(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
The notion of structured $(\infty,1)$-topos is a generalization of the notion of a locally ringed space and locally ringed topos generalized to (∞,1)-toposes. This is a way to formalize higher geometry/derived geometry-structure on little (∞,1)-toposs.
So a structured $(\infty,1)$-topos is an (∞,1)-topos $\mathcal{X}$ equipped with an (∞,1)-structure sheaf $\mathcal{O}$ that we think of as the collection of functions on $\mathcal{X}$ that preserve extra geometric structure – for instance continuous structure or smooth structure.
Being an $\infty$-function $\infty$-algebra, $\mathcal{O}(\mathcal{U})$ is an algebra over an (∞,1)-algebraic theory $\mathcal{G}$, called the (pre)geometry (for structured (∞,1)-toposes), since this encodes the nature of the extra geometric structure on $\mathcal{X}$.
Formally therefore a geometric structure $(\infty,1)$-sheaf of $\mathcal{X}$ is a product/limit-preserving (∞,1)-functor
Here we think of $\mathcal{X} = Sh_{(\infty,1)}(C)$ as being the (∞,1)-sheaf (∞,1)-topos on some (∞,1)-site $C$ and for any $V \in \mathcal{G}$ we think of
as being the (∞,1)-sheaf of structure-preserving functions on $C$ with values in $V$.
Let $S$ be a geometry and let $Sh(S)$ be the (∞,1)-topos of (∞,1)-sheaves on $S$.
Notice that if $S = Op(X)$ is the nerve of the category of open subsets of some topological space $X$, then $Sh(X) \coloneqq Sh(S)$ 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 $Op(X)$) of quantities modeled on some $(\infty,1)$-category $V$ to be an (∞,1)-functor
to be thought of as the assignment to each value space $v \in V$ of an $(\infty,1)$-sheaf of $v$-valued functions on $S$ (on $X$ if $S = Op(X)$).
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.
An admissiblility structure on an $(\infty,1)$-category $V$ is
a choice of sub (∞,1)-category $V^{ad} \hookrightarrow V$, whose morphisms are to be called the admissible morphisms, such that
for every admissible morphism $U \to X$ and any morphism $X' \to X$ there is a diagram
with $U' \to X'$ admissible;
for every diagram of the form
with $X \to Z$ and $Y \to Z$ admissible, also $X \to Y$ is admissible.
a Grothendieck topology on $V$ which has the property that it is generated from a coverage consisting of admissible morphisms.
This is StrSh, def 1.2.1 in view of remark 1.2.4 below that.
An $(\infty,1)$-category $V$ equipped with an admissiblility structure is a geometry if it is essentially small, admits finite limits and is idempotent complete.
The admissible morphisms in an admissibility structure are roughly to be thought of as those morphisms that behave as open immersions ,
(structure sheaf)
Let $\mathcal{G}$ be a geometry and $\mathcal{X}$ an $(\infty,1)$-topos An (∞,1)-functor
is a $\mathcal{G}$-structure on $\mathcal{X}$ or $\mathcal{G}$-structure sheaf on $\mathcal{X}$ if
it is a left exact (∞,1)-functor;
it respects gluing in $\mathcal{G}$ in that for $\{U_i \to V\}_i$ a covering sieve consisting of admissible morphism, the induced morphism
is an effective epimorphism in $\mathcal{X}$.
Write $Str_{\mathcal{G}}(\mathcal{X}) \subset Func(\mathcal{G},\mathcal{X})$ for the full subcategory of such morphisms of the (∞,1)-category of (∞,1)-functors.
Without the condition on preservations of covers, the above defined the ∞-algebras over the essentially algebraic (∞,1)-theory $\mathcal{G}$. The preservation of covers encodes the local $\mathcal{G}$-algebras.
Therefore we shall equivalently write
$\mathcal{G}Alg_{loc}(\mathcal{X})$ is the $(\infty,1)$-category of algebras over an $(\infty,1)$-geometric theory.
This is discussed below.
By the (∞,1)-Yoneda lemma, a cover-preserving functor $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ Yoneda extends equivalently to a (∞,1)-colimit-preserving (∞,1)-functor
By the adjoint (∞,1)-functor theorem this has a right adjoint (∞,1)-functor and if $\mathcal{O}$ preserves finite (∞,1)-limits then so does its extension. Therefore local $\mathcal{G}$-∞-algebras in $\mathcal{X}$ are equivalent to (∞,1)-geometric morphisms
This means that structure sheaves $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ are equivalently encoded in geometric morphisms to the (∞,1)-category of (∞,1)-sheaves on the geometry.
