Tangent Restriction Categories

• Tangent restriction categories are restriction categories endowed with a tangent structure where the tangent functor preserves restriction idempotents and structural maps remain total.
• They integrate differential geometry with partial map theory via restriction pullbacks, enabling rigorous formulations for vector fields, differential objects, and bundles.
• The framework uses split completions and pullback-extension methods to transfer tangent data, addressing limitations in classical smooth manifold constructions.

Searching arXiv for the cited papers to ground the article in current primary sources. {"query":"id:(Lanfranchi, 19 Sep 2025) OR id:(Cruttwell et al., 28 Feb 2025) OR id:(Lanfranchi, 21 Jan 2026) OR id:(Cockett et al., 22 Sep 2025)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} arXiv search completed. Tangent restriction categories are restriction categories equipped with tangent structure. In the formulation used in "The formal theory of tangentads" (Lanfranchi, 19 Sep 2025), a tangent restriction category is a restriction category with a functor $T$ preserving restriction idempotents, together with structural total natural transformations

$$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$

that satisfy the same compatibility axioms as in a tangent category, except that the $n$-fold pullbacks of $p$ along itself and the universality of the vertical lift are replaced by restriction pullbacks preserved by $T$. This places partiality and differential structure in a single formal setting: total maps carry the ordinary tangent data, while restriction idempotents encode domains of definition.

1. Axiomatic setting

Restriction categories are used through the standard Cockett–Lack framework: for each morphism $f:A\to B$ there is a restriction idempotent $\bar f:A\to A$, total maps are those with $\bar f=\mathrm{id}_A$, and restriction pullbacks exist in the sense required for restriction-limit constructions. The tangent-category side consists of a functor $T:X\to X$ and the structural maps $p,z,s,l,c$, with $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$0 forming a $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$1-additive bundle and with coherence conditions for vertical lift and canonical flip. The paper recalls, in particular, the coassociativity and flip-compatibility equations

$$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$2

together with the symmetric braiding identities for $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$3 and the local linearity pullback square (Lanfranchi, 19 Sep 2025).

The restriction version keeps these structural maps but changes the limit theory. Example 4.6 defines tangent restriction categories by requiring that $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$4 preserve restriction idempotents, that $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$5 be total, and that the “tangent limits” be formulated as restriction pullbacks preserved by $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$6 (Lanfranchi, 19 Sep 2025). In the terminology of Part II, tangent restriction categories are restriction categories equipped with a tangent endofunctor $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$7 that preserves restriction idempotents and with structural total natural transformations satisfying the usual tangent axioms “but with pullbacks replaced by restriction pullbacks” (Lanfranchi, 21 Jan 2026). This replacement is the decisive modification: it allows tangent structure to coexist with genuine partiality rather than forcing all relevant maps to be total.

A related motivation comes from the pullback problem in differential geometry. "Pullbacks in tangent categories and tangent display maps" stresses that the category of smooth manifolds lacks many pullbacks, and that even existing pullbacks may fail to be preserved by the tangent bundle functor. The paper isolates tangent display maps as maps along which all required pullbacks exist, are preserved by all iterates $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$8, are stable under pullback and under $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$9, and whose pullbacks along arbitrary morphisms remain tangent display maps (Cruttwell et al., 28 Feb 2025). This perspective explains why restriction-pullback formulations are natural in the partial setting.

2. Split restriction categories, tangentads, and the formal framework

A central structural point is that tangent restriction categories are not themselves tangentads. The entry into the tangentads formalism is through split restriction categories: tangent split restriction categories are tangentads in the 2-category $n$0, and every tangent restriction category embeds in a tangent split restriction category, hence into a tangentad (Lanfranchi, 19 Sep 2025). Part II states the same result as Proposition 4.35 and Lemma 4.36: tangent split restriction categories are precisely the tangentads in the 2-category $n$1, and every tangent restriction category embeds into a tangent split restriction category (Lanfranchi, 21 Jan 2026).

This split/non-split distinction determines the method of construction. In the split case, one works directly inside the ambient 2-category of split restriction categories, where tangentads machinery applies. In the general case, one passes to the split completion and then pulls structures back along the unit of the completion. The paper formulates this systematically by a pullback-extension context with

$n$2

and a natural 2-transformation

$n$3

Vector fields and differential objects are then recovered as 2-pullbacks along $n$4, while differential bundles require a variant in the 2-category $n$5 of pairs of parallel morphisms because both the base and total-space projections must be tracked (Lanfranchi, 19 Sep 2025).

This suggests that the formal theory of tangentads does not replace tangent restriction categories; rather, it supplies a transfer mechanism. Split restriction tangent categories are the tangentadic core, and general restriction categories are handled by embedding and descent. The same pattern is used for vector fields in Part I, and for differential objects, differential bundles, and connections in Part II (Lanfranchi, 21 Jan 2026).

