nLab Lie differentiation

Idea

Lie differentiation is the process reverse to Lie integration. It sends a Lie group to its Lie algebra and more generally a Lie groupoid to its Lie algebroid and a smooth ∞-group to its L-∞ algebra.

• For the moment, for ordinary Lie theory see at Lie's three theorems.

• For infinity-Lie theory see at synthetic differential infinity-groupoid – Lie differentiation .

Definition

A formalization of the notion Lie differentiation in higher geometry has been given in (Lurie), inspired by and building on results discussed at model structure for L-∞ algebras. This we discuss in

• For deformation contexts

We then specialize this to those deformation contexts, def. , that arise in the formalization of higher differential geometry by differential cohesion:

• For cohesive contexts

This is the context in which one has a natural formulation of ordinary Lie differentiation of ordinary Lie groups to Lie algebras and its generalization to the Lie differentiation of smooth ∞-groups to L-∞ algebras. See the discussion of Examples below for more.

For deformation contexts

Definition

A deformation context is an (∞,1)-category $Sp_*$ such that

1. it is a presentable (∞,1)-category;

2. it contains an initial object

and

• equipped with a set of objects

  $$\{E_\alpha \in Stab(Sp_*^{op})\}$$

  in the stabilization of the opposite (∞,1)-category $Sp_*^{op}$.

This is (Lurie, def. 1.1.3) together with the assumption of a terminal object in $Sp_*^{op}$ stated on p.9 (and later implicialy used).

Remark

Definition is meant to be read as follows:

We think of $Sp_*$ as an (∞,1)-category of pointed spaces in some higher geometry. The point is the initial object. We think of the formal duals of the objects $\{E_\alpha\}_\alpha$ as a set of generating infinitesimally thickened points (points in formal geometry).

The following construction generates the “jets” induced by the generating infinitesimally thickened points.

Definition

Given a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, we say

• a morphism in $Sp_*^{op}$ is an elementary morphism if it is the homotopy fiber to a map into $\Omega^{\infty -n}E_\alpha$ for some $n \in \mathbb{Z}$ and some $\alpha$;

• a morphism in $Sp_*^{op}$ is a small morphism if it is the composite of finitely many elementary morphisms.

We write

$$Sp_*^{inf} \hookrightarrow Sp_*$$

for the full sub-(∞,1)-category on those objects $A$ for which the essentially unique map $A \to *$ is small.

(Lurie, def. 1.1.8)

Definition

Given a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, def. , the (∞,1)-category of formal moduli problems over it is the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-presheaves over $Sp_*^{inf}$

$$FormalModuli^{Sp_*} \hookrightarrow PSh_\infty(Sp_*^{inf})$$

on those (∞,1)-functors $X \colon (Sp_*^{inf})^{op} \to \infty Grpd$ such that

1. over the terminal object they are contractible: $X(*) \simeq *$;

2. they sends (∞,1)-colimits in $Sp_*^{inf}$ to (∞,1)-limits in ∞Grpd.

(Lurie, def. 1.1.14)

(Lurie, remark 1.1.7).

Remark

This means that a “formal deformation problem” is a space in higher geometry whose geometric structure is detected by the “test spaces” in $Sp_*^{inf}$ in a way that respects gluing (descent) in $Sp_*^{inf}$ as given by (∞,1)-colimits there. The first condition requires that there is an essentially unique such probe by the point, hence that these higher geometric space have essentially a single global point. This is the condition that reflects the infinitesimal nature of the deformation problem.

We will often just write $Sp_*$ for a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, when the objects $\{E_\alpha\}$ are understood.

Proposition

The (∞,1)-category $FormalModuli^{Sp_*}$ of formal moduli problems is a presentable (∞,1)-category. Moreover it is a reflective sub-(∞,1)-category of the (∞,1)-category of (∞,1)-presheaves

$$FormalModuli^{Sp_*} \stackrel{\leftarrow}{\hookrightarrow} PSh_{\infty}(Sp_*) \,.$$

Proposition

Given a deformation context $Sp_*$, the restricted (∞,1)-Yoneda embedding gives an (∞,1)-functor

$$Lie \;\colon\; Sp_* \to FormalModuli^{Sp_*} \,.$$

Remark

For $(X,x) \in Sp_*$, the object $Lie(X,x)$ represents the formal neighbourhood of the basepoint $x$ of $X$ as seen by the infinitesimally thickened points dual to the $\{E_\alpha\}$.

Hence we may call this the operation of Lie differentiation of spaces in $Sp_*$ around their given base point.

In the archetypical implementation of these axiomatics, discuss below, there is an equivalence of (∞,1)-categories of formal moduli problems with L-∞ algebras and the Lie differentiation of the delooping/moduli ∞-stack $\mathbf{B}G$ of a smooth ∞-group $G$ is its L-∞ algebra $\mathfrak{g}$: $Lie(\mathbf{B}G) \simeq \mathbf{B}\mathfrak{g}$.

Proposition

The Lie differentiation functor

$$Lie \; \colon \; Sp_* \to FormalModuli^{Sp_*}$$

of prop. preserves (∞,1)-limits.

