[Paper Review] Affine geometric spaces in tangent categories

Richard Blute, G. S. H. Cruttwell+1 more|arXiv (Cornell University)|Jul 25, 2018

Homotopy and Cohomology in Algebraic Topology30 references3 citations

TL;DR

This paper introduces affine geometric spaces within tangent categories by characterizing them via flat torsion-free connections, showing that the category of such spaces inherits a tangent category structure. It establishes a 2-comonad on the 2-category of tangent categories, with affine tangent categories as Eilenberg-Moore coalgebras, and provides a novel characterization of flat torsion-free connections as morphisms from T(TM) to TM compatible with induced geometric structures.

ABSTRACT

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having an affine structure on a manifold is equivalent to having a flat torsion-free connection on its tangent bundle. This equivalence allows us to define a category of affine objects associated to a tangent category and we show that the resulting category is also a tangent category, as are several related categories. As a consequence of some of these ideas we also give two new characterizations of flat torsion-free connections. We also consider 2-categorical structure associated to the category of tangent categories and demonstrate that assignment of the tangent category of affine objects to a tangent category induces a 2-comonad. Finally, following work of Jubin, we consider monads and comonads on the category of affine objects associated to a tangent category. We show that there is a rich theory of monads and comonads in this setting as well as various distributive laws and mixed distributive laws relating these monads and comonads. Even in the category of smooth manifolds, several of these results are new or fill in gaps in the existing literature.

Motivation & Objective

• To define and study affine geometric spaces in the context of tangent categories, generalizing classical affine manifolds.
• To show that the category of affine geometric spaces inherits a tangent category structure from the base category.
• To provide a new categorical characterization of flat torsion-free connections using morphisms between iterated tangent bundles.
• To develop a 2-categorical framework for tangent categories, identifying a 2-comonad structure on the assignment of affine geometric spaces.
• To lay the foundation for abstracting higher-order structures like bimonads from smooth manifolds to tangent categories.

Proposed method

• Use the equivalence between affine manifolds and manifolds with flat torsion-free connections (Auslander-Markus theorem) to define affine geometric spaces in tangent categories.
• Define geometric spaces as objects equipped with a connection on their tangent bundle, and affine geometric spaces as those with flat and torsion-free connections.
• Construct the category of affine geometric spaces as a full subcategory of geometric spaces, showing it inherits tangent structure via lifting from the base category.
• Prove that the assignment of affine geometric spaces to a tangent category forms a 2-comonad on the 2-category of tangent categories.
• Leverage technical lemmas on morphisms between tangent categories to ensure compatibility of structures under functors and natural transformations.
• Use diagrammatic reasoning and functorial lifting to verify that the induced structures on T(TM) and TM satisfy the required axioms for the new characterization of connections.

Experimental results

Research questions

• RQ1 How can affine manifolds be axiomatized within the framework of tangent categories?
• RQ2 Under what conditions does the category of geometric spaces (equipped with connections) inherit a tangent category structure?
• RQ3 Can flat torsion-free connections be characterized categorically as morphisms between iterated tangent bundles?
• RQ4 What 2-categorical structure arises from assigning the category of affine geometric spaces to each tangent category?
• RQ5 How do monad and comonad structures on the tangent functor in the category of smooth manifolds generalize to tangent categories?

Key findings

• The category of affine geometric spaces in a tangent category with endemic fibre products is itself a tangent category, with structure lifted from the base category.
• A flat torsion-free connection K on an object M can be equivalently characterized as a morphism K: T(TM) → TM that commutes with the canonical geometric structures induced by K.
• The assignment of the tangent category of affine geometric spaces to any tangent category forms a 2-comonad on the 2-category of tangent categories.
• Affine tangent categories are identified as Eilenberg-Moore coalgebras for this 2-comonad, providing a categorical characterization of structures with flat, torsion-free connections.
• The construction preserves strictness and faithfulness of the forgetful functor, ensuring coherence with the base tangent category structure.
• The results generalize Jubin’s findings on monad and comonad structures on the tangent functor over affine manifolds, suggesting a broader abstract framework for such structures.