Tangent Bibundles: Theory and Applications

• Tangent bibundles are generalized tangent bundle structures that simultaneously encode multiple differential layers and manage singularities with advanced categorical techniques.
• They enable explicit analysis in singular and Poisson geometry by extending classical de Rham theory and vector field computations via constructions like bᵏ-tangent bundles.
• Their applications span sub-Riemannian, Carnot, and algebraic geometry, providing robust frameworks for both analytic methods and higher-categorical investigations.

A tangent bibundle is a geometric or categorical structure that synthesizes or simultaneously encodes multiple tangent (or differential) bundle-like objects, typically in situations where classical linear or smooth tangent bundle theory is either generalized, abstracted, or made to interact with singularities, filtrations, or higher categorical layers. The notion manifests across a spectrum of contemporary research themes including singular and higher-order tangent bundles, sub-Riemannian (Carnot) geometry, filtered diffeological contexts, the categorical theory of tangent structures and fibrations, and advanced settings such as tangent infinity-categories and tangent structures for algebraic geometry.

1. Higher-Order and Singular Tangent Bibundles

One principal source of tangent bibundles arises in the paper of geometric structures exhibiting either higher-order tangency or singularities along hypersurfaces.

bᵏ-Tangent Bundles and their Duals

For a manifold $M$ with a distinguished hypersurface $Z$, the $b^k$-tangent bundle $b^k TM$ is formed so that its sections are vector fields tangent to $Z$ to order $k$. The structure is formalized on a $b^k$-manifold $(M, Z, j_Z)$, where $j_Z$ is a $(k-1)$-jet of the defining function of $Z$. The local model is

$$b^k T_{p}M = \begin{cases} T_{p}M & p \notin Z \ T_{p}Z \oplus \langle y^k \partial_{y} \rangle & p \in Z \end{cases}$$

with dual bundle $b^k T^{*}M$ having fibers

$$b^k T^*_{p}M = \begin{cases} T^*_{p}M & p \notin Z \ T^*_{p}Z \oplus \langle y^{-k} dy \rangle & p \in Z \end{cases}$$

This "tangent bibundle" structure—pairing $b^k TM$ and $b^k T^*M$—supplies the differential complex for a generalized de Rham theory. The cohomology decomposes as

$b^k H^*(M) \cong H^*(M) \oplus (H^{*-1}(Z))^k$

This geometric infrastructure is essential in Poisson geometry of singular type and for the analysis of pseudodifferential operators with prescribed degeneracy along $Z$ (Scott, 2013).

Logarithmic and b-tangent Bundles; bᵐ–Tangent Cohomology

The logarithmic and $b$-tangent bundles (and their higher-order $b^m$ generalizations) provide analytic and homological control over singularities. Locally, sections of the $b^m$-tangent bundle are generated by $x^m \partial_x$ and the usual derivatives $\partial_{y_1},..., \partial_{y_{n-1}}$ in adapted coordinates near $Z = \{x = 0\}$. These packages "absorb" the singular structure, enabling symplectic or Poisson geometric formulations (e.g., $b^m$-symplectic forms).

The isomorphism type of these singular tangent bundles can be refined: for $m=2k$ ($k \in \mathbb{N}$), $b^{2k}T M \cong TM$; for $m=2k+1$, $b^{2k+1}T M \cong b T M$ (Miranda et al., 25 Feb 2025). The associated Poincaré–Hopf theorem for the Euler class of the $b^m$-tangent bundle exhibits computation via indices of zeros of $b$-vector fields, with the sign structure depending subtly on the parity of $m$ and the combinatorics of the gluing data.

2. Tangent Bibundles in Filtered, Diffeological, and Sub-Riemannian Contexts

Carnot Manifolds: Tangent Group Bibundles

A Carnot manifold $(M, H)$ features a filtration $H_1 \subset H_2 \subset... \subset H_r = TM$ compatible with Lie brackets. At each point $a$, the graded algebra $g_M(a)=\bigoplus_j H_j(a)/H_{j-1}(a)$ exponentiates to the (generally nonabelian) tangent group $G_M(a)$, and these groups assemble into the tangent group bundle $G_M \to M$ (Choi et al., 2015). This bundle—together with the canonical Carnot differential—constitutes a tangent bibundle in the sense that it both refines the classical tangent bundle (which is abelian) and encodes intricate infinitesimal group structures crucial for analysis on sub-Riemannian manifolds.

The tangent groupoid construction unifies, in a smooth deformation, the pair groupoid $M \times M$ (for nonzero scale) and the tangent group bundle $G_M$ (for scale zero): $$\mathcal{G} = G_M \cup (M \times M \times \mathbb{R}^*)$$ The interplay of these structures is foundational for the symbolic and index theory of hypoelliptic operators.

Diffeological Bundles: Exact Sequences and Enhanced Smoothness

For a smooth diffeological bundle $T: E \to B$ with fiber $F$, an exact sequence of internal tangent spaces

$T_e(F) \to T_e(E) \to T_b(B) \to 0$

links the tangent structures of fiber, total space, and base (Christensen et al., 2015). If $E$ and $B$ are filtered, this extends to a short exact sequence, and, crucially, if $X$ is filtered, both Hector’s and the dvs tangent bundles coincide and are honest diffeological vector bundles—furnishing a tangent bibundle structure on $E$ over $B$ with strong vector space properties.

