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dg-geometry

Idea

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 $\mathrm{\infty }$-algebra on free loop space objects.

The $(\mathrm{\infty },1)$-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 $k$ be a field of characteristic 0, or more generally a commutative $\mathbb{Q}$-algebra.

Definition

Proposition/Definition

Write

• ${\mathrm{cdgAlg}}_k$ for the category of graded-commutative cochain dg-algebras (meaning: with differential of degree +1) in arbitrary degree;

• ${\mathrm{cdgAlk}}_k^{-}$ for the full subcategory on objects with vanishing cochain cohomology in positive degree, $H^{•\ge 1}(-)=0$.

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)

Proposition/Definition

Let

$$C\hookrightarrow (({\mathrm{cdgAlg}}_k^{-}{)}^{\mathrm{op}}{)}^{\circ }$$C \hookrightarrow ((cdgAlg_k^-)^{op})^\circ

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

Write

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

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

$$(𝒪⊣j):({\mathrm{cdgAlg}}_k^{\mathrm{op}}{)}^{\circ }\stackrel{\stackrel{𝒪}{\leftarrow }}{\underset{j}{\to }}H$$(\mathcal{O} \dashv j) : (cdgAlg_k^{op})^\circ \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{j}{\to}} \mathbf{H}

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

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

Properties

Proposition

The inclusion

$${\mathrm{cdgAlg}}_k^{-}\hookrightarrow {\mathrm{cdgAlg}}_k$$cdgAlg^-_k \hookrightarrow cdgAlg_k

is a homotopical context in the sense of (ToënVezzosi, def. 1.1.0.11).

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

Corollary

$({\mathrm{cdgAlg}}_k,{\otimes }_k)$ is a symmetric monoidal model category.

Corollary

For $B\in ({\mathrm{dgcAlg}}_k{)}_{\mathrm{proj}}$ a cofibrant object, the tensor product with $B$ preserves weak equivalences.

This follows from (ToënVezzosi, assumption 1.1.0.4).

Corollary

The inclusion

$$({\mathrm{cdgAlg}}_k^{-}{)}^{\mathrm{op}}\hookrightarrow ({\mathrm{cdgAlg}}_k{)}^{\mathrm{op}}$$(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

preserves homotopy limits, hence the induced inclusion

$$(({\mathrm{cdgAlg}}_k^{-}{)}^{\mathrm{op}}{)}^{\circ }\hookrightarrow (({\mathrm{cdgAlg}}_k{)}^{\mathrm{op}}{)}^{\circ }$$((cdgAlg_k^-)^{op})^\circ \hookrightarrow ((cdgAlg_k)^{op})^\circ

preserves (∞,1)-limits.

This follows from (ToënVezzosi, assumption 1.1.0.6).

Definition

For $A\in {\mathrm{cdgAlg}}_k$ a dg-algebra, write

• $A\mathrm{Mod}$ for its category of dg-modules;

  This is naturally a symmetric monoidal category.

• ${\mathrm{cdgAlg}}_A:=\mathrm{CMon}(A\mathrm{Mod})$ for the category of commutative monoids in $A\mathrm{Mod}$, the category of cdg-$A$-algebras.

Corollary

For any $A\in {\mathrm{cdgAlg}}_k$ say a morphism in ${\mathrm{cdgAlg}}_A$ is

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

• a fibration precisely if it is degreewise surjective.

This makes ${\mathrm{cdgAlg}}_A$ into a model category that is

• combinatorial;

• proper.

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

$${\mathrm{cdgAlg}}_A\simeq A/{\mathrm{cdgAlg}}_k$$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 1.1.0.4, remark on p. 18).

Corollary

For $B\in {\mathrm{cdgAlg}}_A$ cofibrant with respect to the model structure in cor 4, the tensor product (base change) functor

$$B{\otimes }_A(-):A\mathrm{Mod}\to B\mathrm{Mod}$$B \otimes_A (-) : A Mod \to B Mod

preserves weak equivalences.

This is (ToënVezzosi, assumption 1.1.0.4).

Proposition

The monoidal Dold-Kan correspondence provides a Quillen equivalence

$$({\Gamma }^{\mathrm{cmon}}⊣N_{•}):{\mathrm{cAlg}}_k^{{\Delta }^{\mathrm{op}}}\stackrel{\stackrel{{\Gamma }^{\mathrm{cmon}}}{\leftarrow }}{\underset{N_{•}}{\to }}{\mathrm{cdgAlg}}_k^{+}$$(\Gamma^{cmon} \dashv N_\bullet) : cAlg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}} cdgAlg_k^+

(since $k$ is assumed to be of characteristic 0). Under this equivalence we have that $U\in {\mathrm{cAlg}}_k\hookrightarrow {\mathrm{cAlg}}_k^{{\Delta }^{\mathrm{op}}}\hookrightarrow H$ is $𝒪$-perfect:

$$𝒪(X^K)\simeq K\cdot 𝒪(X)$$\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.

Proof

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

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

is simply the formal dual of $U^K$, which is $K\cdot 𝒪(U)$ formed in ${\mathrm{cdgAlg}}_k$ by formal duality. By the above proposition the inclusion ${\mathrm{cdgAlg}}_k^{-}\hookrightarrow {\mathrm{cdgAlg}}_k$ preserves this $(\mathrm{\infty },1)$-colimit.

Over formal duals of general cdg-algebras

(…)

Applications

• dg-manifold, dg-scheme

• Hochschild cohomology: section Higher order HH modeled on cdg-algebras;

• rational homotopy theory: function algebras on ∞-stacks

• perfect ∞-stacks and their geometric ∞-function theory

• BRST-BV complex

References

Various model category presentations of dg-geometry are presented in

• Bertrand Toën, Gabriele Vezzosi, Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv)

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

• Ben-Zvi, John Francis, David Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv)

The $(𝒪⊣\mathrm{Spec})$-adjunction for dg-geometry is studied in

• David Ben-Zvi, David Nadler, Loop Spaces and Connections (arXiv:1002.3636)

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

• Vladimir Hinich, Homological algebra of homotopy algebras Communications in algebra, 25(10). 3291-3323 (1997)(arXiv:q-alg/9702015, Erratum (arXiv:math/0309453))

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

• Kai Behrend, Differential graded schemes I: prefect resolving algebras (arXiv:0212225)

• Kai Behrend, Differential Graded Schemes II: The 2-category of Differential Graded Schemes (arXiv:0212226)