Non-Isomorphic Tangent Functors

• Non-Isomorphic Tangent Functors are endofunctors on tangent categories that produce distinct infinitesimal geometric and algebraic structures through differing representability and adjunction properties.
• They are classified using techniques such as Kan extensions, duality of corepresentable methods, and algebraic constructions (e.g., Kähler differentials), which determine unique tangent fiber behaviors.
• The study of these functors informs obstruction theory and deformation analysis in contexts like affine schemes, diffeological spaces, and ∞-toposes, offering actionable insights into singular and infinitesimal geometries.

Non-Isomorphic Tangent Functors are endofunctors on categories equipped with canonical tangent bundle data that, while satisfying abstract tangent category axioms, yield genuinely distinct geometric or algebraic structures. The paper of such functors encompasses combinatorial, algebraic, categorical, topological, and homotopic techniques, with explicit classification results in algebraic geometry, diffeology, and infinity-toposes. In these domains, one finds that the distinction between corepresentable and representable tangent bundle constructions, and between left-Kan and right-Kan extensions, gives rise to large families of non-isomorphic tangent functors, each shaping infinitesimal geometry in a different way.

1. Abstract Framework and Representability in Tangent Categories

A tangent category is a category $\mathcal{X}$ equipped with an endofunctor $T\colon \mathcal{X}\to\mathcal{X}$ and a web of natural transformations: projection $p_M$, zero $z_M$, sum $s_M$, vertical lift $l_M$, and canonical flip $c_M$, all satisfying axioms abstracting tangent bundles on smooth manifolds. The functor $T$ is representable if there exists an object $D$ (infinitesimal object) such that $T(-)\cong(-)^D$. In algebraic geometry, for the category of affine schemes $\mathbf{Aff}_R$, the classical tangent structure is realized via Kähler differentials and is represented by the dual-numbers scheme $\Spec R[\varepsilon]/(\varepsilon^2)$ (Lanfranchi et al., 14 May 2025).

2. Algebraic Classification: Affine Schemes and Solid Algebras

For affine schemes, the core problem is the classification of representable tangent structures. Tangentoids, objects $D$ in a symmetric monoidal category equipped with the dual tangent-category structure, correspond to commutative associative solid non-unital $R$-algebras $M$ where the multiplication $\alpha: M\otimes_R M\overset{\sim}{\to} M$ is an isomorphism. A representable tangent functor $T$ on $\mathbf{Aff}_R$ is then the right adjoint of $D\otimes-:\mathbf{cAlg}_R\to\mathbf{cAlg}_R$, where $D = R\oplus M$ and $M$ is finitely generated projective (Lanfranchi et al., 14 May 2025).

In the case where $R$ is a principal ideal domain (PID), all finitely generated projective modules are free, so $M\cong R^n$, with solidity forcing $n^2=n$ and so $n=0$ or $1$. Thus, the only representable tangent functors are the trivial one ($T(A)=A$) and the one induced by Kähler differentials ($T(A) = A[\varepsilon]/(\varepsilon^2)$). These functors are non-isomorphic, as their representing objects are not isomorphic, and their tangent fibers differ in dimension except in the trivial case.

Tangent Functor	Representing Object	Fiber Behavior
Trivial	$\Spec R$	Identity functor
Kähler differential	$\Spec R[\varepsilon]/(\varepsilon^2)$	$\Spec(k\oplus T_x^*A)$

3. Diffeological Spaces: Uncountably Many Non-Isomorphic Tangent Functors

In diffeological geometry, tangent functors can be defined via Kan extensions and germ derivations. The most classical are the internal tangent functor (left Kan extension), external/germ derivation functor (right Kan), and the Vincent-type functor. These can be parameterized by test spaces $(Y,y)$ to yield, for each base point $(X,x)$, the $(Y,y)$-internal and $(Y,y)$-right tangent functors. Varying the test space across irrational tori $T_\alpha$ and orbit spaces $H_n$ yields uncountably many pairwise non-isomorphic tangent functors (Taho, 22 Nov 2025). The non-isomorphism is witnessed by differing behavior on tangent fibers—dimensionality and vanishing—controlled by arithmetic conditions (e.g., slopes of tori, rank of orbit spaces).