Proof

By prop. the (∞,1)-limits in $FormalModuli^{Sp_*}$ may be computed in $PSh_\infty(Sp_*)$. There the statement is that of the (∞,1)-Yoneda embedding, or rather just the statement that the (∞,1)-hom (∞,1)-functor $Sp_*(D,-)$ preserves $(\infty,1)$-limits.

For cohesive contexts

under construction

Let $\mathbf{H}_{th}$ be a cohesive (∞,1)-topos $(ʃ \dashv \flat \dashv \sharp)$ equipped with differential cohesion $(Red \dashv ʃ_{inf} \dashv \flat_{inf})$.

Definition

A set of objects $\{D_\alpha \in \mathbf{H}_{th}\}_\alpha$ is said to exhibit the differential structure or exhibit the infinitesimal thickening of $\mathbf{H}_{th}$ if the localization

$$L_{\{D_\alpha\}} \mathbf{H}_{th} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H}_{th}$$

of $\mathbf{H}_{th}$ at the morphisms of the form $D_\alpha \times X \to X$ is exhibited by the infinitesimal shape modality $ʃ_{inf} \coloneqq i^* i_*$

$$\mathbf{H} \simeq L_{\{D_\alpha\}} \mathbf{H}_{th} \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}} \mathbf{H}_{th} \,.$$

Remark

Def. expresses the infinitesimal analog of the notion of objects exhibiting cohesion, see at structures in cohesion – A1-homotopy and the continuum, hence an infinitesimal notion of A1-homotopy theory.

Proposition

If objects $\{D_\alpha \in \mathbf{H}_{th}\}$ exhibit the differential cohesion of $\mathbf{H}_{th}$, then they are essentially uniquely pointed.

Proof

The localizing objects are in particular themselves local objects so that $ʃ_{inf} D_\alpha \simeq *$. By the $(Red \dashv ʃ_{inf})$-adjunctions this means that

$$\begin{aligned} \mathbf{H}_{th}(*, D_\alpha) & \simeq \mathbf{H}_{th}(Red(*), D_\alpha) \\ & \simeq \mathbf{H}_{th}(*, ʃ_{inf} D_\alpha) \\ & \simeq \mathbf{H}_{th}(*, *) \\ & \simeq * \end{aligned} \,.$$

We now consider $(\mathbf{H}_{th}^{\ast/}, \{D_\alpha\})$ as a deformation context, def. .

Definition

Write

$$Lie \;\colon\; \mathbf{H}_{th}^{\ast/} \to FormalModuli^{\mathbf{H}_{th}^{*/}} \hookrightarrow PSh_\infty(\mathbf{H}_{th}^{*/})$$

for the Lie differentiaon (∞,1)-functor, def. , which sends $(x \colon * \to X) \in \mathbf{H}_{th}$ to

$$Lie(X,x) \;\colon\; D \mapsto \mathbf{H}^{\ast/}(D,(X,x)) \,.$$

Examples

Examples of contexts for Lie differentiation

dg-Geometry

dg-geometry (the running example in (Lurie)).

Synthetic-differential $\infty$-groupoids

• SynthDiff∞Grpd

(…)

synthetic differential infinity-groupoid – Lie differentiation

(…)

Examples of Lie differentiation

Of a Lie group

(…)

Of a Lie groupoid

Given a Lie groupoid $G_1\Rightarrow G_0$, we take the vector bundle $ker Ts|_{G_0}$ restricted on $G_0$, then we show that there is a Lie algebroid structure on $A:=ker Ts|_{G_0} \to G_0$. First of all, the anchor map is given by $ker Ts_{G_0} \xrightarrow{Tt} G_0$. Secondly, to define the Lie bracket, one shows that a section $X$ of $A$ may be right translated to a vector field on $G_1$, which is right invariant. Then Jacobi identity implies that right invariant vector fields are closed under Lie bracket. Thus Lie brackets on vector fields on $G_1$ induces a Lie bracket on sections of $A$.

Related concepts

Examples of sequences of local structures

geometry	point	first order infinitesimal	$\subset$	formal = arbitrary order infinitesimal	$\subset$	local = stalkwise	$\subset$	finite
			$\leftarrow$ differentiation		integration $\to$
smooth functions		derivative		Taylor series		germ		smooth function
curve (path)		tangent vector		jet		germ of curve		curve
smooth space		infinitesimal neighbourhood		formal neighbourhood		germ of a space		open neighbourhood
function algebra		square-0 ring extension		nilpotent ring extension/formal completion				ring extension
arithmetic geometry	$\mathbb{F}_p$ finite field			$\mathbb{Z}_p$ p-adic integers		$\mathbb{Z}_{(p)}$ localization at (p)		$\mathbb{Z}$ integers
Lie theory		Lie algebra		formal group		local Lie group		Lie group
symplectic geometry		Poisson manifold		formal deformation quantization		local strict deformation quantization		strict deformation quantization

References

Lie differentiation of Lie n-groupoids was first considered in generality in

• Pavol Ševera, $L_\infty$ algebras as 1-jets of simplicial manifolds (and a bit beyond) (arXiv:0612349)

See also theorem 8.28 of

• Du Li, Higher Groupoid Actions, Bibundles, and Differentiation (arXiv:1512.04209)

Lie differentiation in deformation contexts is formulated in section 1 of

• Jacob Lurie, Formal moduli problems