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descent for L-infinity algebras

Context

\infty-Lie theory

∞-Lie theory

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

Locality and descent

under construction

so far these are notes taken in talks by Ezra Getzler

Contents

L L_\infty-algebras

The notion of L-infinity algebra is something that naturally arises in deformation theory and in descent problems.

A dg-Lie algebra is a chain complex of vector spaces

V 1dV 0dV 1dV 2 \cdots \to V^{-1} \stackrel{d}{\to} V^0 \stackrel{d}{\to} V^1 \stackrel{d}{\to} V^2 \to \cdots

and is equipped with a bracket operation

[,]:V iV jV i+j [-,-] : V^i \otimes V^j \to V^{i + j}

which is

  • bilinear

  • graded antisymmetric: [x,y]=(1) deg(x)deg(y)[y,x][x,y] = - (-1)^{deg(x) deg(y)} [y,x]

  • satisfies the graded Jacobi identity [x,[y,z]]=[[x,y],z]+(1) deg(x)deg(y)[y,[x,z]][x,[y,z]] = [[x,y],z] + (-1)^{deg(x) deg(y)} [y,[x,z]].

  • is graded Leibnitz: d[x,y]=[dx,y]+(1) deg(x)[x,dy]d[x,y] = [d x,y] + (-1)^{deg(x)} [x, d y]. (aka a graded derivation)

Note If deg(x)deg(x) is odd, then [x,x][x,x] need not vanish. (see also super Lie algebra).

Let LL be a dg-Lie algebra, degreewise finite dimensional (“of finite type” in the language of rational homotopy theory).

We can form its Chevalley-Eilenberg algebra (see there for details) of cochains

CE(L)=( L[1] *,d) CE(L) = (\wedge^\bullet L[1]^*, d)

a quasi-free dga.

(N.B. In full generality, read this as

CE(L)=(( L[1]) *,δ) CE(L) = ((\wedge^\bullet L[1])^*, \delta)

and regard

L[1] \wedge^\bullet L[1]

as a graded commutative COalgebra.)

The underlying graded algebra we may dually think of as functions on some space, a (so-called) formal graded manifold.

The total differential

δ=δ 1+δ 2 \delta = \delta_1 + \delta_2

where the first is dd and the second is the dual of the bracket: [,] *[-,-]^* extended as a graded derivation.

Being a derivation, dually we may think of this as a vector field on our formal smooth manifold. This is sometimes called an NQ-supermanifold.

Definition

An L-infinity algebra (see there) of finite type is the evident generalization of this inroduced by Jim Stasheff and Tom Lada

what kind of link is needed here??

in the early 1990s:

An L-infinity algebra is equivalent to ( in the degreewise finite dimensional case) a free graded-commutative algebra equipped with a differential of degree +1.

Now the differential corresponds to a sequence of nn-ary brackets. For n=1n = 1 this is the differential on the complex, for n=2n = 2 this is the binary bracket from above, and then there are higher brackets.

Morphisms of L L_\infty-algebras

One can consider two notions of morphisms: strict ones and general ones.

A strict one would be a linear map of the underlying vector spaces that strictly preserves all the brackets.

A general definitin of morphisms is: in terms of the dual dg-algebras just a morphism of these, going the opposite way. In the dual formulation this is due to Lada and Stasheff.

We may also think of this as a morphism of NQ-supermanifolds.

All this arose in this form probably most vivedly in the BFV-BRST formalism? or in the BV-BRST formalism.

So in components such a morphism f:LKf : L \to K of L L_\infty-algebras consists of nn-ary maps

f k:L i 1L i kK i 1++1k f_k : L^{i_1} \otimes \cdots L^{i_k} \to K^{i_1 + \cdots + 1 - k}

(where the shift in the indices is due to the numbering convention here only).

The homotopical category of L L_\infty-algebra

We will now describe on the category of L L_\infty-algebras the structure of a category of fibrant objects.

The issue is that the category of L L_\infty-alghebras as defined above has not all products and coproducts.

But we can turn it into a category of fibrant objects.

A notion of category of fibrant objects

This is analogous to (in fact an example of the same general fact) how Kan complexes inside all simplicial sets are the fibrant objects of the model structure on simplicial sets but do not form among themselves a model category but a category of fibrant objects.

See Kan complex for more…

We now look at the axioms for our category of fibrant objects. It is a slight variant of those in BrownAHT described at category of fibrant objects and draws bit from work of Dwyer and Kan.

