Tangent Monads: Categorical & Differential Structures

• Tangent monads are monads defined in the 2-category of tangentads, integrating both tangent structure and monadic operations.
• They encapsulate the differential and geometric properties of tangent bundles in settings like cartesian differential categories and smooth manifolds.
• Their framework extends to algebraic constructions via Eilenberg–Moore and Kleisli categories and connects with affine structures and functor calculus.

Tangent monads are monads that arise from, or are compatible with, categorical tangent structure. In tangent category theory, the term denotes a monad internal to the $2$-category of tangentads, equivalently a tangentad internal to the $2$-category of monads. In cartesian differential and differential $\lambda$-categories, the same expression also refers to the canonical strong commutative monad carried by the tangent bundle functor $T$. A further, explicitly qualified usage appears in functor calculus, where monads built from iterated homotopy cofibers of cubical diagrams are described as “tangent-like” because they extract derivative-like interaction data from functors (Lanfranchi, 24 Mar 2025, Manzyuk, 2012, Bauer et al., 2024).

1. Formal definition in tangent category theory

A tangent category, or more generally a tangentad, is organized around an endomorphism $T$ equipped with projection, zero, sum, vertical lift, and canonical flip,

$$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$

optionally together with negatives $n:T\Rightarrow T$. In the formal theory, a tangentad in a strict $2$-category $\mathcal K$ is an object $X$ together with a strong monoidal functor

$2$0

preserving the fundamental tangent limits pointwise; when $2$1, this recovers ordinary tangent categories (Lanfranchi, 24 Mar 2025).

A tangent monad in $2$2 is then a monad in the $2$3-category $2$4 of tangentads. Concretely, it consists of a tangentad $2$5, a monad $2$6 on the underlying object $2$7, and a distributive law

$2$8

such that $2$9 is a lax tangent morphism and $\lambda$0 is compatible with the monad structure via

$\lambda$1

The formal identification

$\lambda$2

shows that “tangent monad” has two equivalent readings: a monad living in the tangent world, or a tangent structure living in the monadic world. A common misconception is to treat the phrase as referring only to the canonical tangent bundle functor; in the tangentad literature it is a genuinely internal $\lambda$3-categorical notion (Lanfranchi, 24 Mar 2025).

The same formalism also isolates a canonical tangent monad already present in every tangentad. The tangent bundle $\lambda$4 itself carries a monad structure with unit $\lambda$5 and multiplication

$\lambda$6

while the canonical flip $\lambda$7 is a distributive law of monads and the vertical lift $\lambda$8 is a monad morphism. In this sense, the tangent structure does not merely support tangent monads; it canonically produces one (Lanfranchi, 24 Mar 2025).

2. Canonical tangent bundle monads in differential categories and on manifolds

In a cartesian differential category, Manzyuk constructs a canonical tangent functor by

$\lambda$9

With the paper’s direction/base convention, the unit and multiplication are

$T$0

This yields a monad $T$1 on any cartesian differential category. The monad is strong, with tensorial strength

$T$2

and commutative; moreover, the natural transformation

$T$3

is a distributive law of the monad $T$4 over itself. In a differential $T$5-category, the same tangent monad additionally extends to a closed functor and to a $T$6-enriched functor compatible with currying and evaluation (Manzyuk, 2012).

In the category of smooth manifolds, the tangent functor is likewise the functor part of a monad, but the coordinate convention is the usual base/direction one. The unit is the zero section $T$7, and the multiplication is

$T$8

with local expression

$T$9

In the smooth and analytic categories this monad is unique among monads with functor part the tangent functor, and there is no comonad structure on the tangent functor. The uniqueness is a rigidity statement about natural transformations $T$0, which are forced to be combinations of the two canonical projections (Jubin, 2014).

These constructions identify a stable core idea: a tangent monad may be the monadic packaging of the tangent bundle itself. The tangent bundle functor carries first-order information, while the multiplication $T$1 collapses iterated tangent data into a single tangent layer. That interpretation is explicit in both the abstract differential setting and the manifold setting, even though the coordinate conventions differ (Manzyuk, 2012, Jubin, 2014).

3. Algebras, Kleisli categories, and Eilenberg–Moore tangent structure

The formal tangentad theory lifts Street’s algebra construction from ordinary monads to tangent monads. If a $T$2-category $T$3 admits Eilenberg–Moore objects, then $T$4 also admits them. Hence every tangent monad has an algebra object inside the tangent world, provided the underlying monad does. For $T$5, a tangent monad $T$6 on a tangent category $T$7 has an Eilenberg–Moore tangent category $T$8, where

$T$9

defines the lifted algebra structure on $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$0, and the forgetful functor $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$1 is strict tangent (Lanfranchi, 24 Mar 2025).

A parallel bridge appears in the theory of cartesian differential monads. A cartesian differential monad is a monad whose functor is a strong Cartesian differential functor and whose unit and multiplication are $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$2-linear. Its Kleisli category inherits a cartesian differential structure by lifting the base differential combinator, and every cartesian differential monad is a tangent monad relative to the canonical tangent structure on a cartesian differential category. On the Eilenberg–Moore side, the category of algebras of a cartesian differential monad is a cartesian tangent category, and the forgetful functor strictly preserves tangent structure. The tangent bundle monad and reader monads are the basic examples emphasized in this framework (Lemay, 2023).