Formally we have:
For $\mathcal{G}$ a geometry, precomposition of the inverse image functor with the (∞,1)-Yoneda embedding $y : \mathcal{G} \to Sh_{(\infty,1)}(\mathcal{G})$ induces an equivalence of (∞,1)-categories
between the (∞,1)-category of (∞,1)-geometric morphisms from $\mathcal{X}$ to $Sh_{(\infty,1)}(\mathcal{G})$ and the $(\infty,1)$-category of local $\mathcal{G}$-∞-algebras in $\mathcal{X}$.
This is (StrSp, prop 1.42).
This follows from the general fact, discussed in the section Local yoneda embedding at (∞,1)-Yoneda lemma that the essential image of the $(\infty,1)$-functor
is spanned by the left exact and cover preserving functors.
We may think of this as saying that $Sh_{(\infty,1)}(\mathcal{G})$ is the $(\infty,1)$-classifying topos for the $(\infty,1)$-geometric theory of local ∞-algebras over the essentially algebraic (∞,1)-theory $\mathcal{G}$.
The $(\infty,1)$-category $Str_{\mathcal{G}}(\mathcal{X})$ of $\mathcal{G}$-structure sheaves on an $(\infty,1)$-topos $\mathcal{X}$ does not depend on the admissibility structure of $\mathcal{G}$, but only on the Grothendieck topology induced by it.
(See StrSp, remark below prop. 1.4.2).
The admissibility structure does serve to allow the following definition of local morphisms of structure sheaves.
(local morphism of structure sheaves)
A natural transformation $\eta : \mathcal{O} \to \mathcal{O}' : \mathcal{G} \to \mathcal{X}$ of structure sheaves is local if for every admissible morphism $U \to X$ in $\mathcal{G}$ the naturality diagram
is a pullback square in $\mathcal{X}$.
Write
for the sub-(∞,1)-category of $\mathcal{G}$-structures on $\mathcal{X}$ spanned by local transformations between them.
Alternatively, the local transformations can be characterized as follows
it turns out the local transformations are the right half of a factorization system on $Str_{\mathcal{G}}(\mathcal{X})$, and that this factorization system depends functorially on $\mathcal{X}$, in that for every geometric morphism $\mathcal{X} \to \mathcal{Y}$ the induced $Str_{\mathcal{G}}(\mathcal{X}) \to Str_{\mathcal{G}}(\mathcal{Y})$ respects these factorization systems. (theorem 1.3.1)
This one can turn around, to characterize local transformations (and hence admissibility structures on $\mathcal{G}$) in terms of functorial factorization systems on classifying $(\infty,1)$-toposes (def. 1.4.3):
For $\mathcal{K}$ an $(\infty,1)$-topos, declare that a geometric structure on $\mathcal{K}$ is a choice of factorization systems on $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ that is functorial in $\mathcal{X}$ . Given such we have another way of saying “local transformation”: this is the non-full subcategory $Str^{loc}_{\mathcal{K}}(\mathcal{X})$ of $Topos_{geom}(\mathcal{X}, \mathcal{K})^{op}$ on all objects and on the right part of the factorization system.
And this is indeed the same kind of datum as an admissibility structure on a geometry (example 1.4.4): in the case that $\mathcal{K} = Sh(\mathcal{G})$ is the classifying topos for the geometry $\mathcal{G}$, the defining equivalence $Topos_{geom}(\mathcal{X}, Sh(\mathcal{G}))^{op} \stackrel{\simeq}{\to} Str_{\mathcal{G}}(\mathcal{X})$ identifies the two sub-categories of local transformations, $Str^{loc}_{\mathcal{G}}(\mathcal{X})$ and $Str^{loc}_{Sh(\mathcal{G})}(\mathcal{X})$.
Let $(\infty,1)Toposes \subset$ (∞,1)Cat be the sub (∞,1)-category of (∞,1)-toposes: objects are (∞,1)-toposes, morphisms are geometric morphisms.
Write $LTop \coloneqq (\infty,1)Toposes^{op}$.
($(\infty,1)$-category of $\mathcal{G}$-structured $(\infty,1)$-toposes)
For $\mathcal{G}$ a geometry, the $(\infty,1)$-category of $\mathcal{G}$-structured $(\infty,1)$-toposes
is defined as follows.