Let CC be a category. The axioms used here are the following.

  1. There is a subcategory WCW \subset C whose morphisms are called weak equivalences, such that this makes CC into a category with weak equivalences.

  2. There is another subcategory FCF \subset C, whose morphisms are called fibrations (and those that are also in WW are called acyclic fibrations) , such that

    • it contains all isomorphisms;

    • the pullback of a fibration is again a fibration.

    • the pullback of an acyclic fibrations is an acyclic fibration.

  3. CC has all products and in particular a terminal object **.

Filtered L L_\infty-algebras as a Getzler-category of fibrant objects

Write 𝕃\mathbb{L} for the category of filtered L-infinity algebras

Let L L^\bullet be a graded vector space

a decreasing filtrration on it is

L=F 0LF 1L L = F^0 L \supset F^1 L \supset \cdots

such that LL is the limit over this

Llim L/F 1K L \simeq \lim_{\leftarrow} L/F^1 K

i.e. if (x iF iL)(x_i \in F^i L) then i=0 x i\sum_{i=0}^\infty x_i exists

something missing here

[,,] k [-,- , \dots ]_k

has filtration degree 0 if k>0k \gt 0 or filtration degree 1 if k=0k = 0.

The differential dx=[x] 1d x = [x]_1 has the property

namely?

Then gr(d)gr(d) is a true differential gr(L)gr(L)

Morphisms:

f k:L kN f_k : L^{\otimes k} \to N

where we take f kf_k to have filtration degree 0 for k>0k \gt 0 and filtration degree 1 for k=0k = 0.

So gr(f 1)gr(f_1) is a morphism of complexes from gr(L)gr(L) to gr(N)gr(N).

Definition A morphism ff is a weak equivalence if gr(f)gr(f) is a quasi-isomorphism of complexess.

It is a fibration if gr(f 1)gr(f_1) is surjective.

Theorem This defines the structure of a (Getzler-version of a) category of fibrant objects as defined above.

Given CC be a Getzler-category of fibrant objects.

Define a new Getzler-category of fibrant objects sCs C as follows:

  • The objects of sCs C are simplicial objects in CC, subject to a condition stated in the following item.

  • As in the Reedy model structure, for X X_\bullet a simplicial objects let M kX M_k X_\bullet be the corresponding matching object , defined by the pullback diagram

    M kX (X k1) k+1 (X k2) (k+12) (M k+1X ) k+1. \array{ M_k X_\bullet &\to& (X_{k-1})^{k+1} \\ \downarrow && \downarrow \\ (X_{k-2})^{\left(k+1 \atop 2\right)}&\to& (M_{k+1} X_\bullet)^{k+1} } \,.

    Here the right vertical morphism is assumed to be a fibration, hence so is the left vertical morphism.

    So M kX M_k X_\bullet comes with a map X kM kX X_k \to M_k X_\bullet. We assume that this is a fibration. This allows us to define M k+1X M_{k+1} X_\bullet and to continue the induction.

    So this defines a Reedy fibrant object .

So the objects of sCs C are Reedy fibrant objects X X_\bullet and morphisms are morphisms of simplicial objects.

The weak equivalences in sCs C are taken to be the levelwise weak equivalences.

The fibrations are taken to be the Reedy fibrations, as in the Reedy model structure, i.e. those morphisms X Y X_\bullet \to Y_\bullet such that X kY k× M kY M kX X_k \to Y_k \times_{M_k Y_\bullet} M_k X_\bullet is a fibration for all kk.

So this is just the full subcategory of the Reedy model structure on [Δ op],C[\Delta^{}op], C on the fibrant objects.

There is still a fourth axiom for Getzler-cats of fibrant objects to be stated, which is the existence of path space objects. We take this to be gven by a path space functor

P:CsC P : C \to s C

which is such that

  1. for all XX the face maps of (PX) (P X)_\bullet are weak equivalences;

  2. PP preserves fibrations and acyclic fibrations;

  3. (PX) 0(P X)_0 is naturally isomorphic to XX.

Examples

If CC is the category of Kan complexes, then P kX=sSet(Δ[k],X)P_k X = sSet(\Delta[k],X).

Proposition

For our category \mathcal{L} of filtered L L_\infty-algebras we may set

P kL=L complΩ (Δ k)=lim LΩ/F iLΩ, P_k L = L \otimes_{compl} \Omega^\bullet(\Delta^k) = \lim_{\leftarrow} L \otimes \Omega / F^i L \otimes \Omega \,,

where compl\otimes_{compl} denotes the completed tensor product, more commonly denoted ^\hat \otimes.

