Important Classes of Morphisms and the Relative Cotangent Sequence in Tangent Categories

Problem and Motivation

Tangent categories provide a minimal semantic framework for studying differential geometric reasoning, originally developed by Rosick´y and later expanded by Cockett and Cruttwell. While the theory has been successfully applied to smooth manifolds, algebraic geometry, commutative algebra, and Cartesian differential categories (CDCs), a systematic understanding of how fundamental classes of morphisms—specifically immersions, submersions, local diffeomorphisms, and unramified maps—behave across these diverse settings has been lacking.

Previous work established definitions for submersions and local diffeomorphisms in the context of smooth manifolds, but their characterization in other critical tangent categories, such as affine schemes or commutative algebras, remained unclear. Furthermore, the relationship between these classes and the concept of "unramified" maps (which coincide with immersions in differential geometry but may differ in general tangent categories) required a unified, categorical treatment. The paper addresses the need to formalize these concepts within the abstract framework of tangent categories, study their stability and permanence properties, and elucidate their concrete realizations in specific mathematical domains.

Methodology

The authors employ a systematic, category-theoretic approach grounded in the axioms of tangent categories. The core methodology involves:

1. Horizontal Descent: A central technical tool introduced is the "horizontal descent" $\theta_f$ of a morphism $f: X \to Y$. This is defined as the unique map from the tangent bundle $TX$ to the pullback of the tangent bundle $TY$ along $f$ (the "horizontal bundle"), provided the pullback exists. The paper investigates the properties of $\theta_f$ extensively, using it to simplify the definitions and characterizations of various morphism classes.
2. Carrable Morphisms: The study relies on the notion of "carrable" morphisms (maps that admit pullbacks along differential bundle projections which are preserved by the tangent functor). Specifically, the authors focus on $p$-carrable (pullback along the tangent projection) and $0$-carrable (pullback along the zero section) morphisms to ensure the existence of necessary structures like the horizontal bundle and the relative tangent bundle.
3. Relative Cotangent Sequence: By combining the horizontal descent with the relative tangent bundle (defined via a $0$-carrable pullback), the authors construct a "relative cotangent sequence" in an arbitrary tangent category. This sequence generalizes the classical exact sequence of Kähler differentials found in algebraic geometry.
4. Linearization: The authors distinguish between "strong" definitions (based on general categorical properties like prepullbacks) and "linear" definitions (based on the category of differential bundles, $DBun(X)$). They utilize the enrichment of $DBun(X)$ in commutative monoids (and Abelian groups in Rosick´y tangent categories) to relate monomorphisms to trivial kernels.

Key Contributions and Results

The paper provides a comprehensive classification and characterization of four primary classes of morphisms in tangent categories:

• T-Immersions:

  • Strong T-Immersions: Defined as maps where the naturality square of the tangent projection is a "T-prepullback." In smooth manifolds ($SMan$), these correspond to standard immersions. In algebraic geometry ($CAlg_R^{op}$), they correspond to formally unramified morphisms.
  • T-Immersions (Linear): Defined as $p$-carrable maps where the horizontal descent $\theta_f$ is a linear T-monomorphism in $DBun(X)$.
  • Key Result: In a Rosick´y tangent category (one with negation), the notions of strong T-immersion, T-immersion, and T-unramified morphism coincide. In general tangent categories without negation, these classes are distinct (e.g., in the category of commutative monoids, a map can be T-unramified but not a strong T-immersion).

• T-Submersions:

  • Defined as $p$- and $0$-carrable maps where the horizontal descent $\theta_f$ is a linear T-coequalizer (regular T-epimorphism) in $DBun(X)$. This captures the idea of a "locally surjective differential."
  • Split T-Submersions: Defined via the naturality square being a "T-weak pullback." The paper proves that for $p$-carrable maps, being a split T-submersion is equivalent to $\theta_f$ admitting a section.
  • Key Result: In Rosick´y tangent categories and categories of schemes, a split T-submersion admits a linear section, not just a set-theoretic one. This connects the categorical definition to the splitting of the relative cotangent sequence.

• T-´Etale Morphisms (Local Diffeomorphisms):

  • Defined as maps where the naturality square of the tangent projection is a T-pullback.
  • Key Result: A map is T-´etale if and only if it is both a strong T-immersion and a split T-submersion. For $p$-carrable maps, this is equivalent to the horizontal descent $\theta_f$ being an isomorphism.
  • Distinction: The paper clarifies that T-´etale maps in algebraic geometry are not identical to formally ´etale maps. While formally ´etale maps are T-´etale, there exist T-´etale maps (e.g., certain closed immersions in schemes) that are not formally ´etale.

• The Relative Cotangent Sequence:

  • The paper constructs a de Rham relative cotangent sequence $X \to TX/Y \to TX \to f^*(TY)$ in any tangent category where $f$ is $0$- and $p$-carrable.
  • This sequence is shown to be exact in $DBun(X)$, providing a unified framework that recovers the classical cotangent sequences for smooth manifolds and schemes.

Concrete Characterizations

The paper explicitly maps these abstract definitions to specific categories:

• Smooth Manifolds ($SMan$): T-immersions are standard immersions; T-submersions are standard submersions; T-´etale maps are local diffeomorphisms.
• Commutative Algebras ($CAlg_R$): T-immersions are injective algebra homomorphisms; T-submersions are surjective homomorphisms with linear sections; T-´etale maps are isomorphisms.
• Affine Schemes ($CAlg_R^{op}$): T-immersions are formally unramified morphisms; T-submersions correspond to formally smooth morphisms relative to the base ring; T-´etale maps correspond to formally unramified and formally smooth relative maps (distinct from formally ´etale).
• Cartesian Differential Categories (CDCs): T-immersions are maps with injective derivatives (and higher derivatives); T-submersions are maps with surjective derivatives (retracts); T-´etale maps are maps with invertible derivatives.

Significance and Claims

The authors claim that this work provides a "deep and systematic study" that unifies the understanding of differential geometric concepts across algebraic geometry, differential geometry, and computer science. The significance lies in:

1. Unification: It demonstrates how the same categorical machinery (horizontal descent, relative cotangent sequences) yields the correct, known definitions in specific contexts while revealing subtle distinctions in general settings (e.g., the divergence between immersions and unramified maps in non-Rosick´y categories).
2. New Tools: The introduction of horizontal descent as a primary tool simplifies the verification of submersion and immersion properties, replacing complex weak pullback conditions with section or monomorphism conditions on $\theta_f$.
3. Foundational for Future Work: The paper positions these results as essential prerequisites for future research, specifically mentioning an upcoming project to define and study "schemes" in tangent categories and the existence of a Zariski topology for families of tangent categories. The authors note that understanding these morphism classes is critical because open immersions in algebraic geometry are, in particular, T-monic T-´etale maps.

The paper does not propose new experimental applications or future implications beyond the stated theoretical goals of refining tangent category theory and enabling the study of schemes within this framework. It remains a rigorous theoretical contribution to category theory and its applications to geometry.