(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
The notion of structured -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 -topos is an (∞,1)-topos equipped with an (∞,1)-structure sheaf that we think of as the collection of functions on that preserve extra geometric structure – for instance continuous structure or smooth structure.
Being an -function -algebra, is an algebra over an (∞,1)-algebraic theory , called the (pre)geometry (for structured (∞,1)-toposes), since this encodes the nature of the extra geometric structure on .
Formally therefore a geometric structure -sheaf of is a product/limit-preserving (∞,1)-functor
Here we think of as being the (∞,1)-sheaf (∞,1)-topos on some (∞,1)-site and for any we think of
as being the (∞,1)-sheaf of structure-preserving functions on with values in .
Let be a geometry and let be the (∞,1)-topos of (∞,1)-sheaves on .
Notice that if is the nerve of the category of open subsets of some topological space , then is the (∞,1)-category of (∞,1)-sheaves on , as in the above motivating introduction.
We want to define a structure sheaf on (for instance on ) of quantities modeled on some -category to be an (∞,1)-functor
to be thought of as the assignment to each value space of an -sheaf of -valued functions on (on if ).
But since we are taking care of the sheaf condition on , we also want to allow a similar kind of co-sheaf condition on . In order to do so, is taken to be equipped with extra structure encoding covers in , and is then required to respect this structure suitably.
Geometries and admissibility structure
An admissiblility structure on an -category is
a choice of sub (∞,1)-category , whose morphisms are to be called the admissible morphisms, such that
for every admissible morphism and any morphism there is a diagram
for every diagram of the form
with and admissible, also is admissible.
a Grothendieck topology on 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 -category 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 sheaves: local -algebras
Let be a geometry and an -topos An (∞,1)-functor
is a -structure on or -structure sheaf on if
it is a left exact (∞,1)-functor;
it respects gluing in in that for a covering sieve consisting of admissible morphism, the induced morphism
is an effective epimorphism in .
Write for the full subcategory of such morphisms of the (∞,1)-category of (∞,1)-functors.
This is discussed below.
As algebras over geometric -theories
By the (∞,1)-Yoneda lemma, a cover-preserving functor 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 preserves finite (∞,1)-limits then so does its extension. Therefore local -∞-algebras in are equivalent to (∞,1)-geometric morphisms
This means that structure sheaves are equivalently encoded in geometric morphisms to the (∞,1)-category of (∞,1)-sheaves on the geometry.
Formally we have:
For a geometry, precomposition of the inverse image functor with the (∞,1)-Yoneda embedding induces an equivalence of (∞,1)-categories
between the (∞,1)-category of (∞,1)-geometric morphisms from to and the -category of local -∞-algebras in .
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 -functor
is spanned by the left exact and cover preserving functors.
Local morphisms between structure sheaves
(See StrSp, remark below prop. 1.4.2).
The admissibility structure does serve to allow the following definition of local morphisms of structure sheaves.
In terms of admissibility structures
(local morphism of structure sheaves)
A natural transformation of structure sheaves is local if for every admissible morphism in the naturality diagram
is a pullback square in .
for the sub-(∞,1)-category of -structures on spanned by local transformations between them.
In terms of classifying -toposes
Alternatively, the local transformations can be characterized as follows
it turns out the local transformations are the right half of a factorization system on , and that this factorization system depends functorially on , in that for every geometric morphism the induced respects these factorization systems. (theorem 1.3.1)
This one can turn around, to characterize local transformations (and hence admissibility structures on ) in terms of functorial factorization systems on classifying -toposes (def. 1.4.3):
For an -topos, declare that a geometric structure on is a choice of factorization systems on that is functorial in . Given such we have another way of saying “local transformation”: this is the non-full subcategory of 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 is the classifying topos for the geometry , the defining equivalence identifies the two sub-categories of local transformations, and .
The -category of structured -toposes
Let (∞,1)Cat be the sub (∞,1)-category of (∞,1)-toposes: objects are (∞,1)-toposes, morphisms are geometric morphisms.
(-category of -structured -toposes)
For a geometry, the -category of -structured -toposes
is defined as follows.
It is the sub (∞,1)-category
where is the coCartesian fibration associated by the (∞,1)-Grothendieck construction to the inclusion functor , spanned by the following objects and morphisms:
objects are -structures on some -topos :
an object in is an object together with a functor into the fiber of over that object; but that fiber is itself, so an object in the fiber product is a functor and this is in if it is a -structure on ;
morphisms are local morphisms of -structures:
for the image of in , is in precisely if for every admissible morphism in the square
is a pullback square in .
This is StrSp, def 1.4.8
The spectrum construction
For 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 a geometry, let be the corresponding discrete geometry. We have a canonical morphism .
for the composite.