Family	Indexing Example	Non-Isomorphism Criterion
$(T_\alpha,*)$-int	Slopes $\alpha$	$\alpha\not\sim\beta$ by $\SL(2,\mathbb{Z})$ action
$(H_m,0)$-ext	Rank $m$	$m\leq n$ vs. $m > n$ on $H_n$

Moreover, classical functors (internal, external) and their $(Y,y)$-generalizations admit commuting diagrams of natural transformations but fail to be isomorphic unless the space is smoothly regular—i.e., admits smooth bump functions separating points. Outside this case, the limitation of colimit vs limit, and the necessity of global extension by zero, yield the proliferation of non-isomorphic tangent functors (Taho, 7 Jun 2024).

4. Singular and Logarithmic Tangent Bundles: Isomorphism Criteria

In singular geometry, logarithmic and $b^m$-tangent bundles generalize the tangent bundle to handle singularities along hypersurfaces. For a $b^m$-manifold $(M,Z)$, the $b^m$-tangent bundle is locally framed by $$\{ x_1^m\partial_{x_1}, \partial_{x_2}, \dots, \partial_{x_n} \}$$. The fundamental theorem is that, up to isomorphism, $b^m$-functors split into two classes: even $m$ yields isomorphism with the ordinary tangent bundle (${}^{b^{2k}}T M\cong T M$), odd $m$ with the $b$-tangent bundle (${}^{b^{2k+1}}T M\cong {}^bT M$) (Miranda et al., 25 Feb 2025). The obstruction-theoretic and characteristic-class criteria, especially through Stiefel–Whitney and Euler class computations, block unexpected isomorphisms; for spheres, only odd-dimensional $b$-spheres have $b$-tangent bundles isomorphic to the standard tangent bundle.

5. Infinity-Toposes: Dual Tangent Bundle Functors

For presentable $\infty$-toposes, Lurie’s tangent bundle functor $T_X$ and its right adjoint $T_X^R$ both satisfy the tangent category axioms but are not isomorphic except in trivial cases. $T_X$ stabilizes over-categories (Goodwillie calculus), while $T_X^R$ exponentiates by the universal infinitesimal object. Their non-isomorphism echoes categorical non-self-duality of the infinitesimals. Explicit computations for injective $\infty$-toposes show the tangent fibers differ fundamentally: Goodwillie-excisive functor categories vs exponential objects (Ching, 2021).

Functor	Construction	Geometry Encoded
$T_X$	Stabilization (corepresentable)	Goodwillie tangent calculus
$T_X^R$	Mapping out (representable)	Exponential by universal tangent

6. Criteria, Examples, and Pathologies

The existence of non-isomorphic tangent functors is governed by algebraic classification (solid algebras, co-exponentiability), topological regularity (bump functions in diffeology), adjunction types (corepresentable vs representable), and bundle isomorphism obstructions (characteristic classes, clutching maps). Concrete examples—quotient diffeologies, $b$-spheres, and test spaces—provide explicit witnesses to non-isomorphism, and any sufficiently exotic geometric or categorical context will admit such non-uniqueness except under strong regularity or finiteness conditions. The following table synthesizes the key non-isomorphism mechanisms:

Mechanism	Context	Non-Isomorphism Witness
Adjoint vs corepresentable	$\infty$-toposes	Dual non-self-duality
Co-exponentiable infinitesimals	Affine schemes	Distinct tangent fibers
Colimit vs limit	Diffeological spaces	Nonregular topologies
Clutching-map obstruction	$b$-spheres	Degree of attaching map

7. Implications and Future Directions

Non-isomorphic tangent functors encode finer invariants of infinitesimal geometry, singular structure, and categorical enrichment. Their paper elucidates the relationship between synthetic, algebraic, and homotopical models of differentiability, and applies directly to obstruction theory, classification problems, and formal deformation theory. The proliferation of distinct tangent functors in generalized categories suggests robust analogues of tangent bundles with applications ranging from singular symplectic geometry to higher Kodaira–Spencer classes and the comparison of formal Goodwillie-type deformations versus geometric deformations in spectral algebraic geometry (Miranda et al., 25 Feb 2025, Lanfranchi et al., 14 May 2025, Taho, 22 Nov 2025, Ching, 2021, Taho, 7 Jun 2024).