Editing Lie algebroid (section)

=== Morphisms ===
A '''Lie algebroid morphism''' between two Lie algebroids <math>(A_1, [\cdot,\cdot]_{A_1}, \rho_1)</math> and <math>(A_2, [\cdot,\cdot]_{A_2}, \rho_2)</math> with the same base <math>M</math> is a vector bundle morphism <math>\phi: A_1 \to A_2</math> which is compatible with the Lie brackets, i.e. <math>\phi ([\alpha,\beta]_{A_1}) = [\phi(\alpha),\phi(\beta)]_{A_2}</math> for every <math>\alpha,\beta \in \Gamma(A_1)</math>, and with the anchors, i.e. <math>\rho_2 \circ \phi = \rho_1</math>.

A similar notion can be formulated for morphisms with different bases, but the compatibility with the Lie brackets becomes more involved.<ref>{{Cite book|last=Mackenzie|first=Kirill C. H.|url=https://www.cambridge.org/core/books/general-theory-of-lie-groupoids-and-lie-algebroids/DA70C6FAF52F88FB471F62DD68049608|title=General Theory of Lie Groupoids and Lie Algebroids|date=2005|publisher=Cambridge University Press|isbn=978-0-521-49928-6|series=London Mathematical Society Lecture Note Series|location=Cambridge|doi=10.1017/cbo9781107325883}}</ref> Equivalently, one can ask that the graph of <math>\phi: A_1 \to A_2</math> to be a subalgebroid of the direct product <math>A_1 \times A_2</math> (introduced below).<ref>Eckhard Meinrenken, [http://www.math.toronto.edu/mein/teaching/MAT1341_LieGroupoids/Groupoids.pdf Lie groupoids and Lie algebroids], Lecture notes, fall 2017</ref>

Lie algebroids together with their morphisms form a [[Category (mathematics)|category]].