nLab derived geometry


This entry is about a certain way of formalizing higher geometry. For variants and more background, see there.


Higher geometry

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

structures in a cohesive (∞,1)-topos


Idea and motivation

For TT a Lawvere theory, the geometry modeled on TT is encoded in the sheaf topos on formal duals of T-algebras. For instance for T=T = CartSp we have that TT-algebras are smooth algebras and the geometry modeled on them is synthetic differential geometry.

This statement generalizes to (∞,1)-category theory: for TT an (∞,1)-algebraic theory and CTAlg opC \subset T Alg^{op} a (∞,1)-site of formal duals of \infty-algebras over TT, one says that the (∞,1)-topos H=Sh(C)\mathbf{H} = \infty Sh(C) over CC encodes derived geometry modeled on TT. The objects of H\mathbf{H} are also called derived stacks on CC.

The term “derived” here is meant to specifically contrast with ∞-stacks on just a 1-categorical site. Even if TT happens to be just an ordinary Lawvere theory, regarded as a 1-truncated (,1)(\infty,1)-theory, the \infty-topos over TAlg opT Alg_\infty^{op} behaves considerably differently from that over just TAlg opT Alg^{op}.

An object XX in derived geometry has both an ∞-groupoid of internal symmetries as well as an \infty-algebra of functions:

  1. passing from a sheaf topos over a site to the \infty-stacks over that site makes colimits behave well in cohomology. For instance a singular quotient becomes an orbifold.

  2. passing to derived geometry by making the site a genuine (,1)(\infty,1)-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 XTAlg opX \in T Alg^{op} an ordinary space, its free loop space object computed in the underived \infty-topos over TAlg opT Alg^{op} exists, but simply coincides with XX, because XX is 0-truncated in there. But the free loop space object X\mathcal{L}X of XX computed in the \infty-topos over TAlg opT Alg_\infty^{op} may be very rich: its \infty-function algebra is the Hochschild homology of XX. Moreover, the functions on X\mathcal{L}X that are invariant under the canonical internal circle-action are the closed Kähler differential forms on XX.

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

  1. talk about some relevant leftover topics from the later chapters of Higher Topos Theory, continuing where we left of last time?;

  2. then discuss some of the specifics of \infty-sites for derived geometry;

  3. 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 we move along.

\infty-Sites from \infty-algebraic theories

Next, in order to handle TAlg\infty T Alg and to compare it to other known structures it is useful to present it in terms of a model category. This is the topic of the next part.

Models for \infty-algebras over ordinary algebraic theories

Derived geometry over C \infty-C^\infty-rings

  • Nov 27, Dec 3, 2010

    derived loop spaces

    Analogous to how a good deal of the phenomenology of stacks is exhibited already by the weak quotient *//G*//G we have that much of the phenomenology of derived geometry is exhibited already by free loop space objects X\mathcal{L}X of a derived space XX: the (∞,1)-pullback of the diagonal on XX along itself

    X:=X× X×XX. \mathcal{L}X := X \times_{X \times X} X \,.

    When XX is an ordinary object in the inclusion (,1)Sh(TAlg op)(,1)Sh(TAlg op)(\infty,1)Sh(T Alg^{op}) \hookrightarrow (\infty,1)Sh(T Alg_\infty^{op}) one calls X\mathcal{L}X for emphasis also the derived loop space of XX.

    Since this pullback is maximally non-transversal, the derived free loop space is quite different from the ordinary free loop space object computed in (,1)Sh(TAlg op)(\infty,1)Sh(T Alg^{op}). Notably when XX is 0-truncated (just a plain space, no groupoidal morphisms, no derived resoluton) its underived loop space object is just XX itself and hence uninteresting, whereas its derived loop space is very rich:

    one finds that the function algebra on X\mathcal{L}X is the complex that computes the Hochschild homology of the function algebra of XX:

    C(X)HH (C(X)). C(\mathcal{L}X) \simeq \mathbf{HH}_\bullet(C(X)) \,.

    To some extent this is just the tautological dual reformulation of Hochschild homology as the (,1)(\infty,1)-categorical (derived) tensor product

    HH (A):=A AAA. \mathbf{HH}_\bullet(A) := A \otimes_{A \otimes A} A \,.

    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 TT-algebras puts an \infty-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

    derived smooth manifolds

    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 / C C^\infty-rings. Therefore its \infty-algebras are modeled by simplicial C C^\infty-rings.

    Spaces locally ringed in such smooth \infty-algebras are called derived smooth manifolds .



A systematic description of derived geometry using model category-theoretic tools was first undertaken in

This generalizes the Brown-Jardine-Joyal-model structure on simplicial presheaves to a model structure on sSet-enriched presheaves over an sSet-site.

A proposal for the precise set of the scene of derived geometry in general abstract (∞,1)-category theory-terms is

The main point of this is a formalization and identification of special tame objects inside the collection of all derived stacks, namely the derived schemes and structured (∞,1)-toposes.

The relevance of derived loop spaces in derived was notably amplified in a series of articles by David Ben-Zvi and David Nadler,

The application of derived geometry to the construction of tmf is described in

The article

discussed the \infty-version of C-oo-rings – the algebras over the Lawvere theory CartSp – and the corresponding locally ringed spaces: derived smooth manifolds.

Some of the above material is taken from the seminar notes

Survey in the context of BV-formalism:

Last revised on January 4, 2024 at 12:45:38. See the history of this page for a list of all contributions to it.