Synthetic Tangent Bundle Construction

• Synthetic Tangent Bundle Construction is a categorical framework in Synthetic Differential Geometry that models tangent spaces via infinitesimal objects and internal homs.
• It employs Weil algebras and the concept of microlinearity to generalize classical tangent bundles, ensuring smooth, functorial behavior in a cartesian closed topos.
• The construction underlies advanced applications in Lie theory, jet bundles, and homotopy type theory, offering a robust axiomatic basis for modern differential geometry.

A synthetic tangent bundle construction formalizes the tangent bundle of a space in the framework of Synthetic Differential Geometry (SDG) using infinitesimal objects, typically spectra of Weil algebras, within a cartesian closed topos. This construction generalizes the classical tangent bundle to a setting where nilpotent infinitesimals exist and all maps are smooth, allowing rigorous manipulation of infinitesimal neighborhoods without reliance on analytic limits. The synthetic approach provides not only a categorical and combinatorial presentation of tangent bundles but also forms the axiomatic core for more abstract tangent structure theories and underpins the development of differential geometry in generalized settings (including smooth toposes, Frölicher spaces, type theories, and higher categories).

1. Foundations: Topos, Weil Algebras, and Infinitesimal Objects

In SDG, the foundational environment is a cartesian closed topos—typically a well-adapted model such as the Cahiers topos or the topos of sheaves on smooth loci—endowed with a commutative ring object $R$ playing the role of the real line. Key to the synthetic construction is the existence of infinitesimal objects:

• For each $n \geq 1$, the infinitesimal object $D_n := \{ x \in R \mid x^{n+1} = 0 \}$, generalizing the dual numbers $D = D_1 = \{ d \in R \mid d^2 = 0 \}$.
• More generally, for every Weil algebra $W$ (a finite local commutative $R$-algebra with a nilpotent maximal ideal), the spectrum $\mathrm{Spec}_R W$ provides an infinitesimal object.

The Kock–Lawvere axiom postulates that every map $f : D \to R$ is affine; more generally, for each Weil algebra $W$, the functor $M \mapsto M^{\mathrm{Spec}_R W}$ is exact on microlinear objects, mirroring the behavior of polynomial functions on infinitesimals. This directly enables representability of tangent vectors and higher-order jets as maps from infinitesimal spaces into a given synthetic space (Kock, 2016, Leung, 2016, Nishimura, 2010).

2. Microlinearity and Categorical Properties

Microlinearity is the categorical regularity property required of spaces in SDG to ensure well-behaved infinitesimal extensions. An object $M$ is microlinear if, for any finite diagram of Weil algebras, the base change diagram

$\xymatrix{ M^{D_{W_3}} \ar[r] \ar[d] & M^{D_{W_1}} \ar[d] \ M^{D_{W_2}} \ar[r] & M }$

is a pullback in the topos whenever the diagram of $W_i$ is a pushout in Weil with $D_{W_i} = \mathrm{Spec}(W_i)$ (Kock, 2016). This ensures that tangent bundle fibers, jet spaces, and other infinitesimal constructions retain the correct categorical and geometric structure.

In the category of microlinear objects, the synthetic tangent functor is

$T(M) := M^D,$

where $M^D$ denotes the internal hom (exponential object) in the topos, and the projection $T(M) \to M$ is evaluation at $0 \in D$.

3. Synthetic Tangent Bundle Construction

The tangent bundle of $M$ is defined as the exponential object $M^D$, assigning to each base point $x \in M$ the set of infinitesimal arrows $t: D \to M$ with $t(0)=x$. Explicitly, in charts, $T_x M$ consists of maps $d \mapsto x + d v$, so tangent vectors at $x$ are realized as infinitesimal displacements.

For any microlinear $M$, the construction satisfies the following:

• $T(M) = M^D$ forms a bundle over $M$ by projection at $d=0$.
• $T(M)$ carries a canonical vector bundle structure; addition, zero section, scalar multiplication, and negation are induced by algebraic operations on $D$ (e.g., $d_1 + d_2$, $-d$) (Nishimura, 2010, Leung, 2016).
• The tangent functor $T : M \mapsto M^D$ lifts morphisms by precomposition: for $f: M \to N$, $T(f) = f^D: M^D \to N^D$.
• Differential operators are synthetically recovered as $T(f)(t) = f \circ t$ for $t: D \to M$.