This fits into an adjunction
This is (Lurie, theorem xyz).
Classes of examples
Canonical structure sheaves on objects in the big topos
For a geometry let
be the (∞,1)-category of (∞,1)-sheaves on . This is the big topos of higher geometry modeled on . By the above discussion it is also the classifying topos of -structure sheaves on toposes:
a -valued structure sheaf on an (∞,1)-topos is equivalently an (∞,1)-geometric morphism
in that , where is the (∞,1)-Yoneda embedding.
Notice that for every object its little topos-incarnation is the over-(∞,1)-topos . This canonically sits over by its etale geometric morphism
So we have
The little topos of every object in the big topos over is canonically equipped with a -structure sheaf
We want to show that for the (∞,1)-sheaf may indeed be thought of as the “sheaf of -valued functions on ”.
Notice that for any we have that .
Now assume first that is itself representable. Then by the discussion at over-(∞,1)-topos we have that is a lucalization of , where is the big site of . Under this equivalence (more details on this at over-topos) we have that identifies with the presheaf given by
This is the “sheaf of -valued functions on ”.
Structure sheaves of continuous functions
Consider an ordinary topological space and the ordinary category of sheaves on its category of open subsets. Let 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 to the sheaf of continuous functions with values in :
By general properties of the hom-functor, this respects limits. The gluing condition says that for an open cover of by two patches, the morphism of sheaves
is an epimorphism of sheaves. This means that for each point the map of stalks
is an epimorphism of sets. But this just says that given any function on a neighbourhood of , there is a smaller neighbourhood such that the restriction factors either through or through . This is clearly the case by the fact that form an open cover. (A neighbourhood of exists which is contained in or in , so take its preimage under as ).
Locally ringed spaces
StrSh, remark 2.5.11
Let be a topological space as before, but consider now the geometry to be the opposite category of commutative rings, where a covering family of is a family of maps of the form with generating the unit ideal in . So we think of as an the open subset of on wich the function 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 (making 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 is given by a ring homomorphism such that is invertible in . (We think of this as the pullback of functions on to functions on by a map that lands only in the open subset where the functoin is non-vanishing).
So the condition that the above is an epimorphism on small enough says that for every ring homomorphism the value of on at least one of the is invertible element in .
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 to the identity in
in (the partition of unity of functions on ) and hence in That for this invertible finite sum at least one of the summands is invertible is the condition that is a local ring .
So a ringed space has a structure sheaf in the above sense if it is a locally ringed space.
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 with its category of open subsets is a ringed space or is a ringed site if it is equipped with a sheaf with values in the category of rings. For one thinks of as the ring of allowed functions on .
If for the moment we ignore the technicality that 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 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 is in fact a presheaf on with values in a co-presheaf on
or equivalently a generalized quantity on with values in presheaves on :
Since rings can be identified with left-exact functors , 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.
Derived ringed spaces
Now formulate the previous example according to the above definition:
Let be the category of finitely generated commutative rings There is a standard admissibility structure on that makes it a geometry in the above sense.
Then for a topological space an -functor to -sheaves on is a sheaf of local commutative rings on . (StrSh, example 1.2.13)
To generalize this to derived structure sheaves we replace the category of rings here with the -category of simplicial rings.
Definition (StrSh def 4.1.1)
The -category of simplicial commutative rings over an ordinary commutative ring is
the -category of (∞,1)-presheaves on commutative -algebras of the form .
Derived smooth manifolds
(StrSh, example 4.5.2)
Every ordinary smooth manifold becomes canonically a generalized space with structure sheaf as follows:
Let 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 the -topos of -sheaves on , the obvious -functor
which for every co-test manifold is the sheaf
is a -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.
Limits and colimits
Let be a geometry for structured (infinity,1)-toposes. Write (∞,1)Topos for the forgetful (∞,1)-functor from -structured -toposes to their underlying -topos.
The -category has all cofiltered (∞,1)-limits and the forgetful functor preserves these.
This appears as (Lurie, corl 1.5.4).
Embedding into the ambient big -topos
The canonical inclusion
of locally representable structured (∞,1)-toposes by
is a full and faithful (∞,1)-functor.
This is (Lurie, theorem, 2.4.1).
ringed space, locally ringed space
ringed site, locally ringed site
ringed topos, locally ringed topos
locally algebra-ed topos
locally algebra-ed (∞,1)-topos
geometry (for structured (∞,1)-toposes), structured -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 -toposes was introduced in
Analogous precursor discussion in 1-category theory, hence for ringed toposes is in
- Monique Hakim, Topos annelés et schémas relatifs, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 64, Springer, Berlin, New York (1972).
The special case of “smoothly structured spaces” (derived smooth manifolds) is discussed in