nLab
structured (infinity,1)-topos

Idea

A structured (∞,1)-topos is a generalization of a ringed space or rather of a ringed site: a generalized space equipped with a structure sheaf taking values in generalized quantities.

A structured $(\mathrm{\infty }\mathrm{,1})$-topos that is constrained to locally look like an object in a prescribed category of test spaces is a generalized scheme.

At the bottom of it, a structured $(\mathrm{\infty }\mathrm{,1})$-topos is a (∞,1)-functor

$$X:S^{\mathrm{op}}\times R\to V,$$X : S^{op}\times R \to V \,,

where

• $V$ is the suitable enriching category, i.e. SSet for the full (∞,1)-categorical version;

• and for $U\in S$ and $C\in R$ the object $X(U,C)$ is to be thought of as the collection of morphisms from $U$ to $C$ that decompose into a morphism from $U$ into the generalized space and another morphism from that to $C$.

Here the generalization is in the sense described at space and quantity:

• spaces modeled on test spaces in some category $S$ are presheaves $X$ on $S$: $X(U)$ is the collection of probes of $X$ by $U\in S$.

• quantities (meaning: function algebras) modeled on value spaces in some category $S$ are co-presheaves $A$ on $S$: $A(U)$ is the collection of the quantities with values in $U\in S$.

In the context of structured $(\mathrm{\infty }\mathrm{,1})$-toposes $S$ is called a geometry (for structured (∞,1)-toposes).

Combined, this allows to give an analogous general way to think of the notion of a space equipped with a structure sheaf. A structured generalized space is such a generalization.

The basic idea is most simple:

First of all a space with structure sheaf (for instance a ringed space) is supposed to be nothing but

• an ordinary space $X$,

• equipped with a presheaf $O_X$ that takes values in some category of quantities

$$O_X:(\mathrm{Op}(X){)}^{\mathrm{op}}\to \mathrm{Quantities}$$O_X : (Op(X))^{op} \to Quantities

where we think – for $U\subset X$ an object in the category of open subsets of $X$ – of $O_X(U)$ as the collection of admissable functions on $U\subset X$. For instance $O_X(U)$ might be

• the set of all continuous functions from $U$ to $ℝ$

• or, if $X$ has a smooth structure, the set of all smooth functions to $ℝ$

• or, if $X$ has a complex structure, the set of all holomorphic functions to $ℂ$

etc. In any case, the choice of the structure sheaf $O_X$ is a choice of which maps out of $X$ one wants to concentrate on.

Now, taking the idea of space and quantity seriously, we should regard the category $\mathrm{Quantities}$ that $O_X$ takes values in itself as a category of co-presheaves on some category of co-test spaces.

For instance if we let $\mathrm{Cartesian}$ be the category whose objects are real cartesian spaces ${ℝ}^n$ and whose morphisms are continuous maps between these, then every continuous real-valued function on $U\subset X$ naturally forms the co-presheaf $C(U):{ℝ}^n\mapsto C(U,{ℝ}^n)$. In this case our structure sheaf $O_X$ is therefore actually a co-presheaf-valued presheaf:

$$O_X:\mathrm{Op}(X{)}^{\mathrm{op}}\to [\mathrm{Cartesian},\mathrm{Set}]$$O_X : Op(X)^{op} \to [Cartesian, Set]

$$O_X(U)({ℝ}^n):=C(U,{ℝ}^n).$$O_X(U)(\mathbb{R}^n) := C(U,\mathbb{R}^n) \,.

But equivalently, this is a presheaf-valued copresheaf

$$O_X:\mathrm{Cartesian}\to [\mathrm{Op}(X{)}^{\mathrm{op}},\mathrm{Set}]=\mathrm{PSh}(X).$$O_X : Cartesian \to [Op(X)^{op}, Set] = PSh(X)\,.

If we want to impose the condition that we actually want a structure-sheaf, we have a sheaf-valued copresheaf

$$O_X:\mathrm{Cartesian}\to \mathrm{Sh}(X).$$O_X : Cartesian \to Sh(X) \,.

Notice that $\mathrm{Sh}(X)$ is a (Grothendieck-)topos, just as Set itself is. So we can think of $O_X$ as a generalized quantity in the sense of space and quantity which takes values not in the topos Set but in some other topos, such as $\mathrm{Sh}(S)$.

There are other perspectives possible on a structure sheaf $O_X$, but this is the one whose full generalization we describe in the following.

