higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
space $\leftarrow$ Isbell duality $\rightarrow$ algebra
The term ‘space’ is quite generic and can mean many different things depending on context. Often we will link it because we want to include many of these contexts at once!
Probably the default spaces are topological spaces (as defined by Bourbaki), with continuous maps between them. There are many variants, from convergence spaces (general) to metric (metrisable) spaces (specific), all of which fall under the topic of topology. An important variation gives locales; many of the theorems from topology that require the axiom of choice for Bourbaki spaces become constructively valid for locales.
Often one restricts to nice topological spaces or to a nice category of spaces; these are not always the same kind of restriction! There are many examples under those two headings; m-cofibrant spaces in particular allow one to identify homotopy equivalence with weak homotopy equivalence. More generally in homotopy theory, one uses the homotopy hypothesis to identify spaces with $\infty$-groupoids, of which there are several models. Since one common model is simplicial sets, some homotopy theorists use “space” to mean “simplicial set.”
Often one is also interested in spaces with additional structure. For instance, smooth manifolds and generalized smooth spaces are spaces with smooth structure, and algebraic varieties and schemes are spaces with algebraic structure. One can also speak more generally of spaces “modeled” on any suitable starting category; in this way one often comes to consider sheaves on a suitable site (and variations such as presheaves, copresheaves, or objects of an Isbell envelope) as spaces. Generalized smooth spaces and schemes are two commonly encountered examples of spaces “modeled” on a starting category in this way.
One can also categorify the concept of space. As space is dual to quantity, space and quantity can be categorified together; see $\infty$-space and compare $\infty$-quantity. Other types of categorified spaces include Grothendieck toposes (which are categorified locales) and stacks (which are categorified sheaves) and even their $\infty$-versions ($(\infty,1)$-toposes and $\infty$-stacks).
In Structured Spaces, Jacob Lurie provides a coherent general picture of generalized notions of space 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 $\mathcal{T}$ 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
$\mathcal{T} = \mathbb{L}$, the category of smooth loci;
without smooth structure
$\mathcal{T} = (C Ring^{fin})^{op}$, the formal dual of CRing: the category of (finitely generated) algebraic affine schemes;
$\mathcal{T} = (sC Ring^{fin})^{op}$, the formal dual of simplicial objects in CRing;
$\mathcal{T} = (E_\infty Ring^{fin})^{op}$, the formal dual of E-∞ rings: the category of (finitely generated) algebraic derived affine schemes.
These (∞,1)-categories $\mathcal{T}$ 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 $\mathcal{T}$ a (pre-)geometry .
Every pregeometry $\mathcal{T}$ gives rise to a geometry $\mathcal{G}$ – it’s geometric envelope . Roughly speaking, this contains not just the original test spaces but also their derived version. More on this below.
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 probeable 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 general abstract description 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.