∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
An -Lie algebroid is a smooth ∞-groupoid (or rather a synthetic-differential ∞-groupoid) all whose k-morphisms for all have infinitesimal extension (are infinitesimal neighbours of an identity -morphism).
-Lie algebroids are to ∞-Lie groupoids as Lie algebras are to Lie groups:
∞-Lie groupoid - -Lie algebroid .
We discuss -Lie algebroids in the cohesive (∞,1)-topos SynthDiff∞Grpd of synthetic differential ∞-groupoids. This is an infinitesimal cohesive neigbourhood of the cohesive -topos Smooth∞Grpd of smooth ∞-groupoids, which is exhibited by the infinitesimal path (∞,1)-geometric morphism
We consider presentations of the general abstract definition of -Lie algebroids by constructing in the standard model structure-presentation of by simplicial presheaves on CartSp certain classes of simplicial presheaves in the image of semi-free differential graded algebras under the monoidal Dold-Kan correspondence. This amounts to identifying the traditional description of of Lie algebras, Lie algebroids and L-∞ algebras by their Chevalley-Eilenberg algebras as a convenient characterization of the corresponding cosimplicial algebras whose formal dual simplicial presheaves are manifest presentations of infinitesimal smooth ∞-groupoids.
Let
be the full subcategory on the opposite category of cochain dg-algebras over on those dg-algebras that are
graded-commutative;
concentrated in non-negative degree (the differential being of degree +1 );
in degree 0 of the form for SmoothMfd;
semifree: their underlying graded algebra is isomorphic to an exterior algebra on a -graded locally free projective -module
of finite rank;
We call this the category of -algebroids.
More in detail, an object may be identified (non-canonically) with a pair , where
is a smooth manifold – called the base space of the -algebroid ;
is the module of smooth sections of an -graded vector bundle of degreewise finite rank;
is a semifree dga on – a Chevalley-Eilenberg algebra – where
with the th summand on the right being in degree .
An -algebroid with base space the point is an L-∞ algebra , or rather is the delooping of an -algebra. We write for -algebroids over the point. They form the full subcategory
We now construct an embedding of into .
The functor
of the Dold-Kan correspondence from non-negatively graded cochain complexes of vector spaces to cosimplicial vector spaces is a lax monoidal functor and hence induces (see monoidal Dold-Kan correspondence) a functor (which we shall denote by the same symbol)
from non-negatively graded cochain dg-algebras to cosimplicial algebras (over ).
Write
for the restriction of the above along the inclusion :
for the underlying cosimplicial vector space of is given by
and the product of the -algebra structure on the right is given on homogeneous elements in the tensor product by
(Notice that is indeed a commutative cosimplicial algebra, since and in are by definition in the same degree.)
To define the cosimplicial structure, let be the canonical basis for and consider also the basis given by
Then for a morphism in the simplex category, set
and extend this skew-multilinearly to a map . In terms of all this the action of on homogeneous elements in the cosimplicial algebra is defined by
This is due to (CastiglioniCortinas, (1), (2), (20), (22)).
We shall refine the image of to cosimplicial smooth algebras. Let CartSp be the category of Cartesian spaces and smooth functions between them, regarded as a Lawvere theory. Write
for its category of algebras: these are the smooth algebras.
Notice that there is a canonical forgetful functor
to the category of comutative associative algebras over the real numbers.
There is a unique factorization of the functor from def. through the forgetful functor such that for any over base space the degree-0 algebra of smooth functions lifts to its canonical structure as a smooth algebra
Observe that for each the algebra is a finite nilpotent extension of . The claim then follows with using Hadamard's lemma to write every smooth function of sums as a finite Taylor expansion with a smooth rest term. See the examples at smooth algebra for more details on this kind of argument.
Write for the composite (∞,1)-functor
where the first morphism is the monoidal Dold-Kan correspondence as in prop. , the second is the external degreewise Yoneda embedding and is any fibrant-cofibrant resolution functor in the local model structure on simplicial presheaves. The last equivalence holds as discussed there and at models for ∞-stack (∞,1)-toposes.
We do not consider the standard model structure on dg-algebras and do not consider itself as a model category and do not consider an (∞,1)-category spanned by it. Instead, the functor only serves to exhibit a class of objects in , which below in the section Models for the abstract axioms we show are indeed -Lie algebroids by the general abstract definition, . All the homotopy theory of objects in is that of after this embedding.
We may abstractly formalize this in an (infinity,1)-topos with differential cohesion as follows.
Recall that a groupoid object in an (infinity,1)-category is equivalently an 1-epimorphism , thought of as exhibiting an atlas for the groupoid .
