group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
Abstract We sketch some basic ideas (Jacob Lurie’s ideas, that is) about higher geometry motivated from the higher version of the moduli stack of elliptic curves: the derived moduli stack of derived elliptic curves. We survey aspects of the theory of generalized schemes and then sketch how the derived moduli stack of derived elliptic curves is an example of a generalized scheme modeled on the formal dual of E-∞ rings.
For fully appreciating the details of the main theorem here the material discussed in the previous sessions (and a little bit more) is necessary, but our exposition of generalized schemes is meant to be relatively self-contained (albeit necessarily superficial).
This are the entries on the previous sessions:
A Survey of Elliptic Cohomology - formal groups and cohomology
A Survey of Elliptic Cohomology - E-infinity rings and derived schemes
A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations
In the context of elliptic cohomology one assigns to every elliptic curve $\phi$ over a ring $R$ a cohomology theory represented by an E-∞ ring spectrum $E_\phi$.
Since, by definition, we may identify the elliptic curve $\phi$ over $R$ with a patch $\phi : Spec R \to \mathcal{M}_{1,1}$ of the moduli stack $\mathcal{M}_{1,1}$ of elliptic curves, this assignment
looks like an E-∞ ring valued structure sheaf on $\mathcal{M}_{1,1}$.
There is a very general theory of higher geometry for generalized spaces with generalized structure sheaves. Using this one may regard the pair
as a structured space that is a “derived” Deligne-Mumford stack.
The central theorem about elliptic cohomology of Jacob Lurie, refining the Goerss-Hopkins-Miller theorem says that
the central theorem, first version
The moduli stack $\mathcal{M}_{1,1}$ of elliptic curves equipped with the E-∞ ring-valued structure sheaf $\mathcal{O}^{Der}$ may be regarded as the derived moduli stack of derived elliptic curves in that for any E-∞ ring $R$ the space of derived stack morphisms
is equivalent to the space of derived elliptic curves over $R$.
After we have looked at some concepts in higher geometry a bit more closely below, we will restate this in slightly nicer fashion.
A sketch of what this theorem means and how it is proven is part of the content of
and goes back to Jacob Lurie’s PhD thesis (listed here).
The general theory for the context of higher geometry invoked here has later been spelled out in
The special case of the general theory that is needed here, where the coefficient objects of structure sheaves are E-∞ rings, is described in
while the general theory of E-∞ rings themselves, in the (∞,1)-category theory context needed here, is developed in
The statement that we are after really lives in the context of higher geometry (often called “derived geometry”). Here is an outline of the central aspects.
The central ingredient which we choose at the beginning to get a theory of higher geometry going is an (∞,1)-category $\mathcal{G}$ whose objects we think of as model spaces : the simplest objects exhibiting the geometric structures that we mean to consider.
Examples for categories of model spaces
with smooth structure
$\mathcal{G} = \mathbb{L}$, the category of smooth loci;
without smooth structure
$\mathcal{G} = (C Ring^{fin})^{op}$, the formal dual of CRing: the category of (finitely generated) algebraic affine schemes;
$\mathcal{G} = (sC Ring^{fin})^{op}$, the formal dual of simplicial objects in CRing;
$\mathcal{G} = (E_\infty Ring^{fin})^{op}$, the formal dual of E-∞ rings: the category of (finitely generated) algebraic derived affine schemes.
These (∞,1)-categories $\mathcal{G}$ are naturally equipped with the structure of a site (and a bit more, which we won’t make explicit for the present purpose). Following Jacob Lurie we call such a $\mathcal{G}$ a geometry .
We want to be talking about generalized spaces modeled on the objects of $\mathcal{G}$. There is a hierarchy of notions of what that may mean:
Hierarchy of generalized spaces modeled on $\mathcal{G}$
We explain what this means from right to left.
