synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
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 $\mathbb{R}^n$, for $n \in \mathbb{N}$. 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 $Pro(\mathcal{C}) \simeq (Ind(\mathcal{C}^{op}))^{op}$ (remark) it is sufficient to show that the functor in question is on opposite categories a fully faithful functor of the form
where $SmothAlg_{\mathbb{R}}$ 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 $i^{op} \colon CartSp^{op} \hookrightarrow (SmthAlg_{\mathbb{R}})_{cpt} \hookrightarrow SmthAlg$, 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 $Ind(i^{op})$ takes formal filtered colimits in $CartSp^{op}$ to the corresponding formal colimits in $SmthAlg_{\mathbb{R}}$ (prop.), while $L$ 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 $\mathcal{K} = \mathbb{N}_{\geq}$.
For $U \in TowCartSp$ a tower of Cartesian spaces (def. ), say that a tower of good open covers of $U$ is a sequence of morphisms $\{U_i \overset{\phi_i}{\to} U\}$ in $TowCartSp$ such that these are the formal sequential limit of a cofiltered diagram of good open covers $\{U_i^k \overset{\phi_i^k}{\to} U^k\}$.
The collection of towers of good open covers on $TowCartSp$, according to def. , constitutes a coverage.
By the definition of coverage (def.) we need to check that for every tower of good open covers $\{U_i \overset{\phi_i}{\to} U\}$ and for every morphism $V \overset{g}{\longrightarrow} U$ in $TowCartSp$, there exists a tower of good open covers $\{V_j \overset{\psi_j}{\longrightarrow} V\}$ of $V$ such that for each index $j$ we may find an index $i$ and a morphism $V_j \overset{}{\to} U_i$ 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 $(g^k)^\ast U^k_i \to V^{n(k)}$ and then refining this to a good open cover $\tilde V^{n(k)}_j \to V^{n(k)}$. 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 $\{\tilde V^{n(k+1)}_j\}$ with the pullback of $\{V^{n(k)}_j\}$ to $V^{n(k+1)}$.
This refines the above commuting cubes to
and hence provides components for the required diagram in $TowCartSp$.
Last revised on September 20, 2017 at 10:31:03. See the history of this page for a list of all contributions to it.