Now an -Lie algebroid is supposed to be an -groupoid which is only infinitesimally extended over its base space . Hence:
A groupoid object is infinitesimal if under the reduction modality (equivalently under the infinitesimal shape modality ) the atlas becomes an equivalence: .
For example the tangent -Lie algebroid of any is the unit of the infinitesimal shape modality.
It follows that every such -Lie algebroid canonically maps to the tangent -Lie algebroid of – the anchor map. The naturality square of the unit exhibits the morphism:
The full subcategory category from def. is equivalent to the traditional definition of the category of L-∞ algebras and “weak morphisms” / “sh-maps” between them.
The full subcategory on the 1-truncated objects is equivalent to the traditional category of Lie algebroids (over smooth manifolds).
In particular the joint intersection on the 1-truncated -algebras is equivalent to the category of ordinary Lie algebras.
This is discussed in detail at L-∞ algebra and Lie algebroid.
Above we have given a general abstract definition, def. , of -Lie algebroids, and then a concrete construction in terms of dg-algebras, def. . Here we discuss that this concrete construction is indeed a presentation for objects satisfying the abstract axioms.
As in the discussion at SynthDiff∞Grpd we now present this cohesive (∞,1)-topos by the hypercompletion of the model structure on simplicial presheaves of formal smooth manifolds.
For and its image in the standard presentation for SynthDiff∞Grpd, we have that
is a cofibrant resolution, where is the fat simplex.
We have
The fat simplex is cofibrant in .
The canonical morphism is a weak equivalence between cofibrant objects in the Reedy model structure .
Because every representable is cofibrant, the object is cofibrant.
Every simplicial presheaf is cofibrant regarded as an object Reedy model structure .
Now the coend over the tensoring
is a Quillen bifunctor (as discussed there) for the projective and injective global model structure on functors on the simplex category and its opposite as indicated. This implies the cofibrancy.
It is also a Quillen bifunctor (as discussed there) for the Reedy model structures
Using the factorization lemma this implies the weak equivalence (this is the argument of the Bousfield-Kan map).
Let be an L-∞ algebra, regarded as an -algebroid over the point by the embedding of def. .
Then SynthDiff∞Grpd is an infinitesimal cohesive object, in that it is geometrically contractible
and has as underlying discrete ∞-groupoid the point
We present now SynthDiff∞Grpd by the model structure on simplicial presheaves . Since CartSp is an ∞-cohesive site we have by the discussion there that is presented by the left derived functor of the degreewise colimit and is presented by the left derived functor of evaluation on the point.
because each is an infinitesimally thickened point, hence representable and hence sent to the point by the colimit functor.
That this is equivalent to the point follows from the fact that is an acylic cofibration in , and that
is a Quillen bifunctor, using that is cofibrant.
Similarily, we have degreewise that
by the fact that an infinitesimally thickened point has a single global point. Therefore the claim for follows analogously.
Let be an -algebroid, def. , over a smooth manifold , regarded as a simplicial presheaf and hence as a presentation for an object in according to def. .
We have an equivalence
Let first be a representable. Then according to prop. we have that
is cofibrant in . Therefore by this proposition on the presentation of infinitesimal neighbourhoods by simplicial presheaves over infinitesimal neighbourhood sites we compute the derived functor
with the notation as used there.
In view of def. we have for all that where is an infinitesimally thickened point. Therefore for all and hence .
For general choose first a cofibrant resolution by a split hypercover that is degreewise a coproduct of representables (which always exists, by the discussion at model structure on simplicial presheaves), then pull back the above discussion to these covers.
Every -algebroid in the sense of def. under the embedding of def. is indeed a formal cohesive ∞-groupoid in the sense of def. .
We discuss the relation between the intrinsic cohomology of -algebroids when regarded as objects of , and the ordinary cohomology of their Chevalley-Eilenberg algebras. For more on this see ∞-Lie algebroid cohomology.
Let be an -algebroid. Then its intrinsic real cohomoloogy in SynthDiff∞Grpd
coincides with its ordinary L-∞ algebra cohomology: the cochain cohomology of its Chevalley-Eilenberg algebra
By this discussion at SynthDiff∞Grpd we have that
Observe that is cofibrant in the Reedy model structure relative to the opposite of the projective model structure on cosimplicial algebras: the map from the latching object in degree in is dually in the projection
hence is a surjection, hence a fibration in and therefore indeed a cofibration in .
