# nLab Lie differentiation

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## 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.

## 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

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

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

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.

###### 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.

###### 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

(…)

synthetic differential infinity-groupoid – Lie differentiation

(…)

### Examples of Lie differentiation

(…)

#### 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 $A$.

Examples of sequences of local structures

geometrypointfirst order infinitesimal$\subset$formal = arbitrary order infinitesimal$\subset$local = stalkwise$\subset$finite
$\leftarrow$ differentiationintegration $\to$
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry$\mathbb{F}_p$ finite field$\mathbb{Z}_p$ p-adic integers$\mathbb{Z}_{(p)}$ localization at (p)$\mathbb{Z}$ integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict 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)