# Contents

## 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 $\left(\infty ,1\right)$-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 $ℚ$-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, ${H}^{•\ge 1}\left(-\right)=0$.

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

$C↪\left(\left({\mathrm{cdgAlg}}_{k}^{-}{\right)}^{\mathrm{op}}{\right)}^{\circ }$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

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

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

$\left(𝒪⊣j\right):\left({\mathrm{cdgAlg}}_{k}^{\mathrm{op}}{\right)}^{\circ }\stackrel{\stackrel{𝒪}{←}}{\underset{j}{\to }}H$(\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 ${\mathrm{cdgAlg}}_{k}^{+}↪{\mathrm{cdgAlg}}_{k}$.

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

#### Properties

###### Proposition

The inclusion

${\mathrm{cdgAlg}}_{k}^{-}↪{\mathrm{cdgAlg}}_{k}$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

$\left({\mathrm{cdgAlg}}_{k},{\otimes }_{k}\right)$ is a symmetric monoidal model category.

###### Corollary

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

This follows from (ToënVezzosi, assumption 1.1.0.4).

###### Corollary

The inclusion

$\left({\mathrm{cdgAlg}}_{k}^{-}{\right)}^{\mathrm{op}}↪\left({\mathrm{cdgAlg}}_{k}{\right)}^{\mathrm{op}}$(cdgAlg_k^-)^{op} \hookrightarrow (cdgAlg_k)^{op}

preserves homotopy limits, hence the induced inclusion

$\left(\left({\mathrm{cdgAlg}}_{k}^{-}{\right)}^{\mathrm{op}}{\right)}^{\circ }↪\left(\left({\mathrm{cdgAlg}}_{k}{\right)}^{\mathrm{op}}{\right)}^{\circ }$((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.

• ${\mathrm{cdgAlg}}_{A}:=\mathrm{CMon}\left(A\mathrm{Mod}\right)$ 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

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.

###### Corollary

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

$B{\otimes }_{A}\left(-\right):A\mathrm{Mod}\to B\mathrm{Mod}$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

$\left({\Gamma }^{\mathrm{cmon}}⊣{N}_{•}\right):{\mathrm{cAlg}}_{k}^{{\Delta }^{\mathrm{op}}}\stackrel{\stackrel{{\Gamma }^{\mathrm{cmon}}}{←}}{\underset{{N}_{•}}{\to }}{\mathrm{cdgAlg}}_{k}^{+}$(\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 $U\in {\mathrm{cAlg}}_{k}↪{\mathrm{cAlg}}_{k}^{{\Delta }^{\mathrm{op}}}↪H$ is $𝒪$-perfect:

$𝒪\left({X}^{K}\right)\simeq K\cdot 𝒪\left(X\right)$\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 $\left(y\left(U\right){\right)}^{K}\simeq y\left({U}^{K}\right)$ is still representable. Therefore

$𝒪\left(\left(y\left(U\right){\right)}^{K}\right)\simeq 𝒪\left({U}^{K}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\in \left({\mathrm{cdgAlg}}_{k}^{-}{\right)}^{\mathrm{op}}↪\left({\mathrm{cdgAlg}}_{k}{\right)}^{\mathrm{op}}$\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 $K\cdot 𝒪\left(U\right)$ formed in ${\mathrm{cdgAlg}}_{k}$ by formal duality. By the above proposition the inclusion ${\mathrm{cdgAlg}}_{k}^{-}↪{\mathrm{cdgAlg}}_{k}$ preserves this $\left(\infty ,1\right)$-colimit.

(…)

## References

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 $\left(𝒪⊣\mathrm{Spec}\right)$-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 (82.113.99.57)