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This paper introduces and studies "first-order locally convex Lie algebroids" within the framework of Bastiani calculus. The authors define these algebroids by imposing a "first-order condition" on locally convex vector bundles, which is automatic in finite dimensions but serves as a regularity hypothesis for the infinite-dimensional case. They define sheaves of Lie algebroid forms and prove that first-order locally convex Lie algebroids form a category. The paper also explores representations and cohomology sheaves, and demonstrates that locally convex Lie groupoids give rise to first-order Lie algebroids. Key applications include proving that current algebroids associated with first-order Banach Lie algebroids are first-order Fréchet Lie algebroids and establishing a Banach Lie II theorem, which integrates morphisms of Lie algebroids of Banach-Lie groupoids into unique Lie groupoid morphisms under certain connectivity hypotheses.

The paper addresses several challenges in extending the theory of Lie algebroids to the setting of locally convex vector bundles:

Delicacy of the locally convex setting: Unlike finite-dimensional vector bundles, locally convex vector bundles do not always behave similarly. Specifically, exterior dual bundles or bundles of continuous alternating multilinear maps may not possess a canonical smooth vector bundle structure, preventing the direct use of standard definitions like Ω(M) = Γ(^A^).
Lack of automatic regularity: The "first-order condition," which is automatic for finite-dimensional Lie algebroids (where brackets are first-order expressions in local coefficients), is not guaranteed for general locally convex vector bundles. This condition is crucial for defining differentials and forms without relying on cotangent bundles or exterior dual bundles.
Higher-order behavior in infinite dimensions: In infinite-dimensional settings, analogous to Poisson geometry, first-order behavior is not automatic. Examples exist of queer Poisson brackets and Banach Lie algebroids where values can depend on higher derivatives, which the paper aims to rule out with its first-order condition.

Definition of first-order locally convex Lie algebroids.
Definition of sheaves of scalar-valued and vector-valued Lie algebroid forms.
Proof that first-order locally convex Lie algebroids form a category.
Study of representations and induced cohomology sheaves.
Demonstration that locally convex Lie groupoids yield first-order Lie algebroids.
Proof that current algebroids associated with first-order Banach Lie algebroids are first-order Fréchet Lie algebroids.
Proof of a Banach Lie II theorem, integrating morphisms of Lie algebroids of Banach-Lie groupoids into unique Lie groupoid morphisms.

The paper develops the theory of first-order locally convex Lie algebroids by:

Introducing the "first-order condition": This condition is a local regularity hypothesis on locally convex vector bundles, ensuring that certain operations (like brackets) behave locally as first-order differential operators.
Defining Lie algebroid forms as sheaves: Instead of using spaces of sections of tensor products, Lie algebroid forms (scalar-valued and vector-valued) are defined as sheaves of fiberwise continuous alternating maps with smooth local representatives. This approach avoids topology choices on spaces of continuous multilinear maps.
Defining morphisms: Morphisms of first-order Lie algebroids are defined via the pullback of Lie algebroid forms, requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials.
Establishing category structure: The paper proves that compositions of these morphisms are again morphisms, thus showing that first-order locally convex Lie algebroids form a category. It also shows that morphisms of locally convex Lie groupoids induce morphisms of their Lie algebroids.
Applications: The framework is applied to prove results about current algebroids and to establish a Banach Lie II theorem for integrating Lie algebroid morphisms to Lie groupoid morphisms.

Framework: Bastiani calculus is used throughout. Model spaces are Hausdorff. Manifolds are generally Hausdorff, with the exception of arrow spaces of Lie groupoids, which are assumed to have Hausdorff bases and source fibers.
First-order condition: This is a key regularity hypothesis introduced in Definition 2.1, ensuring that the Lie bracket has a specific local structure.
Lie algebroid forms: Defined as sheaves of fiberwise continuous alternating maps with smooth local representatives.
Morphisms: Defined by the pullback of forms commuting with Lie algebroid differentials.
Lie II Theorem: Proved for Banach-Lie groupoids, following path-space methods. Admissible algebroid paths are reconstructed by solving ODEs.
Key Equations:
- The definition of a first-order Lie algebroid involves an anchor a: A \to TM and a bracket [\cdot, \cdot]A satisfying specific properties, including a local formula for the bracket: \xi, \eta = a(\xi).\eta^\phi(x) - a(\eta).\xi^\phi(x) + C\phi(x, \xi^\phi(x), \eta^\phi(x)), where \xi^\phi, \eta^\phi are local representatives.
- The definition of a first-order Lie algebroid morphism F: A \to \tilde{A} over f: M \to \tilde{M} involves a condition on the pullback of forms: F^# : f^{-1}\Omega^k_A \to \Omega^k_{\tilde{A}} is a morphism of sheaves of dg-algebras.
Training Settings: Not applicable, as this is a theoretical mathematics paper.
Datasets: Not applicable.
Baselines: Not applicable.

Category of first-order Lie algebroids: The paper establishes that first-order locally convex Lie algebroids, equipped with the defined morphisms, form a category.
Lie algebroids of Lie groupoids: It is shown that locally convex Lie groupoids have associated first-order Lie algebroids, and Lie groupoid morphisms induce morphisms in this category.
Current algebroids: Current algebroids associated with first-order Banach Lie algebroids are proven to be first-order Fréchet Lie algebroids.
Banach Lie II Theorem: A key result is the proof of a Banach Lie II theorem, stating that under source-connected and source-simply connected hypotheses, first-order morphisms between the Lie algebroids of Banach-Lie groupoids integrate to unique Lie groupoid morphisms.
Representations: Properties of representations of Lie algebroids are studied, including their pullback along Lie algebroid morphisms.

No explicit paper-reported limitations were found.

Method evidence: "We define morphisms by requiring the induced pullback on inverse-image sheaves of scalar-valued forms to commute with the Lie algebroid differentials, and prove that first-order locally convex Lie algebroids form a category." (Abstract)
Result evidence: "As applications, we prove that the current algebroid associated with a first-order Banach Lie algebroid is a first-order Fréchet Lie algebroid, and we prove a Lie II theorem in the Banach setting: first-order morphisms between the Lie algebroids of Banach-Lie groupoids under source-connected and source-simply connected hypotheses integrate to unique Lie groupoid morphisms." (Abstract)
Limitation evidence: Not stated in the paper.

A category of locally convex Lie algebroids

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