We now describe the theory of structure sheaves encoded in topos-valued co-presheaves as developed in

• Jacob Lurie, Structured spaces (arXiv)

Definition

Let $S$ be a geometric, i.e. essentially an $(\mathrm{\infty }\mathrm{,1})$-site, i.e. some small (∞,1)-category equipped with a coverage, and let $\mathrm{Sh}(S)$ be the (∞,1)-topos of (∞,1)-category of (∞,1)-sheaves on $S$.

Notice that if $S=\mathrm{Op}(X)$ is the nerve of the category of open subsets of some topological space $X$, then $\mathrm{Sh}(X):=\mathrm{Sh}(S)$ is the (∞,1)-category of (∞,1)-sheaves on $X$, as in the above motivating introduction.

We want to define a structure sheaf on $S$ (for instance on $\mathrm{Op}(X)$) of quantities modeled on some $(\mathrm{\infty }\mathrm{,1})$-category $V$ to be an (∞,1)-functor

$$O_X:V\to \mathrm{Sh}(S)$$O_X : V \to Sh(S)

to be thought of as the assignment to each value space $v\in V$ of an $(\mathrm{\infty }\mathrm{,1})$-sheaf of $v$-valued functions on $S$ (on $X$ if $S=\mathrm{Op}(X)$).

But since we are taking care of the sheaf condition on $S$, we also want to allow a similar kind of co-sheaf condition on $V$. In order to do so, $V$ is taken to be equipped with extra structure encoding covers in $V$, and $O_X$ is then required to respect this structure suitably

Definition (StrSh, def 1.2.1)

An admissablility structure on an $(\mathrm{\infty }\mathrm{,1})$-category $V$ is a Grothendieck topology on $V$ such that it is generated by a system of admissable morphisms (to be defined…)

Definition (StrSh, def 1.2.5)

An $(\mathrm{\infty }\mathrm{,1})$-category $V$ equipped with an admissablility structure is a geometry if it is essentially small, admits finite limits and is idempotent complete.

Definition (StrSh, def 1.2.8)

(structure sheaf)

Let $𝒢$ be a geometry and $𝒳$ an $(\mathrm{\infty }\mathrm{,1})$-topos An (∞,1)-functor

$$O_X:𝒢\to 𝒳$$O_X : \mathcal{G} \to \mathcal{X}

is a $𝒢$-structure on $𝒳$ or $𝒢$-structure sheaf on $𝒳$ if

• it is a left exact functor;

• it respects gluing in $𝒢$ in that for $\{U_i\to V{\}}_i$ a covering sieve consisting of admissible morphism, the induced morphism

  $$\coprod _i{O}_X(U_i)\to O_X(V)$$\coprod_i O_X(U_i) \to O_X(V)

  is an effective epimorphism in $𝒳$.

Write ${\mathrm{Str}}_{𝒢}(𝒳)\subset \mathrm{Func}(𝒢,𝒳)$ for the full subcategory of such morphisms of the (∞,1)-category of (∞,1)-functors.

Urs Schreiber: second attempt to illustrate what this means:

Examples

1. Consider $X$ an ordinary topological space and $\mathrm{Sh}(X)$ the ordinary category of sheaves on its category of open subsets. Let $𝒢=\mathrm{Top}$ be some small version of Top with its usual Grothendieck topology with covering families being open covers. Consider the functor

   $$O_X:\mathrm{Top}\to \mathrm{Sh}(X)$$O_X : Top \to Sh(X)

   that sends a topological space $V$ to the sheaf of continuous functions with values in $V$:

   $$O_X(V):U\mapsto C(U,V)={\mathrm{Hom}}_{\mathrm{Top}}(U,V).$$O_X(V) : U \mapsto C(U,V) = Hom_{Top}(U,V) \,.

   This functor clearly respects limits, just by the general property of the hom. The gluing condition says that for $V_1,V_2\subset V$ an open cover of $V$ by two patches, the morphism of sheaves

   $$O_X(V_1)\coprod O_X(V_2)\to O_X(V)$$O_X(V_1) \coprod O_X(V_2) \to O_X(V)

   is an epimorphism of sheaves. This means that for each point $x\in X$ the map of stalks

   $$O_X(V_1{)}_x\coprod O_X(V_2{)}_x\to O_X(V{)}_x$$O_X(V_1)_x \coprod O_X(V_2)_x \to O_X(V)_x

   is an epimorphism of sets. But this just says that given any function $f:U_x\to V$ on a neighbourhood $U_x$ of $x$, there is a smaller neighbourhood $V_x\subset U_x$ such that the restriction $f{\mid }_{V_x}$ factors either through $V_1$ or through $V_1$. This is clearly the case by the fact that $V_1,V_2$ form an open cover. (A neighbourhood of $f(x)\in V$ exists which is contained in $V_1$ or in $V_2$, so take its preimage under $f$ as $U_x$).

