Tangent Complex of Lie n-Groupoids

• Tangent complex of Lie n-groupoids is a higher categorical structure capturing infinitesimal symmetries via graded kernel bundles of horn projection maps.
• It defines a differential graded Lie n-algebroid that encodes higher brackets and homotopy symmetries, offering insights for integration and deformation theory.
• It underpins applications in mathematical physics, Poisson geometry, and graded symplectic manifolds, ensuring functorial invariance in higher geometric settings.

The tangent complex of Lie $n$-groupoids is a categorical and geometric structure encapsulating the infinitesimal data underlying higher groupoid objects. It generalizes the classical notion of the Lie algebroid of a Lie groupoid to higher categorical degrees ($n$), with the aim of capturing “infinitesimal symmetries” in contexts such as differentiable stacks, graded manifolds, symmetry groupoids in field theory, and diffeological groupoids arising in mathematical physics and geometry.

1. Formal Definition and Abstract Framework

The tangent complex of a Lie $n$-groupoid is constructed as the direct sum of kernel bundles associated to the failure of the horn projection maps on a simplicial manifold $X_\bullet$. For $X_\bullet$ a Lie $n$-groupoid, one sets

$$T(X_\bullet) \cong \ker(Tp^1_0|_{X_0})[1] \oplus \ker(Tp^2_0|_{X_0})[2] \oplus \cdots \oplus \ker(Tp^n_0|_{X_0})[n],$$

where $Tp^k_0|_{X_0}$ is the differential of the $k$-th face map at the base, and the square brackets indicate grading (degree shift) (Li et al., 2023). The construction is functorial, realized via a “differentiation” or “tangent functor” from the category of $n$-groupoids to that of graded manifolds.

In settings admitting categorical tangent structures, such as the framework of Rosický (Aintablian et al., 27 Dec 2024), the tangent complex is the source vertical tangent bundle restricted to the identity, or more generally, it is the object of invariant vector fields closed under the abstract Lie bracket. The underlying vector bundle is generalized to a module in the slice category over the base of the groupoid, and scalar multiplication is by a ring object representing “real numbers.”

2. Algebraic Structure: Lie $n$-Algebroids and Homotopy

The tangent complex of a Lie $n$-groupoid carries the structure of a (possibly differential graded) Lie $n$-algebroid. In particular, it supports higher brackets (often derived from the simplicial identities) that satisfy graded Jacobi or Leibniz rules, thus encoding the infinitesimal “homotopy” symmetry types of the original groupoid (Li et al., 2023, Cueca et al., 10 Oct 2025).

For $n$-racks, the infinitesimal structure is a Leibniz $n$-algebra, whose bracket satisfies

$$[X_1, \ldots, X_{n-1}, [Y_1, \ldots, Y_n]] = \sum_{i=1}^n [Y_1, \ldots, Y_{i-1}, [X_1, \ldots, X_{n-1}, Y_i], Y_{i+1}, \ldots, Y_n],$$

and which generalizes the classical integration problem for Lie algebras to the context of higher and non-associative symmetries (Biyogmam, 2011).

3. Functoriality, Universality, and Decategorification

Differentiation of groupoids is treated purely categorically, using universal constructions, pullbacks, and limits, rather than functional analysis. For Fréchet Lie groups, this approach identifies the Lie algebra (up to sign) with the space of vector fields, entirely via categorical methods, bypassing issues of completeness and avoiding analytic subtleties (Aintablian et al., 27 Dec 2024).

The procedure applies equally to infinite-dimensional and non-compact cases, ensuring the tangent complex (or “abstract Lie algebroid,” Editor's term) is determined functorially. Invariant vector fields are defined axiomatically in the tangent category, and equipped with a Lie bracket that is closed and satisfies a Leibniz rule relative to the base ring morphisms.

4. Cohomology, Deformations, and Morita Invariance

The tangent complex serves as the model for representations up to homotopy, cohomology computations, and deformation theory. In VB-groupoids, its canonical cochain complex computes groupoid cohomology with values in the adjoint representation up to homotopy (Gracia-Saz et al., 2010). The classification of regular 2-term representations up to homotopy produces new cohomological invariants in $H^2(G; \nu \to K)$, measuring obstructions to local trivializability of the groupoid or multiplicativity of horizontal lifts.

Deformation theory for VB-groupoids and VB-algebroids attaches a linear deformation complex (often a differential graded Lie algebra, DGLA) to each such object, with Morita invariance implying that the linear deformation cohomology is truly an algebraic invariant of the associated vector bundle over the differentiable stack (Pastina, 2020). The van Est theorem relates the global deformation cohomology of the groupoid to the infinitesimal theory given by its algebroid.

5. Symplectic Structures and Graded Geometry

The tangent complex integrates naturally into the geometry of graded manifolds and their symplectic structures. On a graded manifold, the tangent complex appears as a sequence of vector bundles equipped with higher order differentials, and is central in defining symplectic $m$-manifolds, Q-manifolds, and the AKSZ construction in mathematical physics (Cueca et al., 10 Oct 2025).

Shifted symplectic structures on Lie $n$-groupoids are defined by closed shifted $2$-forms that induce IM-pairings quasi-isomorphic to the cotangent complex. Morita invariance of these shifted symplectic structures holds among hypercovers of Lie $n$-groupoids, making them well-defined (up to equivalence) on differentiable stacks.

6. Examples and Applications

Table: Examples of Tangent Complexes in Geometric Contexts

Geometric Setting	Tangent Complex Description	Reference
Fréchet Lie group / Diffeomorphism group	Space of vector fields with opposite Lie bracket	(Aintablian et al., 27 Dec 2024)
Lie $1$-groupoid	Lie algebroid (source vertical tangent bundle at identity)	(Burke, 2017)
Lie $2$-groupoid / Courant algebroid	$T[1]M \oplus T^*[1]M \oplus T^*[2]M$ graded bundle structure	(Li et al., 2023)
VB-groupoid tangent bundle	Adjoint representation up to homotopy, with canonical cochain complex	(Gracia-Saz et al., 2010)
Diffeological symmetry groupoid / Field theory	Abstract Lie algebroid module, functorially defined	(Aintablian et al., 27 Dec 2024)

The tangent complex provides a unifying approach to infinitesimal symmetries in mathematical physics, Poisson geometry, Courant algebroids, AMM groupoids (inertia stacks of $BG$), and holonomy groupoids for singular foliations.

7. Nonassociative and Higher Structures

Nonassociative generalizations of Lie groupoids (“loopoids” and “quasiloopoids”) have tangent complexes whose algebraic structure is almost-Lie or skew-algebroid. The tangent bundle of a loopoid is canonically a loopoid, and the “Lie-like” functor assigns skew brackets and anchors reflecting nonassociative phenomena (Grabowski et al., 2021).

In such contexts, the tangent complex for Lie $n$-groupoids can be imagined as descending from iterated applications of the differentiation functor, potentially capturing higher or “up-to-homotopy” (weak) associative structures.

8. Perspectives and Open Directions

The tangent complex of Lie $n$-groupoids underlies the process of integration, deformation, and representation theory in higher geometry, as well as the formulation of symplectic and Lagrangian structures on differentiable stacks (Cueca et al., 10 Oct 2025). Functorial, categorical approaches suggest a uniform theory applicable to both finite- and infinite-dimensional contexts, with significant simplifications for computation and further generalization possible to supergeometry, discrete, and abstract settings.

Key open directions include explicit computation of higher brackets on tangent complexes, systematic extension of Morita invariance and deformation theory to all higher categorical levels, and application to computational and theoretical problems in field theory, quantization, and noncommutative geometry.