higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
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.
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.
Let $k$ be a field of characteristic 0, or more generally a commutative $\mathbb{Q}$-algebra.
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)
Let
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
for the (∞,1)-category of (∞,1)-sheaves on $C$. We have a derived Isbell duality
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.
The inclusion
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
$(cdgAlg_k, \otimes_k)$ is a symmetric monoidal model category.
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).
The inclusion
preserves homotopy limits, hence the induced inclusion
preserves (∞,1)-limits.
This follows from (ToënVezzosi, assumption 1.1.0.6).
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.
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
There is an equivalence of categories with the under category of cdg-algebras under $A$
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).
For $B \in cdgAlg_A$ cofibrant with respect to the model structure in cor 4, the tensor product (base change) functor
preserves weak equivalences.
This is (ToënVezzosi, assumption 1.1.0.4).
The monoidal Dold-Kan correspondence provides a Quillen equivalence
(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:
and this recovers the constructions discussed above in The Hochschild chain complex of an associative algebra.
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
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.
(…)
Hochschild cohomology: section Higher order HH modeled on cdg-algebras;
perfect ∞-stacks and their geometric ∞-function theory
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 $(\mathcal{O} \dashv Spec)$-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
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)