An object $X$ probeable by objects of $\mathcal{G}$ should come with an assignment
of an ∞-groupoid of possible ways to probe $X$ by $U$, for each possible $U$, natural in $U$. More precisely, this should define an object in the (∞,1)-category of (∞,1)-presheaves on $\mathcal{G}$
But for $X$ to be consistently probeable it must be true that probes by $U$ can be reconstructed from overlapping probes of pieces of $U$, as seen by the topology of $\mathcal{G}$. More precisely, this should mean that the (∞,1)-presheaf $X$ is actually an object in an (∞,1)-category of (∞,1)-sheaves on $\mathcal{G}$
Such objects are called ∞-stacks on $\mathcal{G}$. The (∞,1)-category $Sh(\mathcal{G})$ is called an ∞-stack (∞,1)-topos.
A supposedly pedagogical discussion of the general philosophy of ∞-stacks as probebable spaces is at motivation for sheaves, cohomology and higher stacks.
The ∞-stacks on $\mathcal{G}$ that are used in the following are those that satisfy descent on Čech covers. But we will see (∞,1)-toposes of ∞-stacks that may satisfy different descent conditions, in particular with respect to hypercovers. Every ∞-stack (∞,1)-topos has a hypercompletion to one of this form.
For concretely working with hypercomplete (∞,1)-toposes it is often useful to use models for ∞-stack (∞,1)-toposes in terms of the model structure on simplicial presheaves.
This discussion here is glossing over all set-theoretic size issues. See StSp, warning 2.4.5.
Spaces probeable by $\mathcal{G}$ in the above sense can be very general. They need not even have a concrete underlying space , even for general definitions of what that might mean.
(Counter-)Example For $\mathcal{G} =$ Diff, for every $n \in \mathbb{N}$ we have the ∞-stack $\Omega_{cl}^n(-)$ (which happens to be an ordinary sheaf) that assigns to each manifold $U$ the set of closed n-forms on $U$. This is important as a generalized space: it is something like the rational version of the Eilenberg-MacLane space $K(\mathbb{Z}, n)$. But at the same time this is a “wild” space that has exotic properties: for instance for $n=3$ this space has just a single point, just a single curve in it, just a single surface in it, but has many nontrivial probes by 3-dimensional manifolds.
In the classical theory for instance of ringed spaces or diffeological spaces a concrete underlying space is taken to be a topological space. But this in turn is a bit too restrictive for general purposes: a topological space is the same as a localic topos: a category of sheaves on a category of open subsets of a topological space. The obvious generalization of this to higher geometry is: an n-localic (∞,1)-topos $\mathcal{X}$.
This makes us want to say and make precise the statement that
An concrete ∞-stack $X$ is one which has an underlying (∞,1)-topos $\mathcal{X}$:
the collection of $U$-probes of $X$ is a subobject of the collection of (∞,1)-topos-morphisms from $U$ to $\mathcal{X}$:
We think of $\mathcal{X}$ as the (∞,1)-topos of ∞-stacks on a category of open subsets of a would-be space $X$, only that this would be space $X$ might not have an independent existence as a space apart from $\mathcal{X}$. The available entity closest to it is the terminal object ${*}_{\mathcal{X}} \in \mathcal{X}$.
To say that $\mathcal{X}$ is modeled on $\mathcal{G}$ means that among all the ∞-stacks on the would-be space a structure sheaf of functions with values in objects of $\mathcal{G}$ is singled out: for each object $V \in \mathcal{G}$ there is a structure sheaf $\mathcal{O}(-,V) \in \mathcal{X}$, naturally in $V$.
This yields an (∞,1)-functor
We think of $X$ as being a concrete space co-probebale by $\mathcal{G}$ (we can map from the concrete $X$ into objects of $\mathcal{G}$).
Such an $\mathcal{O}$ is a consistent collection of coprobes if coprobes with values in $V$ may be reconstructed from co-probes with values in pieces of $V$.
More precisely:
($\mathcal{G}$-structure, StSp, def. 1.2.8)
An (∞,1)-functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ is a $\mathcal{G}$-valued structure sheaf on the (∞,1)-topos if
it preserves finite limits
and sends covering coproducts $(\coprod_i U_i) \to U$ to effective epimorphisms.