Remark

We may also speak of cofibrant objects in a (Getzler-) category of fibrant objects:

those objects XX such that for all acyclic fibrations f:ABf : A \to B the induced map C(X,A)C(X,B)C(X,A) \to C(X,B) is surjective (i.e. those with left lifting property again acyclic fibrations).

Maurer-Cartan elements

All the above is designed to make the following come out right.

Generally, C(*,X)C(*,X) is the set of points (global elements) of XX.

A morphism from the terminal object into an L L_\infty-algebra is a Maurer-Cartan element in the L L_\infty-algebra.

Such a point is just an element of degree 1 and filtration degree 1 that satisfies the equation

k=0 1k![ω,,ω] k=0. \sum_{k= 0}^{\infty} \frac{1}{k!} [\omega, \cdots, \omega]_k = 0 \,.

In the case of dg-Lie algebras, this is just the familiar Maurer-Cartan equation

dω+12[ω,ω]=0. d \omega + \frac{1}{2}[\omega, \omega] = 0 \,.

We have that C(*,P kX)C(*, P_k X) is a functor from CC to the category of Kan complexes.

For the category of Kan complexes, it is the identity functor.

For filtered L L_\infty-algebras it gives

LMC (L)=MC(LΩ (Δ )) L \mapsto MC_\bullet(L) = MC(L \otimes \Omega^\bullet(\Delta^\bullet))

This functor MC MC_\bullet takes fibrations to fibrations and acyclic fibrations to acyclic fibrations and weak equivalences to weak equivalences.

Other applications to sheaves of L L_\infty-algebras

Evaluate on a Cech-nerve to get a cosimplicial L L_\infty-algebra

L =(L 0,L 1,)L^\bullet = (L^0 , L^1, \cdots)

L k= i 0,,i kL(U i 0U i k)L^k = \prod_{i_0, \cdots, i_k} L(U_{i_0} \cap \cdots \cap U_{i_k})

Tot(L )= kΔL kΩ (Δ k). Tot(L^\bullet) = \int_{k \in \Delta} L^k \otimes \Omega^\bullet(\Delta^k) \,.

If LL is a dg-Lie algebra, then

MC 1(L)={ω 0+ω 1dt|ω 1F 1L 1[t],ω 1F 1L 0[t],d Lω 0+[ω 1,ω 1]=0,d dRω 0+[ω 1,ω 0]=0} MC_1(L) = \left\{ \omega_0 + \omega_1 d t |\quad \omega_1 \in F^1 L^1 [t],\quad \omega_1 \in F^1 L^0 [t] , \quad d_L \omega_0 + [\omega_1, \omega_1] = 0, \quad d_{dR} \omega_0 + [\omega_1, \omega_0] = 0 \right\}

Now define the Deligne groupoid as in Getzler’ integration article.

We find inside the large Kan complex of MC-elements a smaller one that is still equivalent.

γ 1(L){ωMC 1(L)|ω 1isconstant} \gamma_1(L) \left\{ \omega \in MC_1(L) | \omega_1 is constant \right\}

To get this impose a gauge condition known from homological perturbation theory.

A context is

LgfM L \stackrel{\overset{f}{\to}}{\underset{g}{\leftarrow}} M
gf=Id L g \circ f = Id_L
fg=Id(d Mh+hd M) f \circ g = Id - (d_M h + h d_M)
gh=0,hf=0,hh=0 g \circ h = 0, \; h \circ f = 0, \; h \circ h = 0
MC(L){ωMC(M)|hω=0} MC(L) \simeq \{\omega \in MC(M) | h \omega = 0\}

See Kuranishi’s article in Annals to see where the motivation for all this comes from.

Jim Stasheff: citation please and how much does all refer to??

Example Consider the space of Schouten Lie algebras

L k=Γ(X, k+1TX) L^k = \Gamma(X, \wedge^{k+1} T X)

Then MC (L)MC_(L) is the set of Poisson brackets 𝒪()\mathcal{O}(\hbar).

For let PMC(L)P \in MC(L). Then π 1(MC (L),P)\pi_1(MC_\bullet(L), P) is the locally Hamiltonian diffeomorphisms / Hamiltonian diffeos.

π 2(MC (L),P)\pi_2(MC_\bullet(L), P) is the set of Casimir operators of PP.

for k>2k \gt 2 π k(MC (L),P)\pi_k(MC_\bullet(L), P).