Therefore using the Quillen bifunctor property of the coend over the tensoring in reverse to lemma the above is equivalent to
with the fat simplex replaced again by the ordinary simplex. But in brackets this is now by definition the image under the monoidal Dold-Kan correspondence of the Chevalley-Eilenberg algebra
By the Dold-Kan correspondence we have hence
a -Lie algebroid over the point, is an L-∞-algebra;
an -truncated -Lie algebroid is a Lie -algebroid;
an -Lie algebroid the differential of whose Chevalley-Eilenberg algebra is “co-binary”, i.e. , is strict.
So in particular
a 1-Lie algebroid is a Lie algebroid;
a 1-Lie algebroid over the point is a Lie algebra;
a Lie -algebroid over a point is a Lie n-algebra.
a BRST-complex is the Chevalley-Eilenberg algebra of an action--Lie algebroid of the action of an -algebra, see Lie ∞-algebroid representation;
more generally, the complexes appearing in BV-BRST formalism are derived -Lie algebroids, whose Chevalley-Eilenberg algebra may have generators in negative degree.
a symplectic Lie n-algebroid is a Lie -algebroid equipped with a non-degrenerate bilinear invariant polynomial of degree . For low this is
: a symplectic manifold;
We discuss the traditional notion of Lie algebroids in view of their role as presentations for infinitesimal synthetic differential 1-groupoids.
In this section we characterize ordinary Lie algebroids as precisely those synthetic differential -groupoids that under the above presentation are locally on any chart of their base space given by simplicial smooth loci of the form
where is the smooth locus of infinitesimal k-simplices based at the origin in . (These smooth loci have been considered in (Kock, section 1.2)).
The following definition may be either taken as an informal but instructive definition – in which case the next definition is to be taken as the precise one – or in fact it may be already itself be taken as the fully formal and precise definition if one reads it in the internal logic of any smooth topos with line object – which for the present purpose is the Cahiers topos with line object . (For an exposition of the latter perspective see (Kock)).
For , an infinitesimal -simplex in based at the origin is a collection of points in , such that each is an infinitesimal neighbour of the origin
and such that all are infinitesimal neighbours of each other
Write for the space of all such infinitesimal -simplices in .
Equivalently:
For , the smooth algebra
is the unique lift through the forgetful functor of the commutative -algebra generated from many generators
subject to the relations
and
In the above form these relations are the manifest analogs of the conditions and . But by multiplying out the latter set of relations and using the former, we find that jointly they are equivalent to the single set of relations
In this expression the roles of the two sets of indices is manifestly symmetric. Hence another equivalent way to state the relations is to say
and
This appears as (Kock, (1.2.1)).
Since is a Weil algebra in the sense of synthetic differential geometry, its structure as an -algebra extends uniquely to the structure of a smooth algebra (as discussed there) and we may think of as an infinitesimal smooth locus.
For and we have that consists of elements of the form
for and a collection of ordinary covectors and with “” denoting the evident contraction, and where in the last step we used the above relations.
It is noteworthy here that the coefficient of the term which is multilinear in each of the is the wedge product of two covectors and : we may naturally identify the subspace of on those elements that vanish if either or are set to 0 as the space of 2-forms at the origin of .
Of course for this identification to be more than a coincidence we need that this is the beginning of a pattern that holds more generally. But this is indeed the case.
Let be the set of square submatrices of the -matrix . As a set this is isomorphic to the set of pairs of subsets of the same size of and , respectively. For instance the square submatrix labeled by and is
For an submatrix, we write
for the corresponding determinant, given as a product of generators in . Here the sum runs over all permutations of and is the signature of the permutation .
The elements are precisely of the form
for unique . In other words, the map of vector spaces
given by
is an isomorphism.
This is a direct extension of the argument in the above example: a general product of generators in is
By the relations in , this is non-vanishing precisely if none of the -indices repeats and none of the -indices repeats. Furthermore by the relations, for any permutation of elements, this is equal to
It follows that each such element may be written as
where is the sub-determinant given by the subset and as discussed above.
In (Kock, section 1.3) effectively this proposition appears as the “Kock-Lawvere axiom scheme for ” when is regarded as an object of a suitable smooth topos.
For any we have a natural isomorphism of real commutative and hence of smooth algebras
where on the right we have the algebras that appear degreewise in def. , where the product is given on homogeneous elements by
Let be the canonical basis for and the canonical basis for . We claim that an isomorphism is given by the assignment
To see that this defines indeed an algebra homomorphism we need to check that it respects the relations on the generators. For this compute:
The inverse clearly exists, given on generators by
For a 1-truncated object, hence an ordinary Lie algebroid of rank over a base manifold , its image under the map , def. , is such that its restriction to any chart is, up to isomorphism, of the form
Apply prop. in def. , using that by definition is given by the exterior algebra on locally free modules, so that
For any smooth manifold, there is a standard notion of the Lie algebroid which is the tangent Lie algebroid
of . We discuss this from the perspective of infinitesimal groupoids.