It is the sub (∞,1)-category
where $E LTop \to LTop$ is the coCartesian fibration associated by the (∞,1)-Grothendieck construction to the inclusion functor $LTop \hookrightarrow (\infty,1)Cat$, spanned by the following objects and morphisms:
objects are $\mathcal{G}$-structures $\mathcal{O} : \mathcal{G} \to \mathcal{X}$ on some $(\infty,1)$-topos $\mathcal{X}$:
an object in $Func(\mathcal{G}, E LTop) \times_{Func(\mathcal{G}, LTop)} LTop$ is an object $\mathcal{X} \on LTop$ together with a functor $\mathcal{G} \to E LTop|_{\mathcal{X}}$ into the fiber of $E Top$ over that object; but that fiber is $\mathcal{X}$ itself, so an object in the fiber product is a functor $\mathcal{G} \to \mathcal{X}$ and this is in $LTop(\mathcal{G})$ if it is a $\mathcal{G}$-structure on $\mathcal{X}$;
morphisms $\alpha : \mathcal{O} \to \mathcal{O}'$ are local morphisms of $\mathcal{G}$-structures:
for $f^* : \mathcal{X} \to \mathcal{Y}$ the image of $\alpha$ in $LTop$, $\alpha$ is in $LTop(\mathcal{G})$ precisely if for every admissible morphism $U \to X$ in $\mathcal{G}$ the square
is a pullback square in $\mathcal{Y}$.
This is StrSp, def 1.4.8
For $f : \mathcal{G} \to \mathcal{G}'$ a morphism of geometries, let
be the induced functor on categories of structured toposes.
This functor is a left adjoint (∞,1)-functor
This is (Lurie, theorem 2.1.1).
For $\mathcal{G}$ a geometry, let $\mathcal{G}_0$ be the corresponding discrete geometry. We have a canonical morphism $\mathcal{G}_0 \to \mathcal{G}$.
Write
for the composite.
This fits into an adjunction
This is (Lurie, theorem xyz).
For $\mathcal{G}$ a geometry let
be the (∞,1)-category of (∞,1)-sheaves on $\mathcal{G}$. This is the big topos of higher geometry modeled on $\mathcal{G}$. By the above discussion it is also the classifying topos of $\mathcal{G}$-structure sheaves on toposes:
a $\mathcal{G}$-valued structure sheaf $\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}$ on an (∞,1)-topos $\mathcal{X}$ is equivalently an (∞,1)-geometric morphism
in that $\mathcal{O}_{\mathcal{X}} = p^* j$, where $j$ is the (∞,1)-Yoneda embedding.
Notice that for every object $X \in \mathcal{H}$ its little topos-incarnation is the over-(∞,1)-topos $\mathbf{H}/X$. This canonically sits over $\mathbf{H}$ by its etale geometric morphism
So we have
The little topos $\mathcal{X} \coloneqq \mathbf{H}/X$ of every object $X$ in the big topos $\mathbf{H}$ over $\mathcal{G}$ is canonically equipped with a $\mathcal{G}$-structure sheaf
We want to show that for $V \in \mathcal{G}$ the (∞,1)-sheaf $\mathcal{O}_X(V)$ may indeed be thought of as the “sheaf of $V$-valued functions on $X$”.
Notice that for any $V \in \mathbf{H}$ we have that $X^*(F) = (p_2 : V \times X \to X)$.
Now assume first that $X$ is itself representable. Then by the discussion at over-(∞,1)-topos we have that $\mathbf{H}/X$ is a lucalization of $PSh_\infty(\mathcal{G})/X \simeq PSh_\infty(\mathcal{G}/X)$, where $\mathcal{G}/X$ is the big site of $X$. Under this equivalence (more details on this at over-topos) we have that $(V \times X \to X)$ identifies with the presheaf given by
This is the “sheaf of $V$-valued functions on $X$”.
(…)
Consider $X$ an ordinary topological space and $Sh(X)$ the ordinary category of sheaves on its category of open subsets. Let $\mathcal{G} = Top$ be some small version of Top with its usual Grothendieck topology with admissible covering families being open covers. Consider the functor
that sends a topological space $V$ to the sheaf of continuous functions with values in $V$:
By general properties of the hom-functor, this respects limits. The gluing condition says that for $V_1, V_2 \subset V$ an open cover of $V$ by two patches, the morphism of sheaves
is an epimorphism of sheaves. This means that for each point $x \in X$ the map of stalks
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 $W_x \subset U_x$ such that the restriction $f|_{W_x}$ factors either through $V_1$ or through $V_2$. 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 $U_x$).