3. Vector fields, differential objects, and differential bundles

For vector fields, the split restriction construction is explicit. If $n$6 is a tangent split restriction category, then $n$7 has objects $n$8 where $n$9 is total and satisfies $p$0. A morphism $p$1 is a morphism $p$2 in $p$3 such that $p$4 and such that the restriction idempotent commutes with the vector field,

$p$5

The tangent structure lifts by

$p$6

making $p$7 a tangent split restriction category (Lanfranchi, 19 Sep 2025). The general, non-split case is obtained as the 2-pullback of $p$8 along $p$9, yielding the same explicit description of objects and morphisms.

Restriction differential objects are the partial analogue of Euclidean objects. In a Cartesian tangent split restriction category, a restriction differential object is a differential object $T$0 lying in the total subcategory $T$1, with

$T$2

all total, satisfying the usual differential object axioms, but with the universality condition replaced by a restriction pullback preserved by $T$3 (Lanfranchi, 19 Sep 2025). The lifted category $T$4 has these restriction differential objects as objects and linear morphisms respecting restriction as morphisms; in the general restriction case, $T$5 is obtained from $T$6 by pullback-extension, and its objects are exactly those differential objects for which $T$7 is total, equivalently for which the idempotent $T$8 is trivial (Lanfranchi, 21 Jan 2026).

Restriction differential bundles generalize vector bundles in the same manner. A restriction differential bundle in a Cartesian tangent split restriction category is a tuple of total maps

$T$9

where $f:A\to B$0 is the restriction pullback of $f:A\to B$1 along itself, satisfying the usual additive and linearity axioms, with the universality of the vertical lift replaced by the restriction pullback

$f:A\to B$2

In the split case, $f:A\to B$3 is again a tangent split restriction category; in the general case, $f:A\to B$4 is the subcategory of $f:A\to B$5 spanned by those bundles with $f:A\to B$6 and $f:A\to B$7 total (Lanfranchi, 19 Sep 2025). The paper also extends the standard result that differential bundles over the terminal object are equivalent to differential objects, first in the tangentadic setting and then in the restriction setting via pullback-extension (Lanfranchi, 21 Jan 2026).

4. Universal properties and PIE-limit constructions

The formal theory emphasizes universal properties rather than ad hoc definitions. Vector fields are characterized by a PIE-limit construction: in a 2-category with inserters and equifiers, one first forms the inserter of $f:A\to B$8 and then equifies the two induced transformations that enforce the section condition $f:A\to B$9. In the split restriction context this applies because $\bar f:A\to A$0 admits inserters and equifiers, so $\bar f:A\to A$1 can be constructed as the equifier of the inserter $\bar f:A\to A$2 imposing $\bar f:A\to A$3 (Lanfranchi, 19 Sep 2025).

Differential objects and differential bundles are treated analogously at the tangentad level. Part II formulates a universal pointwise differential object in the Hom-tangent category $\bar f:A\to A$4 and a universal pointwise display differential bundle in $\bar f:A\to A$5, with corresponding equivalences

$\bar f:A\to A$6

and

$\bar f:A\to A$7

These universal properties are then specialized to tangent restriction categories by passing through split completion and 2-pullbacks (Lanfranchi, 21 Jan 2026).

The equivalence between differential objects and differential bundles over the terminal object is one of the main structural results. The tangentadic theorem states that if $\bar f:A\to A$8 exists and the 2-pullback of $\bar f:A\to A$9 along the terminal map exists, then $\bar f=\mathrm{id}_A$0. In the restriction setting, this applies directly in split restriction categories and then extends to the general case through the same pullback-extension mechanism (Lanfranchi, 19 Sep 2025). The significance of this result is conceptual rather than merely formal: it shows that the partial version of “Euclidean space” is not an isolated notion, but the terminal-base instance of the partial theory of bundles.

5. Connections, covariant derivative, curvature, and torsion

"The formal theory of tangentads" develops the formal theory of connections at the level of tangentads and states that the restriction constructions of differential bundles, sections, and vector fields are precisely the ingredients needed to transport the connection formalism via the pullback-extension context, provided the required 2-pullbacks exist (Lanfranchi, 19 Sep 2025). Part II makes this precise. A linear connection on a differential bundle $\bar f=\mathrm{id}_A$1 consists of a vertical linear connection $\bar f=\mathrm{id}_A$2 and a horizontal linear connection $\bar f=\mathrm{id}_A$3, with linearity conditions, an orthogonality condition, and a direct-sum decomposition (Lanfranchi, 21 Jan 2026).

In the restriction setting, the same equations are imposed on total maps, while the relevant pullbacks are restriction pullbacks. Part II states that in tangent split restriction categories a restriction linear connection is a pair $\bar f=\mathrm{id}_A$4 of total maps satisfying the linearity, orthogonality, and direct sum equations exactly as in the tangent-category case, and then constructs $\bar f=\mathrm{id}_A$5 for general tangent restriction categories via pullback-extension in $\bar f=\mathrm{id}_A$6 (Lanfranchi, 21 Jan 2026). The horizontal bundle is formed by a restriction pullback, and the totality conditions ensure compatibility with the restriction structure.

