∞-Lie theory (higher geometry)
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
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 .
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.
A deformation context is an (∞,1)-category $Sp_*$ such that
it is a presentable (∞,1)-category;
it contains an initial object
and
equipped with a set of objects
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).
Definition 1 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.
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
for the full sub-(∞,1)-category on those objects $A$ for which the essentially unique map $A \to *$ is small.
Given a deformation context $(Sp_*, \{E_\alpha\}_\alpha)$, 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 $Sp_*^{inf}$
on those (∞,1)-functors $X \colon (Sp_*^{inf})^{op} \to \infty Grpd$ such that
over the terminal object they are contractible: $X(*) \simeq *$;
they sends (∞,1)-colimits in $Sp_*^{inf}$ to (∞,1)-limits in ∞Grpd.
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.
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
Given a deformation context $Sp_*$, the restricted (∞,1)-Yoneda embedding gives an (∞,1)-functor
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 $\mathfrak{g}$: $Lie(\mathbf{B}G) \simeq \mathbf{B}\mathfrak{g}$.
The Lie differentiation functor
of prop. 2 preserves (∞,1)-limits.
By prop. 1 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.
under construction
Let $\mathbf{H}_{th}$ be a cohesive (∞,1)-topos $(ʃ \dashv \flat \dashv \sharp)$ equipped with differential cohesion $(Red \dashv ʃ_{inf} \dashv \flat_{inf})$.
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
of $\mathbf{H}_{th}$ at the morphisms of the form $D_\alpha \times X \to X$ is exhibited by the infinitesimal shape modality $ʃ_{inf} \coloneqq i^* i_*$
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.
If objects $\{D_\alpha \in \mathbf{H}_{th}\}$ exhibit the differential cohesion of $\mathbf{H}_{th}$, then they are essentially uniquely pointed.
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
We now consider $(\mathbf{H}_{th}^{\ast/}, \{D_\alpha\})$ as a deformation context, def. 1.
Write
for the Lie differentiaon (∞,1)-functor, def. 2, which sends $(x \colon * \to X) \in \mathbf{H}_{th}$ to
dg-geometry (the running example in (Lurie)).
(…)
synthetic differential infinity-groupoid – Lie differentiation
(…)
(…)
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
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 |
Lie differentiation of Lie n-groupoids was first considered in generality in
See also theorem 8.28 of
Lie differentiation in deformation contexts is formulated in section 1 of