A pair $(\mathcal{X}, \mathcal{O})$ of an (∞,1)-topos $\mathcal{X}$ equipped with $\mathcal{G}$-valued structure sheaf $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ we call a structured (∞,1)-topos.
In summary:
A concrete ∞-stack $X$ modeled on $\mathcal{G}$ is
an (∞,1)-topos $\mathcal{X}$ (“of $\infty$-stacks on $X$”)
equipped with a $\mathcal{G}$-valued structure sheaf $\mathcal{O}$ in the form of a finite limits and cover preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$.
The fundamental example for structured (∞,1)-toposes are provided by the objects of $\mathcal{G}$ themselves, which are canonically equipped with a $\mathcal{G}$-structure as follows.
Let $f : \mathcal{G} \to \mathcal{G}'$ be a morphism of geometries, then the obvious (∞,1)-functor $f^* : \mathcal{L}Top(\mathcal{G}) \to \mathcal{L}Top(\mathcal{G}')$ admits a left adjoint
called the relative spectrum functor.
For $\mathcal{G}$ any geometry, write $\mathcal{G}_{disc}$ for the geometry obtained from this by forgetting its Grothendieck topology and instead using the discrete topology where only equivalences cover.
Notice that we may identify $\mathcal{G}_{disc}$-structures on the archetypical (∞,1)-topos ∞Grpd, being finite limit-preserving functors $\mathcal{G}_{disc}^{op} \to \infty Grpd$ with ind-objects in $\mathcal{G}^{op}$, hence with the opposite of pro-objects in $\mathcal{G}$. This gives a canonical inclusion
The composite (∞,1)-functor
we call the absolute spectrum functor
This abstract nonsense is reassuring, but we want a more concrete definition of what such $Spec^{\mathcal{G}} U$ is like:
For every $U \in \mathcal{G}$ there is naturally induced a topology on the over category $Pro(\mathcal{G})/U$. Define the (∞,1)-topos
naturally to be thought of as the (∞,1)-topos of ∞-stacks on $U$ .
This is canonically equipped with a (∞,1)-functor
And this is indeed the concrete underlying space produced by the absolute spectrum functor:
StSp, prop. 2.2.11, thm. 2.2.12)
For every $U \in \mathcal{G}$ the pair $(Spec U, \mathcal{O}_{Spec U})$ is indeed a structured (∞,1)-topos and is indeed equivalent to the $Spec^{\mathcal{G}} U$ defined more abstractly above.
Example For $\mathcal{G} = (C Ring^{fin})^{op}$ with the standard topology we have that 0-localic $\mathcal{G}$-structured spaces are locally ringed spaces , while $\mathcal{G}_{disc}$-structured 0-localic spaces are just arbitrary ringed spaces. Applying the above machinery to this situaton gives a spectrum functor that takes a ring $R$ first to the ringed space $({*,R})$ and this then to the locally ringed space $(Spec R, R)$.
We have seen that $\mathcal{G}$-structured (∞,1)-toposes are those general spaces modeled on $\mathcal{G}$ that are well-behaved in that at least they do have an “underlying topological structure” in the form of an underlying (∞,1)-topos. But such concrete spaces may still be very different from the model objects in $\mathcal{G}$.
In parts this is desireable: many objects that one would naturally build out of the objects in $\mathcal{G}$, such as mapping spaces $[\Sigma,X]$, are much more general than objects in $\mathcal{G}$ but do live happily in $\mathcal{L}Top(\mathcal{G})^{op}$.
But in many situations one would like to regard $\mathcal{G}$-structured (∞,1)-toposes that are not globally but locally equivalent to objects in $\mathcal{G}$. This is supposed to be captured by the following definition.