2. StrSh, remark 2.5.11

   Let $X$ be a topological space as before, but consider now the geometry $𝒢={\mathrm{CRing}}^{\mathrm{op}}$ to be the opposite category of commutative rings, where a covering family of $\mathrm{Spec}R\in {\mathrm{CRing}}^{\mathrm{op}}$ is a family of maps of the form $R\to R[\frac{1}{r_i}]$ with $\{r_i\in R{\}}_i$ generating the unit ideal in $R$. So we think of $\mathrm{Spec}R[\frac{1}{r_i}]$ as an the open subset of $\mathrm{Spec}R$ on wich the function $r_i$ does not vanish, and a covering family we think of as an open cover by such open subsets, in direct analogy to the above example.

   Now given a sheaf of rings

   $${\overline{O}}_X\in \mathrm{Sh}(X,\mathrm{CRing})$$\bar O_X \in Sh(X,CRing)

   on $X$ (making $X$ a ringed space), which we may regard as the functor

   $$O_X:{\mathrm{CRing}}^{\mathrm{op}}\to \mathrm{Sh}(X)$$O_X : CRing^{op} \to Sh(X)

   that it represents

   $$O_X(\mathrm{Spec}R):U\mapsto {\mathrm{Hom}}_{{\mathrm{CRing}}^{\mathrm{op}}}({\overline{O}}_X(U),\mathrm{Spec}R)={\mathrm{Hom}}_{\mathrm{CRing}}(R,{\overline{O}}_X(U))$$O_X(Spec R) : U \mapsto Hom_{CRing^{op}}(\bar O_X(U), Spec R) = Hom_{CRing}(R, \bar O_X(U))

   we check what conditions this has to satisfy to qualify as a structure sheaf in the above sense.

   The condition that

   $$\coprod _i{O}_X(\mathrm{Spec}R[\frac{1}{r_i}])\to \mathrm{Spec}R$$\coprod_i O_X(Spec R[\frac{1}{r_i}]) \to Spec R

   is an epimorphism of sheaves again means that it is stalkwise an epimorphism of sets. Now, a ring homomorphism $R[\frac{1}{r_i}]\to {\overline{O}}_X(U)$ is given by a ring homomorphism $f:R\to O_X(U)$ such that $f(r_i)$ is invertible in $O_X(U)$. (We think of this as the pullback of functions on $\mathrm{Spec}R$ to functions on $U$ by a map $U\to \mathrm{Spec}R$ that lands only in the open subset where the functoin $r_i$ is non-vanishing).

   So the condition that the above is an epimorphism on small enough $U$ says that for every ring homomorphism $\varphi :R\to {\overline{O}}_X(U)$ the value of $\varphi$ on at least one of the $r_i$ is invertible element in $O_X(U)$.

   Thinking of this dually this is just the same kind of statement as in the first example, really. But now we can say this more algebraically:

   by assumption there is a linear combination of the $r_i$ to the identity in $R$

   $$1=\sum _i{\alpha }_i{r}_i$$1 = \sum_i \alpha_i r_i

   in $R$ (the partition of unity of functions on $\mathrm{Spec}R$) and hence ${\sum }_i{\alpha }_i\varphi (r_1)=1$ in $(O_X{)}_x$ That for this invertible finite sum at least one of the summands is invertible is the condition that $(O_X{)}_x$ is a local ring .

   So ringed space has a structure sheaf in the above sense if it is a locally ringed space.

Examples

Ordinary ringed spaces

It may be worthwhile to retell the motivating example in the “Idea” introduction above for the maybe more familiar case of ordinary (1-categorical) structure sheaves with values in (unital) rings.

An ordinary topological space $X$ with its category of open subsets $\mathrm{Op}(X)$ is a ringed space or $\mathrm{Op}(X)$ is a ringed site if it is equipped with a sheaf $O_X:\mathrm{Op}(X{)}^{\mathrm{op}}\to \mathrm{Rings}$ with values in the category of rings. For $U\subset X$ one thinks of $O_X(U)$ as the ring of allowed functions on $U$.