Let $X$ be a topological space as before, but consider now the geometry $\mathcal{G} = CRing^{op}$ to be the opposite category of commutative rings, where a covering family of $Spec R \in CRing^{op}$ 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 $Spec R[\frac{1}{r_i}]$ as an the open subset of $Spec R$ 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
on $X$ (making $X$ a ringed space), which we may regard as the functor
that it represents
we check what conditions this has to satisfy to qualify as a structure sheaf in the above sense.
The condition that
is an epimorphism of sheaves again means that it is stalkwise an epimorphism of sets. Now, a ring homomorphism $R[\frac{1}{r_i}] \to \bar O_X(U)$ 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 $Spec R$ to functions on $U$ by a map $U \to Spec R$ 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 $\phi : R \to \bar O_X(U)$ the value of $\phi$ 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 $R$
in $R$ (the partition of unity of functions on $Spec R$) and hence $\sum_i \alpha_i \phi(r_1) = 1$ 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 a ringed space has a structure sheaf in the above sense if it is a locally ringed space.
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 $Op(X)$ is a ringed space or $Op(X)$ is a ringed site if it is equipped with a sheaf $O_X : Op(X)^{op} \to Rings$ 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
on finitely generated rings, which sends
then we find that $O_X$ is in fact a presheaf on $Op(X)$ with values in a co-presheaf on $(Ring^{fin})^{op}$
or equivalently a generalized quantity on $(Ring^{fin})^{op}$ with values in presheaves on $X$:
Since rings can be identified with left-exact functors $(Ring^{fin})^{op}\to Set$, we don’t need to impose any admissibility structure in order to recover the notion of a sheaf of rings, since left-exactness is part of the definition of a “structure sheaf.” We do, however, need an admissibility structure if we want to recover the notion of a sheaf of local rings, as in the previous example above.
Now formulate the previous example according to the above definition:
Let $CRing^{fin}$ be the category of finitely generated commutative rings There is a standard admissibility structure on $(CRing^{fin})^{op}$ that makes it a geometry in the above sense.
Then for $X$ a topological space an $(\infty,1)$-functor $(CRing^{fin})^{op} \to Sh(S)$ to $(infty,1)$-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 $(\infty,1)$-category of simplicial rings.
Definition (StrSh def 4.1.1)
The $(\infty,1)$-category of simplicial commutative rings over an ordinary commutative ring $k$ is
the $(\infty,1)$-category of (∞,1)-presheaves on commutative $k$-algebras of the form $k[x_1, \cdots, x_n]$.
Then…
Every ordinary smooth manifold $X$ becomes canonically a generalized space with structure sheaf as follows:
Let $V \coloneqq Diff$ be some version of the category of smooth manifolds. This becomes a pregeometry in the above sense by taking admissible morphisms to be inclusions of open submanifolds.
Then for $Sh(X) \coloneqq Sh(Op(X))$ the $(\infty,1)$-topos of $(\infty,1)$-sheaves on $X$, the obvious $(\infty,1)$-functor
which for every co-test manifold $v$ is the sheaf
is a $Diff$-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
possible. They describe derived smooth manifolds as described in DerSmooth.
Let $\mathcal{G}$ be a geometry for structured (infinity,1)-toposes. Write $F : (\infty,1)Topos(\mathcal{G}) \to$ (∞,1)Topos for the forgetful (∞,1)-functor from $\mathcal{G}$-structured $(\infty,1)$-toposes to their underlying $(\infty,1)$-topos.
The $(\infty,1)$-category $(\infty,1)Topos(\mathcal{G})$ has all cofiltered (∞,1)-limits and the forgetful functor $F : (\infty,1)Topos(\mathcal{G}) \to (\infty,1)Topos$ preserves these.
This appears as (Lurie, corl 1.5.4).
For $\mathcal{G}$ a geometry (for structured (∞,1)-toposes) write
for its (∞,1)-category of pro-objects.
Write $\widehat{\infty Grpd}$ for the very large (∞,1)-category of large ∞-groupoids and
for the very large (∞,1)-sheaf (∞,1)-topos.
The canonical inclusion
of locally representable structured (∞,1)-toposes by
This is (Lurie, theorem, 2.4.1).
locally algebra-ed (∞,1)-topos
geometry (for structured (∞,1)-toposes), structured $(\infty,1)$-topos , locally representable structured (∞,1)-topos
Analogous structures in the axiomatic context of differential cohesion are discussed in differential cohesion – Structure sheaves.
The notion of structured $(\infty,1)$-toposes was introduced in
Analogous precursor discussion in 1-category theory, hence for ringed toposes is in
The special case of “smoothly structured spaces” (derived smooth manifolds) is discussed in