A structured (∞,1)-topos $(\mathcal{X}, \mathcal{O})$ is a $\mathcal{G}$-generalized scheme if
there exists a collection $\{V_i \in \mathcal{X}\}$
such that
this covers $\mathcal{X}$ in that the canonical morphism
to the terminal object in $\mathcal{X}$ is an effective epimorphism
the structured (∞,1)-toposes
$(\mathcal{X}/V_i, \mathcal{O}|_{V_i})$ induced by the $V_i$ are model spaces in that there exists $\{U_i \in \mathcal{G}\}$ and equivalences
Examples
warning the following statements really pertain to pregeometries, not geometries. for the moment this here is glossing over the difference between the two. See geometry (for structured (∞,1)-toposes) for the details.
ordinary smooth manifolds are 0-localic Diff-generalized schemes ([Structured Spaces|StSp, ex. 4.5.2]])
ordinary schemes are those $(CRing^{fin})^{op}$-generalized schemes whose underlying (∞,1)-topos is 0-localic and whose structure sheaf is 0-truncated ([Structured Spaces|StSp, prop. 4.2.9]])
Deligne-Mumford stacks are 1-localic $(CRing^{fin})_{et}^{op}$-generalized schemes ([Structured Spaces|StSp, prop. 4.2.9]])
This last statement is then the basis for calling a general $(CRing^{fin})_{et}^{op}$-generalized scheme a derived Deligne-Mumford stack
Finally, to make contact with the application to the derived moduli stack of derived elliptic curves, it seems that in Spectral Schemes a derived Deligne-Mumford stack (with derived in the sense of having replaced ordinary commutative rings by E-∞ rings) is gonna be a 1-localic $(E_\infty Ring^{fin})^{op}$-generalized scheme.
With the above machinery for higher geometry in hand, we now set out to describe the particular application that we are interested in: the study of the derived moduli stack of derived elliptic curves.
Lurie’s discussion of the derived moduli stack $(\mathcal{M}_{1,1}, \mathcal{O}^{Der})$ is more than a re-interpretation of the Goerss-Miller-Hopkins theorem. It is in particular a re-derivation of this result, from the following perspective
the central statement, conceptually
Input We have the $(E_\infty Ring)^{op}$-probeable space
Question: Does this happen to even be a $E_\infty Ring^{op}$-generalized scheme?
Answer Yes. It is actually a derived Deligne-Mumford stack.
Let $\mathcal{M}_{1,1}$ be the ordinary moduli stack of elliptic curves.
Using constructions in elliptic cohomology we may associate to each elliptic curve over $R$, i.e. each morphism $\phi : Spec R \to \mathcal{X}$, an E-infinity ring $E_\phi$ – the multiplicative spectrum that represents the elliptic cohomology theory given by $T$.
This gives an $E_\infty$-ring valued structure sheaf
Question What, if anything, is this derived stack a derived moduli stack of?
The big theorem is that the derived space $(\mathcal{X}, \mathcal{O}^{Der})$ classifies derived elliptic curves over $E_\infty$-rings
This is the theorem that we said above we wanted to consider, stated now a little bit more precisely.
(J. Lurie)
For
$A$ any E-∞ ring
and $E(A)$ is the space of derived elliptic curves over $A$ (the realization of the topological category of elliptic curves over $A$).
we have an equivalence
naturally in $A$.
Jacob Lurie writes that the proof proceeds alonmg these steps. Details will be discussed in the next session.
consider the presheaf of preoriented ellitptic curves $E'(A)$ first
observe that this restricted to ordinary rings produces the ordinary moduli stack
notice that every oo-stack with good deformation theory that restricts this way is a derived Deligne-Mumford stack $(\mathcal{X}, \mathcal{O}')$ that assigns connective $E_\infty$-rings over affines
let $\omega$ be the line bundle on $\mathcal{M}_{1,1}$ regarded as a coheren sheaf. There is then from the preorientation of the universal curve over $(\mathcal{M}, \mathcal{O}')$ a morphism
let $\mathcal{O}$ be the sheaf obtained from $\mathcal{O}'$ by inverting $\beta$
show that
for $n = 2 k$ we have an isomorphism $\omega^k \to \pi_{2 k}\mathcal{O}$
for $n = 2 k + 1$ we have an isomorphism $0 \to \pi_{2k +1}\mathcal{O}$
strategy: reduce to neighbourhood of a point
notice that this implies the desired statement