nLab
Lie differentiation

Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Integration theory

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. 1, 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 *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 * opSp_*^{op} stated on p.9 (and later implicialy used).

Remark

Definition 1 is meant to be read as follows:

We think of Sp *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 α} α\{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 α} α)(Sp_*, \{E_\alpha\}_\alpha), we say

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

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

We write

Sp * infSp * Sp_*^{inf} \hookrightarrow Sp_*

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

(Lurie, def. 1.1.8)

Definition

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

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

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

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

  2. they sends (∞,1)-colimits in Sp * infSp_*^{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 * infSp_*^{inf} in a way that respects gluing (descent) in Sp * infSp_*^{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 *Sp_* for a deformation context (Sp *,{E α} α)(Sp_*, \{E_\alpha\}_\alpha), when the objects {E α}\{E_\alpha\} are understood.

Proposition

The (∞,1)-category FormalModuli Sp *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 *PSh (Sp *). FormalModuli^{Sp_*} \stackrel{\leftarrow}{\hookrightarrow} PSh_{\infty}(Sp_*) \,.
Proposition

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

Lie:Sp *FormalModuli Sp *. Lie \;\colon\; Sp_* \to FormalModuli^{Sp_*} \,.
Remark

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

Hence we may call this the operation of Lie differentiation of spaces in Sp *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 BG\mathbf{B}G of a smooth ∞-group GG is its L-∞ algebra 𝔤\mathfrak{g}: Lie(BG)B𝔤Lie(\mathbf{B}G) \simeq \mathbf{B}\mathfrak{g}.

Proposition

The Lie differentiation functor

Lie:Sp *FormalModuli Sp * Lie \; \colon \; Sp_* \to FormalModuli^{Sp_*}

of prop. 2 preserves (∞,1)-limits.

Proof

By prop. 1 the (∞,1)-limits in FormalModuli Sp *FormalModuli^{Sp_*} may be computed in PSh (Sp *)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,)Sp_*(D,-) preserves (,1)(\infty,1)-limits.

For cohesive contexts

under construction

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

Definition

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

L {D α}H thH th L_{\{D_\alpha\}} \mathbf{H}_{th} \stackrel{\leftarrow}{\hookrightarrow} \mathbf{H}_{th}

of H th\mathbf{H}_{th} at the morphisms of the form D α×XXD_\alpha \times X \to X is exhibited by the infinitesimal shape modality Unknown characterUnknown character infi *i *ʃ_{inf} \coloneqq i^* i_*

HL {D α}H thi *i *H th. \mathbf{H} \simeq L_{\{D_\alpha\}} \mathbf{H}_{th} \stackrel{\overset{i^*}{\leftarrow}}{\underset{i_*}{\hookrightarrow}} \mathbf{H}_{th} \,.
Remark

Def. 4 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 αH th}\{D_\alpha \in \mathbf{H}_{th}\} exhibit the differential cohesion of H th\mathbf{H}_{th}, then they are essentially uniquely pointed.

Proof

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

H th(*,D α) H th(Red(*),D α) H th(*,ʃ infD α) H th(*,*) *. \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 (H th */,{D α})(\mathbf{H}_{th}^{\ast/}, \{D_\alpha\}) as a deformation context, def. 1.

Definition

Write

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

for the Lie differentiaon (∞,1)-functor, def. 2, which sends (x:*X)H th(x \colon * \to X) \in \mathbf{H}_{th} to

Lie(X,x):DH */(D,(X,x)). 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 group

(…)

Of a Lie groupoid

Given a Lie groupoid G 1G 0G_1\Rightarrow G_0, we take the vector bundle kerTs| G 0ker Ts|_{G_0} restricted on G 0G_0, then we show that there is a Lie algebroid structure on A:=kerTs| G 0G 0A:=ker Ts|_{G_0} \to G_0. First of all, the anchor map is given by kerTs G 0TtG 0ker Ts_{G_0} \xrightarrow{Tt} G_0. Secondly, to define the Lie bracket, one shows that a section XX of AA may be right translated to a vector field on G 1G_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 1G_1 induces a Lie bracket on sections of AA.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\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 geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\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

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

Revised on December 17, 2015 04:56:41 by Chenchang Zhu (134.76.82.113)