### On Lie *N*-tegration and Rational Homotopy Theory

#### Posted by Urs Schreiber

In *rational homotopy theory* one studies spaces “up to finite ambiguity” as Dennis Sullivan put it, namely by considering all forms of homotopy and (co)homology over the rationals (i.e discarding all torsion information).

For an overview see for instance

Kathryn Hess
*Rational Homotopy Theory: A Brief Introduction*

(2000)

(pdf).

The crucial insight of Dennis Sullivan described in

Dennis Sullivan
*Infinitesimal computations in topology*
*Publications mathématiqeu de l’ I.H.É.S., tome 47 (1977), p. 269-331*

(NUMDAM)

was that all rational spaces are obtained from *integrating Lie $n$-algebras*.

Of course Sullivan didn’t put it that way, nor do many people in rational homotopy theory. Instead they are talking about differential graded commutive algebras, which are freely generated in positive degree, as graded commutative algebras.

Here at the $n$-Café we call (following Jim Stasheff’s suggestion) such dg-algebras “quasi free differential algebras” (qDGCAs) and are fond of the fact that they are dual to codifferential coalgebras, which are the same as $L_\infty$-algebras, which are the same as Lie $n$-algebras, which are $n$-fold categorifications of Lie algebras. For a quick reminder on how this works, see Lie $n$-algebra cohomology. For a bestiary of examples, most of them described in both languages, try Zoo of Lie $n$-algebras. For more try section *Plan*, subsection *The bridge* as well as the section *Lie $n$-algebra cohomology* here.

It was Ezra Getzler who explained that what Sullivan did with qDGCAs was essentially the integration of the corresponding Lie $n$-algebras:

Ezra Getzler
*Lie theory for nilpotent $L_\infty$-algebras*

arXiv:math/0404003

The basic idea of this integration process, vividly but ultra-tersely sketched on the first two pages of

Pavol Ševera
* Some title containing the words “homotopy” and “symplectic”, e.g. this one*

arXiv:math/0105080

and less vividly, but in more detail, described in

André Henriques
* Integrating $L_\infty$ algebras*

arXiv:math/0603563

is that from any Lie $n$-algebra $g_{(n)}$ we form the simplicial space whose set of $k$-simplices is the set of Lie $n$-algebroid morphisms from the tangent algebroid of the standard $k$-simplex to the given Lie $n$-algebroid

$[k] \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T \Delta_k , g_{(n)})$

which in terms of the dual qDGCAs reads

$[k] \mapsto \mathrm{Hom}_{qDGCA}(g_{(n)}^* , \Omega^\bullet(\Delta_k) ) \,,$

where $\Omega^\bullet(X)$ is simply the deRham differential algebra of forms on $X$.

Notice that the Lie $n$-algebroid morphisms appearing here are, morally, the differential version of smooth pseudofunctors

$\Pi_1(\Delta_k) \to \Sigma G_{(n)}$

from the *fundamental groupoid* of the $k$-simplex to the one-object $n$-groupoid of the Lie $n$-group integrating our Lie $n$-algebra: hence nothing but a flat $G_{(n)}$-valued parallel $n$-transport on $\Delta_k$.

For too long to comfortably admit, I didn’t get the intuitive and conceptual gist underlying this construction. While a good pedagogical account still needs to be written, as far as I can see, I personally profited a lot from realizing how the above procedure reproduces ordinary integration of Lie algebras when we keep the nonabelian Stokes theorem in mind (I talked about that
here. For a description of the nonabelian Stokes theorem see section *Parallel $n$-transport*, subsection *2-Functors and differential 2-forms* here) and that we might profitably think of the space built by the above procedure as the fundamental $n$-groupoid

$\Pi_n(\mathbf{BG}_{(n)})$

of the “generalized smooth classifying space” given by the sheaf on manifolds

$\mathbf{BG}_{(n)} : U \mapsto \mathrm{Hom}_{n\mathrm{Lie}}(T U , g_{(n)}) \,.$

This is not (yet) supposed to be a precise statement, but if you are like me in that you need to have the feeling to know *why* we are doing something in order to enjoy doing it, I suggest this as a good working assumption (originally mentioned here).

The central issue of Ezra Getzler’s above article is to reduce the size of $\Pi_n(\mathbf{GB}_{(n)})$ by strictifying it a lot, such that, in particular, it becomes finite dimensional at each stage.

André Henriques instead works with the unrestricted space. His main application was the integration of the String Lie 2-algebra (section *Lie $n$-algebra cohomology*, subsection *String, Chern-Simons and Chern Lie $n$-algebras* here) and the demonstration that it is the 3-connected cover of $\mathrm{Spin}$.

