under the dual [[Dold-Kan correspondence]] $Ab^\Delta \simeq Ch^\bullet_+(Ab)$. The model structure is described in detail at . The structure of a [[simplicial model category]] is described in detail at [[model structure on cosimplicial abelian groups]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[geometric stack]], [[geometric ∞-stack]] \item [[Lie integration]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The Quillen adjunction over abelian $T$-algebras that we consider generalizes that discussed in \begin{itemize}% \item [[nLab:Bertrand Toën]], \emph{Champs affine} (\href{http://arxiv.org/abs/math/0012219}{arXiv:math/0012219}) \end{itemize} 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 \begin{itemize}% \item [[Herman Stel]], \emph{$\infty$-Stacks and their function algebras -- with applications to $\infty$-Lie theory} , master thesis (2010) ([[schreiber:master thesis Stel|web]]) \item [[Herman Stel]], \emph{Cosimplicial $C^\infty$-rings and the de Rham complex of Euclidean space} (\href{http://arxiv.org/abs/1310.7407}{arXiv:1310.7407}) \end{itemize} on which this entry here is based. The considerations in \begin{itemize}% \item [[David Spivak]], \emph{Derived smooth manifolds} Duke Math. J. Volume 153, Number 1 (2010), 55-128. (\href{http://www.uoregon.edu/~dspivak/derived-smooth-manifolds.pdf}{pdf}) \end{itemize} on [[derived smooth manifold]]s 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 \emph{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 \begin{itemize}% \item [[David Ben-Zvi]], [[David Nadler]], \emph{Loop spaces and connections} (\href{http://arxiv.org/abs/1002.3636}{arXiv:1002.3636}) \end{itemize} [[!redirects function algebras on ∞-stacks]] [[!redirects rational homotopy theory in an (∞,1)-topos]] [[!redirects rational homotopy theory in an (infinity,1)-topos]] \end{document}