synthetic differential geometry
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from point-set topology to differentiable manifolds
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Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A pro-manifold is a pro-object in a category of manifolds, i.e. a formal projective limit of manifolds.
Details depend on what exactly is understood by “manifold”, i.e. whether topological manifolds or smooth manifold, etc.
Typically one wants to mean pro-objects in manifolds of finite dimensions, the point being then that a pro-manifold is like an infinite-dimensional manifold but with “mild” infinite dimensionality, expressed by the very fact that it may be presented as a formal projective limit of finite dimensional manifolds.
To amplify this specification, one should properly speak of “pro-(finite dimensional smooth manifolds)”, but beware that people often abbreviate to “pro-manifold” regardless. Also “pro-finite manifold” is in use, which however, strictly speaking, is a misnomer since a “finite manifold” is one with a finite number of points.
An important example of pro-objects in finite-dimensional smooth manifolds are infinite jet bundles. These are the formal projective limits of the underlying finite-order jet bundles.
Write CartSp for the full subcategory of that of smooth manifolds on the Cartesian spaces, i.e. on those of the form , for . Write
for its category of pro-objects, the pro-Cartesian spaces.
The functor which sends a formal cofiltered limit of Cartesian spaces to its actual cofiltered limit of smooth loci is a fully faithful functor, hence constitutes a full subcategory inclusion of pro-Cartesian spaces (def. ) into smooth loci:
Since (remark) it is sufficient to show that the functor in question is on opposite categories a fully faithful functor of the form
where is the category of smooth algebras.
Now, there is the fully faithful functor
(prop.) hence a fully faithful functor
Moreover, the image of the latter is in compact objects , because
is co-representable, hence compact (by the Yoneda lemma and since colimits are computed objectwise prop.).
This implies that the composite
is also fully faithful (prop.).
Here takes formal filtered colimits in to the corresponding formal colimits in (prop.), while takes formal filtered colimits to actual filtered colimits (prop.). Hence this is indeed the functor in question.
under construction
Write
for the full subcategory of the category of pro-Cartesian spaces (def. ) on those pro-objects in CartSp which are presented as formal sequential limits of tower diagrams, i.e. where the indexing category .
For a tower of Cartesian spaces (def. ), say that a tower of good open covers of is a sequence of morphisms in such that these are the formal sequential limit of a cofiltered diagram of good open covers .
The collection of towers of good open covers on , according to def. , constitutes a coverage.
By the definition of coverage (def.) we need to check that for every tower of good open covers and for every morphism in , there exists a tower of good open covers of such that for each index we may find an index and a morphism such as to make a commuting diagram of the form
Now by this prop. the bottom morphism is represented by a sequence of component morphisms
Since ordinary good open covers do form a coverage on CartSp (prop.) each of these component diagrams may be completed
by first forming the pullback open cover and then refining this to a good open cover . By the universal property of the pullback, there are morphisms
that make the evident cube commute
Take
and then inductively define
to be a refinement by a good open cover of the joint refinement of with the pullback of to .
This refines the above commuting cubes to
and hence provides components for the required diagram in .
Last revised on September 20, 2017 at 10:31:03. See the history of this page for a list of all contributions to it.