This entry describes the higher geometry/derived geometry modeled on (∞,1)-sites of formal duals of dg-algebras, bounded or unbounded, over a field of characteristic 0.

The corresponding (∞,1)-topos is the context for classical rational homotopy theory, which arises by forming function algebras on ∞-stacks over constant ∞-stacks. It is also the context in which classical and higher order Hochschild homology of algebras and dg-algebras arises naturally as the function \infty-algebra on free loop space objects.

The (,1)(\infty,1)-toposes

We discuss some basic aspects of the (∞,1)-toposes over (∞,1)-sites of formal duals of cdg-algebras and of cdg-algebras of functions on its objects.

Over formal duals of non-positively graded cdg-algebras

Let kk be a field of characteristic 0, or more generally a commutative \mathbb{Q}-algebra.




  • cdgAlg kcdgAlg_k for the category of graded-commutative cochain dg-algebras (meaning: with differential of degree +1) in arbitrary degree;

  • cdgAlk k cdgAlk_k^- for the full subcategory on objects with vanishing cochain cohomology in positive degree, H 1()=0H^{\bullet \geq 1}(-) = 0.

There are the standard projective model structures on dg-algebras on these categories, whose weak equivalences are the quasi-isomorphisms and whose fibrations are the degreewise surjections.

This is considered in (Toën-Vezzosi, 2.3.1)



C((cdgAlg k ) op) C \hookrightarrow ((cdgAlg_k^-)^{op})^\circ

be a small full sub-(∞,1)-category of the (∞,1)-category presented by this model structure, and let CC be equipped with the structure of a subcanonical (∞,1)-site.


H:=Sh (,1)(C) \mathbf{H} := Sh_{(\infty,1)}(C)

for the (∞,1)-category of (∞,1)-sheaves on CC. We have a derived Isbell duality

(𝒪j):(cdgAlg k op) j𝒪H (\mathcal{O} \dashv j) : (cdgAlg_k^{op})^\circ \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} \mathbf{H}

where the left adjoint (∞,1)-functor 𝒪\mathcal{O} is the Yoneda extension of the inclusion cdgAlg k +cdgAlg kcdgAlg^+_k \hookrightarrow cdgAlg_k.

This is considered in (Ben-ZviNadler). See function algebras on ∞-stacks for details.



The inclusion

cdgAlg k cdgAlg k cdgAlg^-_k \hookrightarrow cdgAlg_k

is a homotopical context in the sense of (ToënVezzosi, def.

This is (ToënVezzosi, lemma 2.3.11). We record the following implications of this statement


(cdgAlg k, k)(cdgAlg_k, \otimes_k) is a symmetric monoidal model category.


For B(dgcAlg k) projB \in (dgcAlg_k)_{proj} a cofibrant object, the tensor product with BB preserves weak equivalences.

This follows from (ToënVezzosi, assumption


The inclusion

(cdgAlg k ) op(cdgAlg k) op (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

preserves homotopy limits, hence the induced inclusion

((cdgAlg k ) op) ((cdgAlg k) op) ((cdgAlg_k^-)^{op})^\circ \hookrightarrow ((cdgAlg_k)^{op})^\circ

preserves (∞,1)-limits.

This follows from (ToënVezzosi, assumption


For AcdgAlg kA \in cdgAlg_k a dg-algebra, write


For any AcdgAlg kA\in cdgAlg_k say a morphism in cdgAlg AcdgAlg_A is

  • a weak equivalence precisely if it is a quasi-isomorphism;

  • a fibration precisely if it is degreewise surjective.

This makes cdgAlg AcdgAlg_A into a model category that is

There is an equivalence of categories with the under category of cdg-algebras under AA

cdgAlg AA/cdgAlg k cdgAlg_A \simeq A/cdgAlg_k

which is a Quillen equivalence with respect to the standard model structure on an under category on the right.

This is (ToënVezzosi, assumption, remark on p. 18).


For BcdgAlg AB \in cdgAlg_A cofibrant with respect to the model structure in cor 4, the tensor product (base change) functor

B A():AModBMod B \otimes_A (-) : A Mod \to B Mod

preserves weak equivalences.

This is (ToënVezzosi, assumption


The monoidal Dold-Kan correspondence provides a Quillen equivalence

(Γ cmonN ):cAlg k Δ opN Γ cmoncdgAlg k + (\Gamma^{cmon} \dashv N_\bullet) : cAlg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}} cdgAlg_k^+

(since kk is assumed to be of characteristic 0). Under this equivalence we have that UcAlg kcAlg k Δ opHU \in cAlg_k \hookrightarrow cAlg_k^{\Delta^{op}} \hookrightarrow \mathbf{H} is 𝒪\mathcal{O}-perfect:

𝒪(X K)K𝒪(X) \mathcal{O} (X^{K}) \simeq K \cdot \mathcal{O}(X)

and this recovers the constructions discussed above in The Hochschild chain complex of an associative algebra.


Since the (∞,1)-Yoneda embedding yy commutes with (∞,1)-limits we have that the powering (y(U)) Ky(U K)(y(U))^{K} \simeq y(U^K) is still representable. Therefore

𝒪((y(U)) K)𝒪(U K)(cdgAlg k ) op(cdgAlg k) op \mathcal{O} ((y(U))^K) \simeq \mathcal{O}(U^K) \;\; \in (cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

is simply the formal dual of U KU^K, which is K𝒪(U)K \cdot \mathcal{O}(U) formed in cdgAlg kcdgAlg_k by formal duality. By the above proposition the inclusion cdgAlg k cdgAlg kcdgAlg_k^- \hookrightarrow cdgAlg_k preserves this (,1)(\infty,1)-colimit.

Over formal duals of general cdg-algebras




Various model category presentations of dg-geometry are presented in

The geometric ∞-function theory of perfect ∞-stacks in dg-geometry, and the corresponding Hochschild cohomology is considered in

The (𝒪Spec)(\mathcal{O} \dashv Spec)-adjunction for dg-geometry is studied in

The basic reference for the model structure on dg-algebras (see there for more details) for the commutative case over a field of characteristic 0 is

Details on the use of this model category structure for modelling dg-spaces are in

Revised on October 17, 2011 13:27:35 by Urs Schreiber (