This entry is about a certain way of formalizing higher geometry. For variants and more background, see there.
derived smooth geometry
structures in a cohesive (∞,1)-topos
For a Lawvere theory, the geometry modeled on is encoded in the sheaf topos on formal duals of T-algebras. For instance for CartSp we have that -algebras are smooth algebras and the geometry modeled on them is synthetic differential geometry.
This statement generalizes to (∞,1)-category theory: for an (∞,1)-algebraic theory and a (∞,1)-site of formal duals of -algebras over , one says that the (∞,1)-topos over encodes derived geometry modeled on . The objects of are also called derived stacks on .
The term “derived” here is meant to specifically contrast with ∞-stacks on just a 1-categorical site. Even if happens to be just an ordinary Lawvere theory, regarded as a 1-truncated -theory, the -topos over behaves considerably different from that over just .
passing to derived geometry by making the site a genuine -category makes limits behave well in cohomology. For instance the intersection pairing of non-transversal smooth manifolds comes out correctly when regarding them as derived smooth manifolds.
A central class of examples for nontransveral pullbacks in derived geometry are derived loop space objects. For an ordinary space, its free loop space object computed in the underived -topos over exists, but simply coincides with , because is 0-truncated in there. But the free loop space object of computed in the -topos over may be very rich: its -function algebra is the Hochschild homology of . Moreover, the functions on that are invariant under the canonical internal circle-action are the closed Kähler differential forms on .
Examples of derived spaces have appeared long ago as the configuration spaces in gauge theory. What is called the BV-BRST complex of a gauge theory is the function algebra on the infinitesimal approximation to a derived orbifold whose internal symmetries are the gauge transformations and whose function complex provides a resolution of the locus of solutions to the physical equations of motion.
More recently, much of the motivation for derived geometry came from the observation that the Goerss-Hopkins-Miller theorem suggests that there is a derived moduli space of derived elliptic currves, that it carries a structure ∞-sheaf of E-∞-rings, and the the global sections of that yield the ring spectrum of the generalized cohomology theory called tmf.
The rough plan is that we
then discuss some of the specifics of -sites for derived geometry;
and finally look at the specific case of derived smooth differential geometry.
The following is the beginning of a detailed schedule of talks. The first few talks have been fixed by now, the later talks will be fixed as me move along.
Oct 15, 2010
The theories of groups, rings, -algebras, C-∞-rings and so forth are examples of algebraic theories. Bill Lawvere famously noticed that these theories are encoded by categories with products, all whose objects are cartesian powers of a generating object . One speaks of Lawvere theories. An algebra of a Lawvere theory is identified with a product-preserving co-presheaf
Here the value is identified with the underlying set of the algebra and for any morphism in the morphism is an -ary operation in the algebra. Functoriality of encodes the compatibilities of all these operations, such as associativity.
The identification of algebras with co-presheaves gives rise to Isbell duality, which naturally identifies the opposite of the category of all -algebras with a category of test spaces: the sheaf topos on a small full subcategory is the context in which geometry over takes place. For CartSp the theory of smooth algebras, this is synthetic differential geometry.
Oct 22, 2010
The idea is evident: we say
Dec 17, 2010
This is the result of
Oct 29, 2010
We know from the model structure on simplicial presheaves that every (∞,1)-functor is modeled by a strict functor sSet. A homotopy -algebra is such a functor that preserves products up to weak equivalence in that for all the canonical morphism
is a weak equivalence.
There is a model structure for such homotopy -algebras, given by left Bousfield localization of the projective model structure on simplicial presheaves at the morphisms , where is the free -algebra on generators.
But it turns out that one can further strictify this: a simplicial T-algebra is a strict functor that also preserves all products up to isomorphism. There is a model structure on simplicial T-algebras, going back to Quillen, where the weak equivalences and the fibrations are objectwise those of simplicial sets.
And there is a Quillen equivalence between these model structures for homotopy -algebras and for simplicial -algebras.
This Quillen equivalence is described in
The result that strict simplicial algebras model all --algebras is also in
Nov 5, Nov 19, 2010
We have seen that the model structure on simplicial T-algebras models -T-algebras. When is an abelian Lawvere theory, so that every -algebra has an underlying abelian group, it is useful to consider the normalized chains complex of a simplicial -algebra: equipped with the cup product this is a dg-algebra (in non-negative degree).
between the model structure on simplicial algebras and that on chain dg-algebras in non-negative degree. This is a monoidal refinement of the standard Dold-Kan correspondence that identifies simplicial abelian groups with chain complexes. The proof is based on general statements about monoidal Quillen adjunctions.
This means that dg-algebras are yet another equivalent model for (certain) -algebras. In applications it is convenient to pass back and forth along this equivalence: the simplicial algebras connect more directly to the abstract theory, but the available toolset of theorems and techniqies for dg-algebras is much larger, and they are generally easier to handle.
Nov 12, 2010
Algebraic theories may also be encoded by operads. Accordingly -algebras may be regarded as ∞-algebras over an (∞,1)-operad. There is again a model category-structure available to present these: the model structure on algebras over an operad.
Nov 27, Dec 3, 2010
Analogous to how a good deal of the phenomenology of stacks is exhibited already by the weak quotient we have that much of the phenomenology of derived geometry is exhibited already by free loop space objects of a derived space : the (∞,1)-pullback of the diagonal on along itself
When is an ordinary object in the inclusion one calls for emphasis also the derived loop space of .
Since this pullback is maximally non-transversal, the derived free loop space is quite different from the ordinary free loop space object computed in . Notably when is 0-truncated (just a plain space, no groupoidal morphisms, no derived resoluton) its underived loop space object is just itself and hence uninteresting, whereas its derived loop space is very rich:
To some extent this is just the tautological dual reformulation of Hochschild homology as the -categorical (derived) tensor product
But there are some noteworthy subtleties. For instance in the traditional literature the derived tensor product is taken in the context of modules, not of algebras. But one can see that forming a fibrant replacement in the model structure on homotopy -algebras puts an -algebra structure back on the Hochschild complex.
This relation between derived loop spaces and Hochschild homology is very fruitful. It gives a transparent conceptual interpretation to many constructions in Hochschild cohomology and makes all this standard theory applicable to the study of derived geometry.
Dec 10 19, 2010
The Lawvere theory that encodes standard models of smooth differential geometry (synthetic differential geometry) is the category CartSp of Cartesian spaces and smooth functions between them. Its algebras are smooth algebras / -rings. Therefore its -algebras are modeled by simplicial -rings.
Spaces locally ringed in such smooth -algebras are called derived smooth manifolds .
A systematic description of derived geometry using model category-theoretic tools was first undertaken in
A proposal for the precise set of the scene of derived geometry in general abstract (∞,1)-category theory-terms is
Loop Spaces and Langlands Parameters (arXiv:0706.0322)
Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry (arXiv:0805.0157)
Loop Spaces and Connections (arXiv:1002.3636)
This article uses Toën’s theory of function algebras on ∞-stacks for showing that the function complex on a derived loop space is under mild conditions the Hochschild homology complex of hence by Hochschild-Kostant-Rosenberg theorem the collection of Kähler differential forms on , and that the functions on that are invariant under the canonical -action on are the closed forms. This also gives a geometric interpretation of the old observation by Maxim Kontsevich and others, that the differential and grading on the de Rham complex may be understood as induced from automorphisms of the odd line.
Loop Spaces and Representations (arXiv:1004.5120)
The application of derived geometry to the construction of tmf is described in
Some of the above material is taken from the seminar notes