For , the infinitesimal singular simplicial complex is the simplicial smooth locus which in terms in degree is the space of -tuples of pairwise infinitesimal neighbour points in
and whose face and degeneracy maps are as for the finite singular simplicial complex.
More explicitly, in terms of the spaces from def. we may identify
where in degree a generalized element of is thought of as a base point and infinitesimal paths starting at that basepoint
The dual cosimplicial algebra is read off from this,
For instance for we have and .
The object is not objectwise a Kan complex: in general the composite of two first order neighbours produces a second order infinitesimal neighbour. Its Kan fibrant replacement may be thought of as the infinitesikmal -groupoid whose morphisms are paths of a finite number of first order infinitesimal steps.
The image of under the embedding from def. is the simplicial smooth locus given by the infinitesimal singular simplicial complex
of .
Moreover, the intrinsic real cohomology of SynthDiff∞Grpd is the de Rham cohomology of
The first statement may be checked locally on any chart where it follows from prop. . Since the Chevalley-Eilenberg algebra of the tangent Lie algebroid is the de Rham complex
Let be a Lie group with Lie algebra . We describe how looks when regarded as a special case of an -Lie algebroid.
Write
for the delooping groupoid of , regarded as an an ∞-Lie groupoid modeled by a simplicial smooth space.
We claim that a morphism
from the tangent Lie algebroid of some CartSp is flat Lie-algebra valued form and how that can be used to find the Lie algebra as the infinitesimal sub--groupoid
inside .
Since is 2-coskeletal (being the nerve of a groupoid) a morphism is fixed already under its 2-truncation
It is clear that factors through the inclusion that sends the unique point of to the neutral element (by respect for the degeneracy maps). Then from that one finds that factors through the inclusion that sends the unique point of to . And evidently these two factorizations are universal, in that every other factorization will uniquely factor through these
The universal object found this way we claim is the Lie algebra in its incarnation as an infinitesimal -Lie groupoid
The normalized cochain complex of the cosimplicial alghebra of functions on this is isomorphic to the ordinary Chevalley-Eilenberg algebra of .
By the above discussion we have that for the subspace of those functions that are in the joint kernel of the co-degeneracy maps is naturally isomorpic to , so that we have a natural isomorphism of vector spaces
By the fact that everything is 2-coskeletal it suffices to check that the differential in first degree
is indeed the dual of the Lie bracket. But the product restricted along to the infinitesimal space linearizes in each of its arguments: for we have
Since the origin here corresponds to the neutral element of and since with one of its arguments the neutral element the operaton is the identity, and since the double derivative produces the Lie bracket (keeping in mind that in ), this is
Accordingly the alternating sum of co-face maps is
as it should be for the Chevalley-Eilenberg algebra of a Lie algebra.
The infinitesimal reasoning involved in this proof is discussed in (Kock, section 6.8).
The term “Lie -algebroid” or “-algebroid” as such is not as yet established in the literature, as most authors working with these objects think of them entirely in terms of dg-algebras or NQ-supermanifolds and either ignore the relation to Lie theory or take it more or less for granted.
Possibly the first explicit appearance of the idea of -Lie algebroids recognized in their full Lie theoretic meaning is
which uses “NQ-supermanifolds”. Of course, as this article also points out, in hindsight one finds that much of this is already implicit in the much older theory of Sullivan models in rational homotopy theory, which is concerned with modelling topological spaces by dg-algebras. That these spaces can be regarded as ∞-groupoids and as ∞-Lie groupoids in particular is clear in hindsight, but was possibly first explicitly realized in the above reference. See also Lie integration, rational homotopy theory in an (∞,1)-topos and function algebras on ∞-stacks.
The explicit term -Lie algebroid / -algebroid as such is due to
Urs Schreiber: On -Lie (2008) [pdf, Schreiber-InfinityLie.pdf]
Hisham Sati, Urs Schreiber, Jim Stasheff, Section A.1 of: Twisted Differential String and Fivebrane Structures, Communications in Mathematical Physics, Volume 315, Issue 1 (2012) pp 169-213 (arXiv:0910.4001, doi:10.1007/s00220-012-1510-3)
The term then appears in
The dual monoidal Dold-Kan correspondence is discussed in
The smooth spaces of infinitesimal simplices are considered in section 1.2 of
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