A category of locally convex Lie algebroids
We study first-order locally convex Lie algebroids in the setting of Bastiani calculus. The first-order condition is automatic in finite rank, but is an additional regularity hypothesis for general locally convex vector bundles. We define sheaves of scalar-valued and vector-valued Lie algebroid forms as fiberwise continuous alternating maps with smooth local representatives. Morphisms are defined by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials. We prove that these algebroids form a category, study representations and cohomology sheaves, and show that the Lie algebroids and morphisms of locally convex Lie groupoids belong to this category. We also give two applications. For a compact manifold $K$, the current algebroid $C^\infty(K,A)\to C^\infty(K,M)$ of a first-order Banach Lie algebroid $A\to M$ is a first-order Fréchet Lie algebroid, provided $M$ admits local addition. Finally, we prove a Banach Lie II theorem: a morphism from the Lie algebroid of a locally convex Lie groupoid to that of a Banach-Lie groupoid, both with Hausdorff source fibers, integrates uniquely to a groupoid morphism when the source groupoid is source-connected and source-simply connected.