∞-Lie theory (higher geometry)
Formal Lie groupoids
A formal smooth -groupoid is an ∞-groupoid equipped with a cohesive structure that subsumes that of smooth ∞-groupoids as well as of infinitesimal -groupoids – ∞-Lie algebroids, hence equipped with “differential cohesion”.
In the cohesive (∞,1)-topos of formal smooth -groupoids the canonical fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos factors through a version relative to Smooth∞Grpd: the infinitesimal path ∞-functor . In traditional terms this is the object modeled by the tangent Lie algebroid and the de Rham space of . The quasicoherent ∞-stacks on are D-modules on .
We consider (∞,1)-sheaves over a “twisted semidirect product” site or (∞,1)-site of
we consider the 1-site, then in
we consider the -site.
This appears as ([Kock 86, (5.1)]).
We say the (∞,1)-topos of formal smooth -groupoids is the (∞,1)-category of (∞,1)-sheaves
We now generalize the 1-category of infinitesimally thickened points to the (∞,1)-category of “derived infinitesimally thickened points”, the formal dual of “small commutative -algebras” from (Hinich, Lurie).
is a dense sub-site of .
Write for the canonical embedding.
We discuss the realization of the general abstract structures in a cohesive (∞,1)-topos in .
Since by the above discussion is strongly -connected relative to Smooth∞Grpd all of these structures that depend only on -connectedness (and not on locality) acquire a relative version.
-Lie algebroids and deformation theory
This subsection is at
Paths and geometric Postnikov towers
We discuss the intrinsic infinitesimal path adjunction realized in .
For we have that
is the reduced smooth locus: the formal dual of the smooth algebra obtained by quotienting out all nilpotent elements in the smooth algebra .
By the model category presentation of of the above proof and using that every representable is cofibrant and fibrant in the local projective model structure on simplicial presheaves we have
(using that is a full and faithful functor).
For a smooth locus, we have that is the corresponding de Rham space, the object in which all infinitesimal neighbour points in are equivalent, characterized by
By the -adjunction relation
Cohomology and principal -bundles
We discuss the intrinsic cohomology in a cohesive (∞,1)-topos realized in .
The canonical line object of the Lawvere theory CartSp is the real line, regarded as an object of the Cahiers topos, and hence of
We shall write also for the underlying additive group
regarded as an abelian ∞-group object in . For write for its -fold delooping.
For and write
for the cohomology group of with coefficients in the canonical line object in degree .
Let be the projective model structure on cosimplicial smooth algebras and let be the prolonged external Yoneda embedding.
This constitutes the right adjoint of a Quillen adjunction
Restricted to simplicial formal smooth manifolds along
the right derived functor of is a full and faithful (∞,1)-functor that factors through the cohomology localization and thus identifies a full reflective sub-(∞,1)-category
The intrinsic -cohomology of any object is computed by the ordinary cochain cohomology of the Moore cochain complex underlying the cosimplicial abelian group of the image under the left derived functor under the Dold-Kan correspondence:
First a remark on the sites. By the above proposition is equivalent to the hypercomplete (∞,1)-topos over formal smooth manifolds. This is presented by the left Bousfield localization of at the ∞-connected morphisms. But a fibrant object in that is also n-truncated for is also fibrant in the hyperlocalization (only for the untruncated object there is an additional condition). Therefore the right Quillen functor claimed above indeed lands in truncated objects in .
The proof of the above statements is given in (Stel), following (Toën). Some details are spelled out at function algebras on ∞-stacks.
Cohomology of Lie groups
Let be a Lie group.
Then the intrinsic group cohomology in Smooth∞Grpd and in of with coefficients in coincides with Segal’s refined Lie group cohomology (Segal, Brylinski).
For the full proof please see here, section 3.4 for the moment.
For a compact Lie group we have for all that
This follows from applying the above result to the fiber sequence induced by the sequence .
This means that the intrinsic cohomology of compact Lie groups in Smooth∞Grpd and coincides for these coefficients with the Segal-Blanc-Brylinski refined Lie group cohomology (Brylinski).
Cohomology of -Lie algebroids
Let be an L-∞ algebroid. Then its intrinsic real cohomology in
coincides with its ordinary L-∞ algebroid cohomology: the cochain cohomology of its Chevalley-Eilenberg algebra
By the above propoposition we have that
By this lemma this is
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 this 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
It follows that a degree- -cocycle on is presented by a morphism
where on the right we have the -algebroid whose -algebra is concentrated in degree . Notice that if is the delooping of an - algebra this is equivalently a morphism of -algebras
de Rham theorem
We consider the de Rham theorem in , with respect to the infinitesimal de Rham cohomology
For all The canonical morphism
is an equivalence.
This means that for all the infinitesimal de Rham cohomology coincides with the ordinary real cohomology of the geometric realization of
Since all representables are formally smooth, we have a colimit
In the presentation over the site we have that
Therefore a morphism is equivalently a morphism such that for all that coincide on we have that all the composites
are equals. If is the point, then is necessarily constant. If is not the point, there is one nontrivial tangent vector in . The composite produces the corresponding tangent vector in . But all these tangent vectors must be equal. Hence must be constant.