Notice that, using Sullivan’s age-old theorem (8.1),v) from the above article, this becomes *essentially* a triviality:

the qDGCA defining the String Lie 2-algebra is simply that of the underlying Lie algebra together with a single additional degree 2-generator $b$ whose differential is required $d b = \mu$ to be the canonical 3-cocycle $\mu = \langle \cdot , [\cdot, \cdot] \rangle$ on the underlying semisimple Lie algebra. But that says nothing but that the generator of the third cohomology becomes cohomologically trivial!

(For more on the Lie 2-algebra cohomology of the String Lie 2-algebra see Cohomology of the String Lie 2-algebra. Danny Stevenson has meanwhile started to check (some terms still need to be done) that the qDGCA computation discussed there is indeed reproduced by a computation of the (rational) cohomology of the total space of the String group as constructed by Henriques and in From Loop groups to 2-groups).

Here I start with having a closer look at parts of this literature by going through Sullivan’s old paper and highlighting the Lie $n$-algebra theory he discusses, without saying so at that time.

p. 277 of Sullivan’s article: *action Lie $n$-algebroids*

The first comment concerns the “twisted cohomology of differential algebras” that Sullivan discussed in section 1, p. 277, and which also plays a major role as a starting point for Ezra Getzler (see the first couple of pages of his introduction). The same phenomenon also appears in various texts on BV-quantization.

I am claiming that we might profitably think of **Lie action algebroids** here, namely Lie algebroids obtainable from differentiating action Lie groupoids obtained from a group acting on something.

Let $V$ be a vector space. Then the Lie algebra

$gl(V)$

is dually described by the qDGCA on $\wedge^\bullet (V \otimes V^*)$ with the differential on the generator with respect to a given basis $\{e_i\}$ ov $V$

$\tensor{a}{_i_^j} := e_i \otimes e^j$

given by

$d_{gl(V)} \tensor{a}{_i_^j} = \sum_k \tensor{a}{_i_^j} \wedge \tensor{a}{_j_^k} \,.$

Now, a representation of any other Lie algebra $g$ on $V$ is nothing but a morphism $\rho : g \to gl(V)$ of Lie algebras. Hence a morphism $\rho^\dual : (\wedge^\bullet V \otimes V^*, d_{gl(V)}) \to (\wedge^\bullet g^*, d_g)$ of the qDGCA described above to the Chevalley-Eilenberg algebra of $g$.

Being a morphism of free graded-commutative algebras means that this is given by a set of elements of $g^*$,

$\tensor{\Theta}{_i_^j} \,,$

one for each generator of $gl(V)^*$. But, by the above definition of $d_{gl(V)}$, respect for the differentials then requires that these generators satisfy

$d_g \tensor{\Theta}{_i_^j} = \sum_k \tensor{\Theta}{_i_^k} \wedge \tensor{\Theta}{_k_^j} \,.$

This is the Maurer-Cartan-like equation which pervades Getzler’s discussion. One notices that a collection of such “Maurer-Cartan elements” $\Theta$ can be canonically used to construct a new *algebroid*, namely that dually given by the Chevalley-Eilenberg algebra of $g$ for Lie algebra cohomology with values in the module $V$. See p. 277 of Sullivan’s article. His “twisted cohomology” is, I claim, to be regarded as the qDGCA dual to the action Lie algebroid of $g$ acting on $V$.

This is the same kind of reasoning underlying the interpretation of the BV formalism which I mentioned in BV-Formalism, Part IV.

p. 278 of Sullivan’s article: *quasi-free graded commutative algebras*

Here in the very last paragraph on that page the condition is first mentioned that we want to restrict attention to differential graded-commutative algebras which are *free* as graded commutative algebras (and generated in non-negative degree).

These are known to be those related dually to “semistrict” Lie $n$-algebras. These are Lie $n$-algebras for which the Jacobi identity is allowed to hold up to coherently weak equivalence, but where the skew symmetry of the bracket functor is required to hold strictly.

One might ask:

a) to what kind of Lie $n$-algebras would, dually, dg-algebras non-free as graded commutative algebras correspond?

b) to what kind of dg-algebras would Lie $n$-algebras with coherently weak skew symmetry correspond.

Neither answer is known. But I am speculating that the answer to a) is b) and the answer to b) is a), essentially.

See the discussion at Categorified Clifford Algebra and weak Lie $n$-Algebras and the table in this comment. This has grown out of thinking about

Dmitry Roytenberg
*On weak Lie 2-algebras *

(pdf)

p. 279 of Sullivan’s article *strict Lie $n$-algebras*

On the top of p. 279 Sullivan restricts attention to those dg-algebras $A$ whose differential satisfies $d A \subset A \wedge A \,.$

This says that only binary operations exist in the dual $L_\infty$-algebra. And this means that the Jacobiator and all its higher coherences vanish: we are dealing with a *strict* Lie $n$-algebra. Equivalently, with a *dg-Lie algebra*.

Sullivan calls qDGCAs satisfying this condition *minimal*.

