nLab descent for L-infinity algebras

under construction

so far these are notes taken in talks by Ezra Getzler

$L_\infty$-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

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

and is equipped with a bracket operation

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

which is

• bilinear

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

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

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

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

Let $L$ 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

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

a quasi-free dga.

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

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

and regard

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

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

$$\delta = \delta_1 + \delta_2$$

where the first is $d$ 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 $n$-ary brackets. For $n = 1$ this is the differential on the complex, for $n = 2$ this is the binary bracket from above, and then there are higher brackets.

Morphisms of $L_\infty$-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 $f : L \to K$ of $L_\infty$-algebras consists of $n$-ary maps

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

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

The homotopical category of $L_\infty$-algebra

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

The issue is that the category of $L_\infty$-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 $C$ be a category. The axioms used here are the following.

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

2. There is another subcategory $F \subset C$, whose morphisms are called fibrations (and those that are also in $W$ 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. $C$ has all products and in particular a terminal object $*$.

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

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

Let $L^\bullet$ be a graded vector space

a decreasing filtrration on it is

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

such that $L$ is the limit over this

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

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

something missing here

$$[-,- , \dots ]_k$$

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

The differential $d x = [x]_1$ has the property

namely?

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

Morphisms:

$$f_k : L^{\otimes k} \to N$$

where we take $f_k$ to have filtration degree 0 for $k \gt 0$ and filtration degree 1 for $k = 0$.

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

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

It is a fibration if $gr(f_1)$ is surjective.

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

Given $C$ be a Getzler-category of fibrant objects.

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

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

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

  $$\array{ M_k X_\bullet &\to& (X_{k-1})^{k+1} \\ \downarrow && \downarrow \\ (X_{k-2})^{\left(k+1 \atop 2\right)}&\to& (M_{k+1} X_\bullet)^{k+1} } \,.$$

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

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

  So this defines a Reedy fibrant object .

So the objects of $s C$ are Reedy fibrant objects $X_\bullet$ and morphisms are morphisms of simplicial objects.

The weak equivalences in $s C$ 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 $X_\bullet \to Y_\bullet$ such that $X_k \to Y_k \times_{M_k Y_\bullet} M_k X_\bullet$ is a fibration for all $k$.

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

$$P : C \to s C$$

which is such that

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

2. $P$ preserves fibrations and acyclic fibrations;

3. $(P X)_0$ is naturally isomorphic to $X$.

Examples

If $C$ is the category of Kan complexes, then $P_k X = sSet(\Delta[k],X)$.

Proposition

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

$$P_k L = L \otimes_{compl} \Omega^\bullet(\Delta^k) = \lim_{\leftarrow} L \otimes \Omega / F^i L \otimes \Omega \,,$$

where $\otimes_{compl}$ 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 $X$ such that for all acyclic fibrations $f : A \to B$ the induced map $C(X,A) \to C(X,B)$ 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, $C(*,X)$ is the set of points (global elements) of $X$.

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

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

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

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

$$d \omega + \frac{1}{2}[\omega, \omega] = 0 \,.$$

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

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

For filtered $L_\infty$-algebras it gives

$$L \mapsto MC_\bullet(L) = MC(L \otimes \Omega^\bullet(\Delta^\bullet))$$

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

Other applications to sheaves of $L_\infty$-algebras

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

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

$L^k = \prod_{i_0, \cdots, i_k} L(U_{i_0} \cap \cdots \cap U_{i_k})$

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

If $L$ is a dg-Lie algebra, then

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

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.

$$\gamma_1(L) \left\{ \omega \in MC_1(L) | \omega_1 is constant \right\}$$

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

A context is

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

$$g \circ f = Id_L$$

$$f \circ g = Id - (d_M h + h d_M)$$

$$g \circ h = 0, \; h \circ f = 0, \; h \circ h = 0$$

$$MC(L) \simeq \{\omega \in MC(M) | h \omega = 0\}$$

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

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

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

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

$\pi_2(MC_\bullet(L), P)$ is the set of Casimir operators of $P$.

for $k \gt 2$ $\pi_k(MC_\bullet(L), P)$.

Descent for $L_\infty$-algebra valued sheaves

Associated to an L-infinity algebra $L$ is a Kan complex whose set of $k$-cells is the set of Maurer-Cartan elements on the $n$-simplex

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

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

On $k$-fold intersections we form

$$L^k = \oplus_{\alpha_0,\cdots, \alpha_k} L(U_{\alpha_0, \cdots, \alpha_k}) \,.$$

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

$$Tot(L^\bullet) = \int_k L^k \otimes \Omega^\bullet(\Delta^k)$$

and check if that is equivalent to the one assigned to $X$.

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

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

with

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

Notice that we have an evident map

$$MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l) \otimes \Omega^\bullet(\Delta^k) ) \to MC(\int_l L^l \otimes \Omega^\bullet(\Delta^l \times \Delta^k))$$

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

It is easy to prove it for abelian $L_\infty$-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

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

between filtered complex-like things,

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

satisfying some conditions.

We can arrange this such that $h$ imposes a certain gauge condition on $L$, or something

I missed some details here…

Theorem

With

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

we have

$$g : MC(M,h) \stackrel{\simeq}{\to} MC(L)$$

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

Proof Along the lines of Kuranishi’s construction:

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

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 $n$-simplex with just simplicial cochains on $\Delta[n]$.

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

This would make the construction even smaller.

What’s the problem?

If $L$ 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

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

References

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

• Ezra Getzler, Lie theory for nilpotent L-infinity algebras (arXiv:0404003)

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