synthetic differential geometry
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(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)
$(\Re \dashv \Im \dashv \&)$
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$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
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Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
A generalization of the notion of foliation of a smooth manifold from manifolds to Lie algebroids.
One of several equivalent definitions of a (regular) foliation of a smooth manifold is
A regular foliation of a smooth manifold $X$ is a wide sub-Lie algebroid of its tangent Lie algebroid, hence a Lie algebroid $\mathcal{P}$ over $X$ with injective anchor map
In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.
A (regular) foliation of a Lie algebroid $A$ is a sub-double Lie algebroid of the tangent double Lie algebroid which is wide over $A$
Foliations of a Lie algbroid $A \to X$ according to def. are in natural bijection to the following data:
an ordinary foliation $\mathcal{P}_0 \hookrightarrow T X$
$\mathcal{P}_1 \hookrightarrow A$ a sub-vector bundle
(this is the joint kernel $\mathcal{P}_1 = ker(p_{T A}) \cap ker(d p_{A})$ naturally identified as a subspace of $A$)
$\nabla$ a flat connection on the quotient bundle $A/\mathcal{P}_1$ partially defined over vector fields in $\mathcal{P}_0$
(this is the induced linear foliation of the total space $A$ regarded as a horizontal-subspace distribution)
such that over every open subset $U \hookrightarrow X$
The sections of $A$ that become $\nabla$-constant in $A/\mathcal{P}_1$ form a sub-Lie algebra of $\Gamma_U(A|_U)$ (an integrable distribution of subspaces);
the sections of $\mathcal{P}_1$ are a Lie ideal inside this sub-Lie algebra;
the image of $\mathcal{P}_1$ under the anchor map is in $\mathcal{P}_0$;
the quotiented anchor map $A/\mathcal{P}_1 \to T X / \mathcal{P}_0$ intertwines $\nabla$ with the $\mathcal{P}_0$-Bott connection.
This is (EH, theorem 7.2).
According to prop. the notion of foliation of a Lie algebroid generalizes the notion of ideal system of a Lie algebroid (Higgins-Mackenzie, Mackenzie): A foliation as in def. comes from an ideal system in this sense precisely of $\mathcal{P}_0$ is a simple foliation (the quotient map exists in smooth manifolds and is a surjective submersion) and the holonomy of $\nabla$ is trivial.
Let $\mathcal{G}_\bullet$ be a Lie groupoid and let
be a foliation of a Lie groupoid, regarded as an internal groupoid in Lie algebroids. Then applying Lie differentiation yields a foliation of the Lie algebroid $Lie(\mathcal{G}_\bullet)$.
Maybe the first discussion of foliations of Lie algebroids appears in
Ideal systems of Lie algebroids have been introduced and studied in
Philip Higgins, Kirill Mackenzie, Algebraic constructions in the category of Lie algebroids . J. Algebra
129 (1990), no. 1, 194–230.MR1037400
Kirill Mackenzie, General theory of Lie groupoids and Lie algebroids Cambridge Univ. Press, Cambridge (2005)
Related discussion is in
Last revised on August 14, 2017 at 06:31:13. See the history of this page for a list of all contributions to it.