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 -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.
Over formal duals of non-positively graded cdg-algebras
Let be a field of characteristic 0, or more generally a commutative -algebra.
for the category of graded-commutative cochain dg-algebras (meaning: with differential of degree +1) in arbitrary degree;
for the full subcategory on objects with vanishing cochain cohomology in positive degree, .
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)
This is considered in (Ben-ZviNadler). See function algebras on ∞-stacks for details.
is a homotopical context in the sense of (ToënVezzosi, def. 18.104.22.168).
This is (ToënVezzosi, lemma 2.3.11). We record the following implications of this statement
For a cofibrant object, the tensor product with preserves weak equivalences.
This follows from (ToënVezzosi, assumption 22.214.171.124).
preserves homotopy limits, hence the induced inclusion
This follows from (ToënVezzosi, assumption 126.96.36.199).
For a dg-algebra, write
This is (ToënVezzosi, assumption 188.8.131.52, remark on p. 18).
For cofibrant with respect to the model structure in cor 4, the tensor product (base change) functor
preserves weak equivalences.
This is (ToënVezzosi, assumption 184.108.40.206).
The monoidal Dold-Kan correspondence provides a Quillen equivalence
(since is assumed to be of characteristic 0). Under this equivalence we have that is -perfect:
and this recovers the constructions discussed above in The Hochschild chain complex of an associative algebra.
Since the (∞,1)-Yoneda embedding commutes with (∞,1)-limits we have that the powering is still representable. Therefore
is simply the formal dual of , which is formed in by formal duality. By the above proposition the inclusion preserves this -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 -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