Manifolds and cobordisms
A smooth manifold is a space that is locally isomorphic to a Cartesian space equipped with its canonical smooth structure.
Traditionally, a smooth manifold is defined as follows.
As special topological manifolds
As special locally ringed spaces
A smooth manifold is equivalently a locally ringed space which is locally isomorphic to the ringed space .
General abstract geometric definition
There is a more fundamental and general abstract way to think of smooth manifolds, which realizes their theory as a special case of general constructions in higher geometry.
In this context one specifies for instance a geometry (for structured (∞,1)-toposes) and then plenty of geometric notions are defined canonically in terms of . The theory of smooth manifolds appears if one takes CartSp.
Alternatively one can specify differential cohesion and proceed as discussed at differential cohesion – structures - Cohesive manifolds (separated).
This is discussed in The geometry CartSp below.
Let CartSp be the category of Cartesian spaces and smooth functions between them. This has finite products and is in fact (the syntactic category) of a Lawvere theory: the theory of smooth algebras.
Moreover, is naturally equipped with the good open cover coverage that makes it a site.
Both properties together make it a pregeometry (for structured (∞,1)-toposes) (if the notion of Grothendieck topology is relaxed to that of coverage in StrSp).
For a topos, a product-preserving functor
is a -algebra in . This makes is -ringed topos. For CartSp this algebra is a smooth algebra in . If has a site of definition , then this is a [sheaf] of smooth algebras on .
If sends covering families in to effective epimorphism we say that it is a local -algebra in , making a -locally ringed topos.
The big topos itself is canonically equipped with such a local -algebra, given by the Yoneda embedding followed by sheafification
It is important in the context of locally representable locally ringed toposes that we regard as equipped with this local -algebra. This is what remembers the site and gives a notion of local representability in the first place.
The big topos is a cohesive topos of generalized smooth spaces. Its concrete sheaves are precisely the diffeological spaces. See there for more details. We now discuss how with regarded as a -structured topos, smooth manifolds are precisely its locally representable objects.
Cartesian spaces as representable objects of
The representables themselves should evidently be locally representable and canonically have the structure of -structured toposes.
Indeed, every object is canonically a CartSp-ringed space, meaning a topological space equipped with a local sheaf of smooth algebras. More generally: every object is canonically incarnated as the -structured (∞,1)-topos
given by the over-(∞,1)-topos of the big (∞,1)-sheaf (∞,1)-topos over and the structure sheaf given by the composite of the (∞,1)-Yoneda embedding and the inverse image of the etale geometric morphism induced by .
Smooth manifolds as locally representable objects of
Say a concrete object in the sheaf topos – a diffeological space – is locally representable if there exists a family of open embeddings with such that the canonical morphism out of the coproduct
is an effective epimorphism in .
Let be the full subcategory on locally representable sheaves.
There is an equivalence of categories
of the category Diff of smooth manifolds with that of locally representable sheaves for the pre-geometry .
Define a functor by sending each smooth manifold to the sheaf over that it naturally represents. By definition of manifold there is an open cover . We claim that is an effective epimorphism, so that this functor indeed lands in . (This is a standard argument of sheaf theory in Diff, we really only need to observe that it goes through over CartSp, too.)
For that we need to show that
is a coequalizer diagram in (that the Cech groupoid of the cover is equivalent to .). Notice that the fiber product here is just the intersection in . By the fact that the sheaf topos is by definition a reflective subcategory of the presheaf topos we have that colimits in are computed as the sheafification of the corresponding colimit in . The colimit in in turn is computed objectwise. Using this, we see that that we have a coequalizer diagram
in , where is the sieve corresponding to the cover: the subfunctor of the functor which assigns to the set of smooth functions that have the property that they factor through any one of the .
Essentially by the definition of the coverage on , it follows that sheafification takes this subfunctor inclusion to an isomorphism. This shows that is indeed the tip of the coequalizer in as above, and hence that it is a locally representable sheaf.
Conversely, suppose that for there is a family of open embeddings such that we have a coequalizer diagram
in , which is the sheafification of the corresponding coequalizer in . By evaluating this on the point, we find that the underlying set of is the coequalizer of the underlying set of the in . Since every plot of factors locally through one of the it follows that is a diffeological space.
It follows that in the pullback diagrams
the object is the diffeological space whose underlying topological space is the intersection of and in the topological space underlying . In particular the inclusions are open embeddings.
As locally representable -structured -toposes
We may switch from regarding smooth manifolds as objects in the big topos to regrading them as toposes themselves, by passing to the over-topos . This remembers the extra (smooth) structure on the topological space by being canonically a locally ringed topos with the structure sheaf of smooth functions on : a CartSp-structured (∞,1)-toposes
For every choice of geometry (for structured (∞,1)-toposes) there is a notion of -locally representable structured (∞,1)-topos (StrSp).
Sketch of proof
The statement says that a smooth manifold may be identified with an ∞-stack on CartSp (an ∞-Lie groupoid) which is represented by a CartSp-structured (∞,1)-topos such that
is a 0-localic (∞,1)-topos;
There exists a family of objects such that the canonical morphism to the terminal object in is a regular epimorphism;
For every there is an equivalence
The second and third condition say in words that is locally equivalent to the ordinary cannonically CartSp-locally ringed space (for the dimension. The first condition then says that these local identifications cover .
A textbook reference is
Discussion of smooth manifolds as colimits of the Cech nerves of their good open covers is also at
The general abstract framework of higher geometry referred to above is discussed in