By characterization of the Kock–Lawvere axiom, for every $f : R \to R$, and $d \in D$, $f(x+d) = f(x) + d f'(x)$, ensuring that the synthetic derivative aligns with the coordinate-free differential (Kock, 2016, Leung, 2016, Nishimura, 2010).

4. Weil Functors, Prolongations, and Higher Jets

The construction generalizes by replacing $D$ with arbitrary infinitesimal objects $D_W = \mathrm{Spec}_R W$ for a Weil algebra $W$. The Weil prolongation of $M$ by $W$ is $M^W := M^{D_W}$, encapsulating higher-order tangent structures and jet bundles:

• $T(M) = M^D$ is the first-order tangent bundle.
• $J^k(M) = M^{D_k}$ is the $k$-th order jet bundle, with $D_k = \{ d \in R \mid d^{k+1} = 0 \}$ (Leung, 2016, Nishimura, 2013).
• For a microlinear Lie group $G$, the $W$-prolongation $G_W = G^{D_W}$ admits a group structure.

This formalism enables a universal tangent structure: the category of Weil algebras is initial among categories supporting such tangent structures. Thus, any tangent structure on a category with finite pullbacks factors through the construction $M \mapsto M^{D_W}$ indexed by $W$ (Leung, 2016).

5. Synthetic Lie Theory and Formal Group Laws

Within SDG, the synthetic tangent bundle underpins a full formal Lie theory:

• For a microlinear Lie group $G$, $T_e G = G^D$ gives the Lie algebra structure, and the identification of $T(G)$ enables synthetic derivation of the Lie bracket, Jacobi identity, and the full Baker–Campbell–Hausdorff and Zassenhaus formulas: all such polynomial expressions truncate finitely due to nilpotency of $D$ (Nishimura, 2013).
• The jet prolongations $G_{D^n}$ form a tower of group-like objects whose limit is a formal group integrating the Lie algebra (Nishimura, 2013).
• The functorial properties of these constructions persist for Lie groupoids and general microlinear spaces, as proven via internal path objects and integral completeness considerations (Burke, 2016).

6. Tangent Structure in Other Frameworks: Type Theory and Frölicher Spaces

The synthetic tangent bundle construction has precise analogues in other categorical frameworks:

• Homotopy Type Theory (HoTT): The tangent bundle is encoded as the dependent type $T_x M := \{ t : D \to \|M\|_0 \mid t(0) = |x|_0 \}$, and the general Kock–Lawvere axiom takes a homotopical form ensuring polynomiality of maps out of infinitesimals (Nishimura, 2016).
• Frölicher Spaces: For a microlinear and Weil-exponentiable Frölicher space $X$, one uses the functor $X \;{data}\;W$ associating to each Weil algebra $W$ the smooth maps $W \to X$, yielding a cartesian closed category supporting tangent bundle and Lie bracket constructions analogous to topos-based SDG (Nishimura, 2010).

In both settings, tangent bundles, vector fields, and bracket operations are constructed entirely in terms of infinitesimal or algebraic data, not by analytic approximation.

7. Synthetic Tangent Bundles and Pre-cohesive, Radical SDG

Recent advances have clarified the interplay between tangent bundle construction, order structure, and the broader categorical context:

• Radically synthetic models construct the object of "reals" $R$ as the endomorphism monoid of a distinguished infinitesimal object $T$; the tangent bundle is recovered as $X^T$ with projection by evaluation at the unique point (Menni, 2024).
• In pre-cohesive toposes supporting bi-directional "radical" reals, the synthetic tangent bundle retains its structural properties, and the axiomatic construction is compatible with the order structure derived from infinitesimal symmetries in $T^T$.

This categorical synthesis opens novel directions for interpreting order, integration, and geometric features in the tangent bundle, making the construction robust across a wide variety of smooth, generalized, or infinitesimal settings.

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References: (Kock, 2016, Leung, 2016, Nishimura, 2013, Nishimura, 2013, Nishimura, 2016, Nishimura, 2010, Menni, 2024, Burke, 2016)