For the tangent functor monad on smooth manifolds, an algebra is a smooth map

$$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$3

satisfying

$$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$4

Writing $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$5, the associated linear endomorphism

$$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$6

satisfies $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$7. The induced distribution

$$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$8

therefore has rank at most $$p:T\Rightarrow \mathrm{id},\qquad z:\mathrm{id}\Rightarrow T,\qquad s:T^2\Rightarrow T,\qquad \ell:T\Rightarrow T^2,\qquad c:T^2\Rightarrow T^2,$$9 in finite dimension. The accessible sets

$n:T\Rightarrow T$0

partition the manifold, and for a tame algebra they are exactly the Stefan orbits of the induced distribution; equivalently, they are the leaves of a foliation whose tangent distribution is precisely $n:T\Rightarrow T$1. For a regular monadic foliation, any linear holonomy map on a transverse space has $n:T\Rightarrow T$2 as an eigenvalue (Jubin, 2014).

4. Lifted geometric structures: vector fields, differential objects, bundles, and connections

Within the formal theory of tangentads, vector fields are not auxiliary data but a universal construction. The tangentad $n:T\Rightarrow T$3 corepresents vector fields on $n:T\Rightarrow T$4, and vector fields form both a Lie algebra and a $n:T\Rightarrow T$5-monad. The unit picks out the zero vector field, while the multiplication adds vector fields fibrewise. Because this construction is $n:T\Rightarrow T$6-functorial, a tangent monad $n:T\Rightarrow T$7 on $n:T\Rightarrow T$8 lifts to a tangent monad on $n:T\Rightarrow T$9; on an object $2$0, the lifted vector field is

$2$1

This makes the interaction between tangent monads and infinitesimal dynamics entirely formal rather than ad hoc (Lanfranchi, 19 Sep 2025).

The same pattern extends to differential objects, differential bundles, and connections. In the tangentad framework, each is defined by a universal property in a Hom-tangent category. For a strong Cartesian tangent monad $2$2 on a Cartesian tangent category $2$3, the monad lifts to a Cartesian tangent monad on the tangent category $2$4 of differential objects, to a tangent monad on the tangent category $2$5 of differential bundles, and to a tangent monad on the tangent category $2$6 of linear connections. The theory also lifts the classical equivalence between differential objects and differential bundles over the terminal object, and reconstructs covariant derivative, curvature, and torsion as tangentad morphisms defined by universal properties rather than by external formulas (Lanfranchi, 21 Jan 2026).

A plausible implication is that tangent monads organize a large part of monadic differential geometry at the level of structure transport. Instead of asking separately whether a monad preserves vector fields, Euclidean objects, vector bundles, or connections, the tangentad approach treats all of these as formal consequences of monadic compatibility with tangent structure (Lanfranchi, 19 Sep 2025, Lanfranchi, 21 Jan 2026).

5. Affine and $2$7-categorical variants

Affine geometry provides a particularly rich test case. An affine geometric space is a pair $2$8 where

$2$9

is a connection; it is affine when the connection is flat and torsion-free. The category $\mathcal K$0 of affine geometric spaces in a tangent category $\mathcal K$1 is itself a tangent category, with lifted tangent functor

$\mathcal K$2

The construction extends to a strict $\mathcal K$3-comonad

$\mathcal K$4

whose counit is the forgetful functor and whose comultiplication sends $\mathcal K$5 to $\mathcal K$6. Thus affine structure is itself organized comonadically at the $\mathcal K$7-categorical level (Blute et al., 2018).

The flat torsion-free condition admits two characterizations with clear monad-like flavor. For a geometric space $\mathcal K$8, flatness and torsion-freeness are equivalent to the associativity-type identity

$\mathcal K$9

and also to the statement that $X$0 is a connection-preserving morphism from $X$1 to $X$2. This suggests that on affine objects the connection behaves like an internal multiplication on the tangent endofunctor, although the paper formulates the result in terms of connections rather than monad axioms (Blute et al., 2018).

The affine manifold case shows that monadic behavior can become substantially richer after passage to affine objects. In the category of smooth manifolds the tangent functor carries a unique monad and no comonad, but in the affine category monads on $X$3 are indexed by scalars $X$4, comonads by scalars $X$5, and mixed distributive laws make the resulting pairs into bimonads; when $X$6, one obtains Hopf monads. On affine manifolds themselves, algebras are classified by square-zero endomorphisms $X$7, locally through formulas of the form

$X$8

and globally via holonomy-covering arguments (Jubin, 2014, Blute et al., 2018).

6. Generalized and adjacent notions

A distinct but related line of work constructs monads from cubical diagrams and homotopy colimits. Starting from a $X$9-module $2$00, one forms the $2$01-cube $2$02 and defines

$2$03

implemented as a homotopy colimit over an extended cubical indexing category. The resulting endofunctor on $2$04 underlies a monad. In the McCarthy dual-calculus case, this recovers the cocross-effect monads, and the paper explicitly interprets these as “tangent-like monads” because they isolate higher interaction terms and codegree information in a derivative-like way. Here “tangent” refers to functor-calculus behavior, not to tangent-category axioms (Bauer et al., 2024).

Another adjacent construction starts from a monadic linear assignment, equivalently an idempotent monad $2$05 preserving finite products and carrying a compatible commutative-monoid or abelian-group structure. Such data induce a tangent bundle functor

$2$06

and differential objects are precisely $2$07-algebras. The abelianization functor on groups is the prototype: it yields a tangent structure on $2$08 with

$2$09

and differential bundles correspond to abelian groups. This literature does not introduce “tangent monad” as its primary term, but it provides a monadic source of tangent structure and therefore belongs to the broader landscape of monadic tangent geometry (Ikonicoff et al., 14 Oct 2025).

Taken together, these strands show that the phrase “tangent monad” is not exhausted by a single construction. It can denote the canonical monad carried by a tangent bundle functor, a monad compatible with a pre-existing tangent structure, or, in qualified derivative-like settings, a monad extracting tangent information from higher-order interaction data. The unifying theme is monadic control of infinitesimal or first-order structure, but the ambient categorical framework determines the precise meaning (Bauer et al., 2024, Ikonicoff et al., 14 Oct 2025).