The derived differential-geometric operators are also formalized. For total vector fields $\bar f=\mathrm{id}_A$7 and total sections $\bar f=\mathrm{id}_A$8, the covariant derivative is

$\bar f=\mathrm{id}_A$9

When negatives exist, curvature and torsion are given by

$T:X\to X$0

The corresponding curvature tensor and torsion operator are obtained by composing these morphisms with iterated tangent lifts of vector fields and sections (Lanfranchi, 21 Jan 2026). Since these formulas use only composition with structural maps, the paper states that they make sense verbatim in tangent restriction categories provided the relevant maps are total and $T:X\to X$1 preserves restriction idempotents. This yields a restriction-compatible version of covariant derivative, curvature, and torsion without changing the formal algebra of the definitions.

Connections are also constructed by PIE limits. Vertical and horizontal linear connections are built by inserters that capture the candidate morphisms $T:X\to X$2 and $T:X\to X$3, followed by equifiers that impose the linearity equations; full linear connections are then formed by a 2-pullback of the vertical and horizontal constructions, followed by equifiers for orthogonality and direct sum (Lanfranchi, 21 Jan 2026). In split restriction categories these PIE limits exist sufficiently to carry out the construction; in general tangent restriction categories the resulting structures are transferred back from the split completion.

6. Canonical constructions from tangent categories and related variants

A different route to tangent restriction categories starts from an ordinary tangent category and extracts canonical partiality. "Pullbacks in tangent categories and tangent display maps" introduces tangent display maps and proves that they form the unique maximal tangent display system, closed under composition and stable under $T:X\to X$4 and $T:X\to X$5-pullbacks (Cruttwell et al., 28 Feb 2025). In the category of smooth manifolds, tangent display maps are precisely submersions. The same paper defines open subobjects as tangent monic display étale maps and proves that $T:X\to X$6, the class of monics underlying open subobjects, is the maximal tangent display system of monics.

From these open subobjects one obtains the canonical split restriction tangent category

$T:X\to X$7

Its objects are those of $T:X\to X$8, and its morphisms $T:X\to X$9 are isomorphism classes of spans $p,z,s,l,c$0 with $p,z,s,l,c$1 an open mono and $p,z,s,l,c$2 arbitrary. The restriction is

$p,z,s,l,c$3

restriction idempotents split, and the tangent functor is defined by

$p,z,s,l,c$4

The canonical embedding $p,z,s,l,c$5 sends a total map $p,z,s,l,c$6 to $p,z,s,l,c$7 and strictly preserves $p,z,s,l,c$8 (Cruttwell et al., 28 Feb 2025). In smooth geometry, this recovers the familiar category of partial smooth maps with open domains, but now equipped with tangent restriction structure.

A further variant is the tangent join restriction category, where compatible local restrictions can be joined to obtain global maps. "Lie groups in tangent join restriction categories" studies this setting for group objects and principal bundles, assuming a restriction-preserving tangent functor and total structural maps $p,z,s,l,c$9 with the tangent axioms expressed using restriction pullbacks (Cockett et al., 22 Sep 2025). The paper proves, for a group object $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$00, a canonical isomorphism

$$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$01

and formulates principal bundles and vertical bundles entirely in the language of join restriction categories. This suggests that tangent restriction structure is flexible enough to support both local-to-global constructions and Lie-theoretic geometry.

7. Assumptions, limitations, and open directions

The restriction theory depends on several standing assumptions. Structural maps $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$02 must be total, and $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$03 must preserve restriction idempotents. The split-to-general extension requires a pullback-extension context $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$04 together with suitable 2-pullbacks, including the diagrams used for vector fields, differential objects, and differential bundles (Lanfranchi, 19 Sep 2025). PIE-limit constructions require inserters and equifiers in the ambient 2-category; this is why the split case is the primary tangentadic case.

The main limitation emphasized in the first tangentads paper is that tangent restriction categories are not themselves tangentads, so the general tangentads machinery cannot be pushed directly beyond the split case. The proposed method is embedding into the split completion and then pulling back the desired structure (Lanfranchi, 19 Sep 2025). Part II carries this method through for differential objects, differential bundles, and linear connections, but the need to verify existence of the corresponding 2-pullbacks remains part of the formal infrastructure (Lanfranchi, 21 Jan 2026).

The connection theory also illustrates the boundary between what is established and what is suggested. The first tangentads paper states that covariant derivative, curvature, and torsion are developed formally for tangentads, and that extending these notions to tangent restriction categories requires verifying the existence of the corresponding pullbacks in $$p_M:TM\to M,\quad z_M:M\to TM,\quad s_M:T_2M\to TM,\quad l_M:TM\to T^2M,\quad c_M:T^2M\to T^2M$$05 and the compatibility of restriction limits with these constructions (Lanfranchi, 19 Sep 2025). A plausible implication is that the restriction setting is now equipped with a uniform transfer principle: once a tangent-geometric construction is formulated tangentadically and the necessary 2-pullbacks exist, it can be transported from tangent split restriction categories to general tangent restriction categories.