Later on he proves that all qDGCAs come from tensoring minimal ones with *free* ones. In my language this should mean: all Lie $n$-algebras come from strict ones tensored with those obtained from weak cokernels for an identity morphism on a Lie $n$-algebra. (Because these are precisely those corresponding to free dg-algebras, as described in Structure of Lie $n$-algebras. (Remind me to update this file with my latest polished version.))

p . 281 *the first hint on Lie $n$-tegration*

After having discussed qDGCAs, Sullivan notices a couple of striking analogies with homotopy theory of spaces. In particular he notices that the fundamental group of a space plays the role of the Lie algebra dual to a qDGCA. This is the first hint for $G_{(n)} = \Pi_{(n)}(U \mapsto \mathrm{Hom}_{qDGCA}(g_{(n)}^*, \Omega^\bullet(U))) \,.$

p. 283: *obstruction theory*

Now we are already discussing obstruction theory. At that time diagrams were not yet fashionable, I guess, but no amount of prose can substutute them:

given qDGCAs $A$, $B$ and $C$, and given morphisms

$\array{ && A \\ && \uparrow \\ C &\leftarrow& B }$

we are asking for the obtruction to lifting to a morpism

$\array{ && A \\ &\swarrow& \uparrow \\ C &\leftarrow& B } \,.$

Notice that, dually, we have a sequence of Lie $n$-algebras

$\array{ k \\ \downarrow \\ g \\ \downarrow \\ b }$ together with a connection taking values in one of them $\array{ &&k \\ &&\downarrow \\ &&g \\ &&\downarrow \\ T X &\to&b }$

and are asking for the obstruction to lifting this $b$-connection to a $g$-connection through our extension

$\array{ &&k \\ &&\downarrow \\ &&g \\ &\nearrow&\downarrow \\ T X &\to&b } \,.$

I need to think about how Sullivan’s component-based answer relates to the one I talk about in section *String and Chern-Simons $n$-transport*, subsection *obstruction theory* here, based on the idea of weak cokernels which has grown out of the discussion of Obstructions for $n$-bundle lifts, Obstructions, Tangent Categories and Lie $n$-tegration , Obstructions to $n$-Bundle Lifts Part II: The BIG Diagram.

section 8: *integration and cohomology*

In section 8 (p. 300) finally the integration process is discussed. Sullivan does not think of his qDGCAs as Lie $n$-algebras, but does emphasize (that’s the whole point) how their cohomology can be used to study the rational cohomology of spaces (“integrating them”, as we would now say).

The striking statement is that at the beginning of p. 301, which later becomes item v) of theorem (8.1) on p. 304:

the cohomology of the space “integrating” a qDGCA is precisely that of the qDGCA.

The familiar example for that is Lie algebra cohomology: the cohomology of the Chevalley-Eilenberg algebra, which is the qDGCA dual to a Lie algebra and noting but what is known as Lie algebra cohomology, is the same as the cohomology of the simply connected compact Lie group integrating it.

The generalization of this fact means that computations as in Cohomology of the String Lie 2-algebra indeed compute the cohomology of the group integrating the String Lie 2-algebra.

And indeed, Danny started to check (looks good so far but some more terms need to be checked), for String, that this is true.

But at some point here I am slightly confused about dimensions, re cohomology of $G$ versus that of $B G$. Need to think about that.

## Re: On Lie N-tegration and Rational Homotopy Theory

I’m not an expert on this, and hopefully Jim Stasheff can clarify, but from what I’ve gathered the equivalence of the homotopy theories of rational homotopy types, comm dgas and dg Lie algebras goes back to Quillen’s 1968 (?) paper on rational homotopy theory, where he is pretty explicit that what we’re doing is a direct generalization of the classical integration of Lie algebras. Certainly the whole modern approach to deformation theory, as explained by Kontsevich (and others, Deligne, Drinfeld, Feigin — don’t know the history unfortunately) is built on this part of Quillen. To use modern language, Lie and Comm are Koszul dual operads, with the Koszul duality realized geometrically by the operations “differentiate” (aka take tangent complex) and “integrate” (aka solve Maurer-Cartan equations). I believe all the other general statements follow from interpreting this correctly (eg resolving the operads themselves to get homotopically correct versions etc).

The work of Schechtman, Hinich, Getzler and others makes this into a nice geometric statement (at least morally, I don’t know if we need some boundedness etc). As Hinich explains it is just “Lie’s theorem”: rationally Lie algebras and formal groups are the same thing, and likewise L

_{∞}algebras can be integrated to geometric objects (formal derived stacks - functions on which give commutative dgas), geometric objects can be differentiated to L_{∞}-algebras (tangent complexes), and this gives an equivalence between the two homotopy theories.(The subtle questions treated by Getzler etc, have to do as far as I understand with the kind of models we can build for the integration, and with the subtleties of positively graded vs arbitrary complexes - Quillen was dealing with spaces, ie connective things, while Getzler isn’t. Henriques is interested in getting more than formal integration, requiring a different set of ideas.)