Descent for L L_\infty-algebra valued sheaves

Associated to an L-infinity algebra LL is a Kan complex whose set of kk-cells is the set of Maurer-Cartan elements on the nn-simplex

MC k(L)=MC(LΩ (Δ k)). MC_k(L) = MC( L \otimes \Omega^\bullet(\Delta^k) ) \,.

Now assume that we have a sheaf L-∞ algebras over a topological space XX. Let {U αX}\{U_\alpha \to X\} be an open cover of XX.

On kk-fold intersections we form

L k= α 0,,α kL(U α 0,,α k). L^k = \oplus_{\alpha_0,\cdots, \alpha_k} L(U_{\alpha_0, \cdots, \alpha_k}) \,.

The problem of descent is to glue all this to a single L L_\infty algebra given by the totalization end

Tot(L )= kL kΩ (Δ k) Tot(L^\bullet) = \int_k L^k \otimes \Omega^\bullet(\Delta^k)

and check if that is equivalent to the one assigned to XX.

We now want to compare the \infty-stack of L L_\infty-algebras and that of the “integration” to the Kan complexes of Maurer-Cartan elements, so compare

MC (Tot(L )) MC_\bullet(Tot(L^\bullet))

with

Tot(MC (L )) Tot(MC_\bullet(L^\bullet))

Notice that we have an evident map

MC( lL lΩ (Δ l)Ω (Δ k))MC( lL lΩ (Δ l×Δ k)) MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l) \otimes \Omega^\bullet(\Delta^k) ) \to MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l \times \Delta^k))

Hinich shows in a special case that this is a homotopy equivalence.

It is easy to prove it for abelian L L_\infty-algebras.

Theorem (Getzler)

This is indeed a homotopy equivalence.

Proof By E.G.‘s own account he has “a terrible proof” but thinks a nicer one using induction should be possible.

Gauge fixing

Recall the notion of “context” from above, which is a collection of maps

LgfMhM L \stackrel{\overset{f}{\to}}{\underset{g}{\leftarrow}} M \stackrel{h}{\to} M

between filtered complex-like things,

meaning?? more general or complex with additional structure??

satisfying some conditions.

We can arrange this such that hh imposes a certain gauge condition on LL, or something

I missed some details here…

Theorem

With

MC(M,h):={ωMC(M)|hω=0} MC(M,h) := \left\{ \omega \in MC(M) | h \omega = 0 \right\}

we have

g:MC(M,h)MC(L) g : MC(M,h) \stackrel{\simeq}{\to} MC(L)
MC (M,h)gMC (L)MC (f)MC (M) MC_\bullet(M,h) \stackrel{g \simeq}{\to} MC_\bullet(L) \stackrel{MC_\bullet(f) \simeq}{\to} MC_\bullet(M)

Proof Along the lines of Kuranishi’s construction:

h([] 0+d Mω+ k=1 1k![ω,,ω] h)=0. h \left( [-]_0 + d_M \omega + \sum_{k = 1}^\infty \frac{1}{k!} [\omega, \cdots , \omega]^h \right) = 0 \,.

So the big \infty-groupoid that drops out of the integration procedure is equivalent to the smaller one which is obtained from it by applying that gauge fixing condition.

It would be nice if in the definition of the MC complex we could replace differential forms on the nn-simplex with just simplicial cochains on Δ[n]\Delta[n].

MC(LC (Δ )). MC(L \otimes C^\bullet(\Delta^\bullet)) \,.

This would make the construction even smaller.

What’s the problem?

If LL is abelian, then this is the Eilenberg-MacLane space which features in the Dold-Kan correspondence.

This is true if one takes care of some things. This is part of the above “terrible proof”.

Because one can prove that using explicit Eilenberg-MacLane’s homotopies that proove the Eilenberg-Zilber theorem in terms of simplicial cochains we have an equivalence

MC(LC (Δ[k]Δ[l]))MC(LC (Δ[k]×Δ[l])) MC( L \otimes C^\bullet(\Delta[k] \otimes \Delta[l]) ) \to MC( L \otimes C^\bullet(\Delta[k] \times \Delta[l]))

References

The discussion of the Deligne groupoid (the \infty-groupoid “integrating” an L L_\infty-algebra) and the gauge condition on the Maurer-Cartan elements is

A reference for the theorem above seems not to be available yet, but I’ll check.

Revised on January 3, 2014 16:42:15 by Urs Schreiber (82.113.98.138)