If for the moment we ignore the technicality that $O_X$ is supposed to be a sheaf and just regard it as a presheaf, and if we furthermore invoke the idea of space and quantity and think of a ring $R$ as a generalized quantity in form of a copresheaf, canonically the co-representable co-presheaf

$$R:({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}Set$$R : (Ring^fin)^{op} \Set

on finitely generated rings, which sends

$$R:R\prime \mapsto \mathrm{Hom}(R,R\prime )$$R : R' \mapsto Hom(R,R')

then we find that $O_X$ is in fact a presheaf on $\mathrm{Op}(X)$ with values in a co-presheaf on $({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$

$$O_X:\mathrm{Op}(X{)}^{\mathrm{op}}\to [({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}},\mathrm{Set}]$$O_X : Op(X)^{op} \to [(Ring^{fin})^{op}, Set]

or equivalently a generalized quantity on $({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ with values in presheaves on $X$:

$$O_X:({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}\to [\mathrm{Op}(X{)}^{\mathrm{op}},\mathrm{Set}].$$O_X : (Ring^{fin})^{op} \to [Op(X)^{op}, Set] \,.

Mike Shulman: What is the admissibility structure on $({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ in this example? It seems to me like you would want to consider only functions preserving finite limits, since that would amount to treating $({\mathrm{Ring}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ as the syntactic category of the theory of rings and you would recover the usual notion of a ring object in $\mathrm{Sh}(X)$. But preserving finite limits doesn’t seem to be a property that could be specified by an admissibility structure.

Derived ringed spaces

Now formulate the previous example according to the above definition:

Let ${\mathrm{CRing}}^{\mathrm{fin}}$ be the category of finitely generated commutative rings There is a standard admissability structure on $({\mathrm{CRing}}^{\mathrm{fin}}{)}^{\mathrm{op}}$ that makes it a geometry in the above sense.

Then for $X$ a topological space an $(\mathrm{\infty }\mathrm{,1})$-functor $({\mathrm{CRing}}^{\mathrm{fin}}{)}^{\mathrm{op}}\to \mathrm{Sh}(S)$ to $(\mathrm{infty}\mathrm{,1})$-sheaves on $X$ is a sheaf of local commutative rings on $X$. (StrSh, example 1.2.13)

To generalize this to derived structure sheaves we replace the category of rings here with the $(\mathrm{\infty }\mathrm{,1})$-category of simplicial rings.

Definition (StrSh def 4.1.1)

The $(\mathrm{\infty }\mathrm{,1})$-category of simplicial commutative rings over an ordinary commutative ring $k$ is

$${\mathrm{SCR}}_k:={\mathrm{PSh}}_{\Sigma }({\mathrm{FreeAlg}}_k)$$SCR_k := PSh_\Sigma(FreeAlg_k)

the $(\mathrm{\infty }\mathrm{,1})$-category of (∞,1)-presheaves on commutative $k$-algebras of the form $k[x_1,\cdots ,x_n]$.

Then…

see also

• generalized scheme

Derived smooth manifolds

(StrSh, example 4.5.2)

Every ordinary smooth manifold $X$ becomes canonically a generalized space with structure sheaf as follows:

Let $V:=\mathrm{Diff}$ be some version of the category of smooth manifolds. This becomes a pregeometry in the above sense by taking admissable morphisms to be inclusions of open submanifolds.

Then for $\mathrm{Sh}(X):=\mathrm{Sh}(\mathrm{Op}(X))$ the $(\mathrm{\infty }\mathrm{,1})$-topos of $(\mathrm{\infty }\mathrm{,1})$-sheaves on $X$, the obvious $(\mathrm{\infty }\mathrm{,1})$-functor

$$O_X:V\to \mathrm{Sh}(X)$$O_X : V \to Sh(X)

which for every co-test manifold $v$ is the sheaf

$$O_X(v):(U\subset X)\mapsto {\mathrm{Hom}}_{\mathrm{Diff}}(U,v)$$O_X(v) : (U \subset X) \mapsto Hom_{Diff}(U,v)

is a $\mathrm{Diff}$-structure sheaf. Notice that this is precisely nothing but the structure sheaf of smooth functions from the introduction above.

The point is that there are other, more fancy structure sheaves

$$O_X:V\to \mathrm{Sh}(X)$$O_X : V \to Sh(X)

possible. They describe derived smooth manifolds as described in DerSmooth.

References

The general theory is developed in

• StrSh Jacob Lurie, Structured Spaces (arXiv)

The special case of “smoothly structured spaces” called derived smooth manifold is discussed in

• David Spivak, Derived smooth manifolds PhD thesis (pdf)

Apart from looking at the special case this article also contains useful introduction and details on the general case.

In the version of this that is available on the arXiv (arXiv) the point of view is more on bi-presheaves, a useful discussion to the relation to structured morphisms here is in section 10.1. there.