3. Abstract Tangent Bibundles in Tangent Categories and Higher Structures

Tangent Categories and Differential Bundles

A tangent category comprises a category $\mathcal{C}$ equipped with an endofunctor $T$ (with projection, unit, addition, vertical lift, and possibly flip), axiomatizing the behavior of the tangent bundle functor (Leung, 2016). A "tangent bibundle" arises when an object simultaneously has compatible differential bundle structures for distinct tangent functors or when a fibration of differential bundles $P: \mathsf{DBunD}(\mathcal{C})\to\mathcal{C}$ inherits a tangent structure "fiberwise," so that each fiber becomes a Cartesian differential category (Cockett et al., 2016).

A principal result in the characterization of differential bundles states that, under suitable limit and preservation conditions, the structure is determined up to isomorphism by its projection and zero section (Ching, 9 Jul 2024). This principle likely generalizes to bibundle settings: a tangent bibundle structure carrying two compatible tangent modules would be characterized by two projection maps and two zero sections, with the rest of the structure uniquely determined.

Connections, Decomposition, and Coordinatization

In categorical settings, a connection on a differential bundle can be equivalently specified using a single vertical connection $K: TE \to E$ such that the canonical fiber product diagram

$\begin{xy} (0,0)*+{TE}="te"; (-20,-20)*+{E}="e"; (0,-20)*+{TM}="tm"; (20,-20)*+{E}="e2"; {\ar_{p_E} "te";"e"} {\ar^{T(q)} "te";"tm"} {\ar^{K} "te";"e2"} \end{xy}$

is a limit diagram. This induces a biproduct decomposition $TE \cong E \oplus_M TM \oplus_M E$ in the additive category of differential bundles, clarifying the geometric structure of tangent bibundles (Lucyshyn-Wright, 2017).

4. Higher-Categorical Tangent Bibundles: Tangent $\infty$-(Bi)bundles and Duality

Goodwillie Tangent Structure and $(\infty,2)$-Bibundles

For a differentiable $\infty$-category $\mathcal{C}$, the Goodwillie tangent bundle is constructed as $T(\mathcal{C}) := \mathrm{Exc}(S, \mathcal{C})$, where $S$ is finite pointed spaces and $\mathrm{Exc}(S, \mathcal{C})$ denotes excisive functors (Bauer et al., 2021). For a Weil algebra $A$, the $A$–tangent is $T^A(\mathcal{C}) = \mathrm{Exc}^A(S^n,\mathcal{C})$. The resulting structure maps (projection, zero section, addition, flip, vertical lift) make the $(\infty,1)$-category of differentiable $\infty$-categories into a tangent $\infty$-category, with stable $\infty$-categories as the appropriate "tangent fibers"—providing, in effect, a higher-categorical tangent bibundle.

Moreover, dual tangent structures arise on $\infty$-toposes: the geometric tangent functor $U(X) = X^{T(S)}$ (exponential cotensor) is dual, via adjunction, to Lurie's Goodwillie tangent functor whose fibers are stabilization categories of slice $\infty$-toposes. On injective $\infty$-toposes, these tangent structures are equivalent under the functor of points (Ching, 2021). This duality of perspectives is a haLLMark of tangent bibundle phenomena at the universal level.

5. Algebraic and Scheme-Theoretic Tangent Bibundles

In algebraic geometry, the notion of a tangent bundle for affine schemes is realized in the category of commutative algebras via the Kähler differential functor, represented by the ring of dual numbers. The abstract tool of tangentoids—objects in a monoidal category (e.g., commutative $R$-algebras) carrying "bundle-like" structure with projection, zero, addition, and vertical lift—classifies representable tangent structures (Lanfranchi et al., 14 May 2025). The representable tangent structures correspond to tangentoids arising from finitely generated projective commutative solid non-unital algebras, with a precise classification over principal ideal domains (PID): only the trivial and dual numbers tangent structures appear.

This algebraic realization of tangent bundle theory, and its extension to bibundle-like contexts via simultaneous actions or structurings (e.g., operadically or along multiple tangent directions), provides categorical, non-linear, and possibly multi-fibered generalizations of geometric tangent bibundles.

6. Implications and Applications

The theory of tangent bibundles as synthesized in these approaches has profound implications:

• In singular and Poisson/symplectic geometry, $b^k$-tangent bibundles enable the classification and analysis of degeneracies, the extension of de Rham cohomology, and the construction of singular pseudodifferential calculi.
• In sub-Riemannian and Carnot geometry, group-valued tangent bibundles and the associated groupoids undergird both geometric analysis and noncommutative index theory.
• The categorical framework of tangent bibundles in tangent categories permits simultaneous encoding of multiple "differentiation directions," supports the theory of connections and curvature, and informs higher-level fibrations (e.g., in display tangent categories) and vertical/horizontal decompositions.
• Higher-categorical tangent bibundles, particularly in the Goodwillie and $\infty$-topos settings, extend these ideas to the calculus of functors and derived geometry, linking deformation theory, homotopy theory, and synthetic differentiable notions.
• The theory provides a robust classification and identification principle by reducing differential (and bibundle) structures to their bundle-indexing data (projections, zero sections) when universal limit conditions are satisfied.

The paper and synthesis of tangent bibundles thus modulate between concrete geometric analysis of singularities and abstract, multi-layered differential structures in both algebraic and categorical settings, generating new techniques and perspectives across differential and algebraic geometry, categorical logic, and topological analysis.