nLab derived geometry

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

Idea and motivation

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

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

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

An object $X$ 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 $(\infty,1)$-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 $X \in T Alg^{op}$ an ordinary space, its free loop space object computed in the underived $\infty$-topos over $T Alg^{op}$ exists, but simply coincides with $X$, because $X$ is 0-truncated in there. But the free loop space object $\mathcal{L}X$ of $X$ computed in the $\infty$-topos over $T Alg_\infty^{op}$ may be very rich: its $\infty$-function algebra is the Hochschild homology of $X$. Moreover, the functions on $\mathcal{L}X$ that are invariant under the canonical internal circle-action are the closed Kähler differential forms on $X$.

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.

Plan

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

• Oct 15, 2010

  algebraic theories / Lawvere theories

  The theories of groups, rings, $k$-algebras, C-∞-rings and so forth are examples of algebraic theories. Bill Lawvere famously noticed that these theories are encoded by categories $T$ with products, all whose objects are cartesian powers of a generating object $x$. One speaks of Lawvere theories. An algebra $A$ of a Lawvere theory is identified with a product-preserving co-presheaf

  $$A : T \to Set \,.$$

  Here the value $A(x)$ is identified with the underlying set of the algebra and for any morphism $f : x^n \to x$ in $T$ the morphism $A(f) : A(x^n) \simeq A(x)^n \to A(x)$ is an $n$-ary operation in the algebra. Functoriality of $A$ 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 $T Alg^{op}$ of the category of all $T$-algebras with a category of test spaces: the sheaf topos $Sh(C)$ on a small full subcategory $C \hookrightarrow T Alg^{op}$ is the context in which geometry over $T$ takes place. For $T =$ CartSp the theory of smooth algebras, this is synthetic differential geometry.

• Oct 22, 2010

  (∞,1)-algebraic theory

  There is an $\infty$-categorical generalization of the notion of Lawvere theory and algebra over a Lawvere theory. The general abstract discussion of this is hidden in section 5.5.8 of Higher Topos Theory .

  The idea is evident: we say

  • an $(\infty,1)$-algebraic theory is an (∞,1)-category $T$ with (∞,1)-products.

  • an $(\infty,1)$-algebra over $T$ is an (∞,1)-functor $A : T \to$ ∞Grpd that preserves these products;

  • the $(\infty,1)$-category of all $T$-algebras is the full sub-(∞,1)-category $\infty T Alg \hookrightarrow \infty Func(T, \infty Grpd)$ of the (∞,1)-category of (∞,1)-copresheaves on those functors that preserve products.

• Dec 17, 2010

  (2,1)-algebraic theory of E-∞ algebras

  An important example of a (∞,1)-algebraic theory beyond ordinary 1-algebraic theories is the (2,1)-category of spans of finite sets. Its algebras turn out to be E-∞ algebras.

  This is the result of

  • James Cranch, Algebraic Theories and $(\infty,1)$-Categories (arXiv)

Next, in order to handle $\infty T Alg$ 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

• Oct 29, 2010

  homotopy T-algebras

  We know from the model structure on simplicial presheaves that every (∞,1)-functor $A : T \to \infty Grpd$ is modeled by a strict functor $T \to$ sSet. A homotopy $T$-algebra is such a functor that preserves products up to weak equivalence in that for all $n \in \mathbb{N}$ the canonical morphism

  $$\prod_{i = 1}^n A(p_i) : A(x^n) \to (A(x))^n$$

  is a weak equivalence.

  There is a model structure for such homotopy $T$-algebras, given by left Bousfield localization of the projective model structure on simplicial presheaves at the morphisms $\coprod_n F_T(1) \to F_T(n)$, where $F_T(n)$ is the free $T$-algebra on $n$ generators.

  But it turns out that one can further strictify this: a simplicial T-algebra is a strict functor $T \to sSet$ 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 $T$-algebras and for simplicial $T$-algebras.

  This Quillen equivalence is described in

  • Bernard Badzioch, Algebraic theories in homotopy theory Annals of Mathematics, 155 (2002), 895-913 (JSTOR)

  The result that strict simplicial algebras model all $\infty$-$T$-algebras is also in

  • Julie Bergner, Rigidification of algebras over multi-sorted theories , Algebraic and Geometric Topoogy 7, 2007.

• Nov 5, Nov 19, 2010

  monoidal Dold-Kan correspondence and model structure on dg-algebras

  We have seen that the model structure on simplicial T-algebras models $\infty$-T-algebras. When $T$ is an abelian Lawvere theory, so that every $T$-algebra has an underlying abelian group, it is useful to consider the normalized chains complex of a simplicial $T$-algebra: equipped with the cup product this is a dg-algebra (in non-negative degree).

  There is a standard model structure on dg-algebras that prevails much of the literature. For instance in its dual form this is the model structure traditionally used in rational homotopy theory.

  The monoidal Dold-Kan correspondence asserts that at least for $T$ a theory of ordinary commutative algebras, there is a Quillen equivalence

  $$T Alg_{proj}^{\Delta^{op}} \simeq dgAlg^+_{proj}$$

  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) $\infty$-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

  model structure on algebras over an operad

  Algebraic theories may also be encoded by operads. Accordingly $\infty$-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.

  Many classical examples of $\infty$-algebras are captured by this, such as A-∞ algebras / A-∞ spaces, E-∞ algebras, L-∞ algebras.

Derived geometry over $\infty-C^\infty$-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 $*//G$ we have that much of the phenomenology of derived geometry is exhibited already by free loop space objects $\mathcal{L}X$ of a derived space $X$: the (∞,1)-pullback of the diagonal on $X$ along itself

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

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

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

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

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

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

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

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

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

Examples

• derived differential geometry

References

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

• Bertrand Toën, Gabriele Vezzosi,

  • Homotopical Algebraic Geometry I: Topos theory (arXiv)

  • Homotopical Algebraic Geometry II: geometric stacks and applications (arXiv)

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

• Jacob Lurie, Structured Spaces

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,

• David Ben-Zvi, David Nadler,

  • 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 $\mathcal{L}X$ is under mild conditions the Hochschild homology complex of $X$ hence by Hochschild-Kostant-Rosenberg theorem the collection of Kähler differential forms on $X$, and that the functions on $\mathcal{L}X$ that are invariant under the canonical $S^1$-action on $\mathcal{L}X$ 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

• Jacob Lurie, A Survey of Elliptic Cohomology

The article

• David Spivak, Derived smooth manifolds (arXiv:0810.5174)

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

• Urs Schreiber, Seminar on derived differential geometry (2010)

Survey in the context of BV-formalism:

• Luigi Alfonsi, §5 in: Higher geometry in physics, in: Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2312.07308]