nLab 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 $\infty$-algebra on free loop space objects.

The $(\infty,1)$-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

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

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

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)

Proposition/Definition

Let

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

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

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

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

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

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

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

Properties

Proposition

The inclusion

$$cdgAlg^-_k \hookrightarrow cdgAlg_k$$

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

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

Corollary

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

This follows from (ToënVezzosi, assumption 1.1.0.4).

Corollary

The inclusion

$$(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}$$

preserves homotopy limits, hence the induced inclusion

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

preserves (∞,1)-limits.

This follows from (ToënVezzosi, assumption 1.1.0.6).

Definition

For $A \in cdgAlg_k$ a dg-algebra, write

• $A Mod$ for its category of dg-modules;

  This is naturally a symmetric monoidal category.

• $cdgAlg_A := CMon(A Mod)$ for the category of commutative monoids in $A Mod$, the category of cdg-$A$-algebras.

Corollary

For any $A\in cdgAlg_k$ say a morphism in $cdgAlg_A$ is

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

• a fibration precisely if it is degreewise surjective.

This makes $cdgAlg_A$ into a model category that is

• combinatorial;

• proper.

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

$$cdgAlg_A \simeq A/cdgAlg_k$$

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 $B \in cdgAlg_A$ cofibrant with respect to the model structure in cor , the tensor product (base change) functor

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

preserves weak equivalences.

This is (ToënVezzosi, assumption 1.1.0.4).

Proposition

The monoidal Dold-Kan correspondence provides a Quillen equivalence

$$(\Gamma^{cmon} \dashv N_\bullet) : cAlg_k^{\Delta^{op}} \stackrel{\overset{\Gamma^{cmon}}{\leftarrow}}{\underset{N_\bullet}{\to}} cdgAlg_k^+$$

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

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

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

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

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