This kind of argument is indicated in (Simpson-Teleman, prop. 3.4).
Let SmoothMfd and write for the tangent Lie algebroid regarded as a simplicial object (see L-infinity algebroid for the details).
Then there is a morphism which is an equivalence in -cohomology.
Formally étale morphisms and cohesive étale -groupoids
We discuss formally étale morphisms and étale objects with respect to the cohesive infinitesimal neighbourhood Smooth∞Grpd .
Let be a degreewise smooth paracompact simplicial manifold, canonically regarded as an object of Smooth∞Grpd.
Then in is presented by the same simplicial manifold.
First consider an ordinary smooth paracompact manifold . It admits a good open cover and the corresponding Cech nerve is a cofibrant resolution of . Therefore the -functor is computed on by evaluating the corresponding simplicial functor (of which it is the derived functor) on .
Since the simplicial functor
is a left adjoint (indeed a left Quillen functor) it preserves the coproducts and coend that the Cech nerve is built from:
Here we used that, by assumption on a good open cover, all the are Cartesian spaces, and that coincides on representables with the inclusion .
Let now be a general simplicial manifold. Assume that in each degree there is a good open cover such that these fit into a simplicial system giving a bisimplicial Cech nerve such that there is a morphism of bisimplicial presheaves
with regarded as simplicially constant in one direction. Each is a cofibrant resolution.
We claim that the coend
is a cofibrant resolution of , where is the fat simplex. From this the proposition follows by again applying the left Quillen functor degreewise and pulling it through all the colimits.
This remaining claim follows from the same argument as used above in prop. 9.
A morphism in , is a formally étale morphism with respect to the infinitesimal cohesion precisely if for all infinitesimally thickened points the diagram
is an -pullback under .
We spell out the case for smooth manifolds. Here we need to to show that
is a pullback in precisely if is a local diffeomorphism. This is a pullback precisely if for all the diagram of sets of plots
is a pullback. Using, by the discussion at ∞-cohesive site, that preserves colimits and restricts to on representables, and using that , this is equivalently the diagram
where the vertical morphisms are given by restriction along the inclusion .
For one direction of the claim it is sufficient to consider this situation for the point and the first order infinitesimal interval. Then is the underlying set of points of the manifold and is the set of tangent vectors, the set of points of the tangent bundle . The pullback
is therefore the set of pairs consisting of a point and a tangent vector . This set is in fiberwise bijection with precisely if for each there is a bijection , hence precisely if is a local diffeomorphism. Therefore being a local diffeo is necessary for being formally étale with respect to the given notion of infinitesimal cohesion.
To see that this is also sufficient notice that this is evident for the case that is in fact a monomorphism, and that since smooth functions are characterized locally, we can reduce the general case to that case.
Let be the inclusion of the smooth manifold of objects. This is an effective epimorphism. It remains to show that this is formally étale with respect to the given cohesive neighbourhood.
By the discussion at (∞,1)-pullback we may compute the characteristic -pullback by an ordinary pullback of a fibration of simplicial presheaves that presents .
By the factorization lemma such is given by
By inspection one see that this morphis is
By prop. 14 both of these need to be étale maps in the ordinary sense. By definition, this is the case precisely if is an étale groupoid.
Formally smooth / formally unramified morphisms
As a direct consequence of prop. 14 we have the following
As in the proof of prop. 14 we find that the pullback is over the infinitesimal interval isomorphic to
and the canonical morphism from into this pullback is
We indicate how to formalize Lie differentiation in the context of formal smooth -groupoids.
be the canonical inclusion. By (Lurie, theorem 0.0.13, remark 0.0.15, also Pridham 07) we have a full inclusion
on those objects whose space of global sections is contractible and which are infinitesimally cohesive (for a somewhat different notion of “infinitesimal cohesion”, beware the terminology). Consider then the -functor
which sends a pointed connected formal smooth -groupoid to the -presheaf of pointed morphisms
By assumption that is connected (and we need to assume that it is geometric, which will gives infinitesimal cohesion by the Artin-Lurie representability theorem) this factors as
The resulting -functor
is Lie differentiation.
For differentiation of smooth groupoids with atlas to L-infinity algebroids this happens under
The site FormalCartSp is discussed in section 5 of
- Anders Kock, Convenient vector spaces embed into the Cahiers topos , Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 1 (1986), p. 3-17 (numdam).
For more on this see at Cahiers topos.
The notion of formally étale maps as obtained here from the general abstract definition in differential cohesion coincided on 0-truncated objects with that defined on p. 82 of
The infinitesimal path ∞-groupoid adjunction and the de Rham theorem for -stacks is discussed at the level of homotopy categories in section 3 of
The -topos is discussed in section 4.4 of
The cohomology localization of and the infinitesimal singular simplicial complex as a presentation for infinitesimal paths objects in is discussed in
- Herman Stel, -Stacks and their function algebras – with applications to -Lie theory , master thesis (2010) (web)
The discussion of the cohomology localization of follows that in another context in
The construction of the infinitesimal path object has been amplified and discussed by Anders Kock under the name combinatorial differential forms, for instance in
The discussion that the normalized cosimplicial algebra of functions on the infinitesimal singular simplicial complex is the de Rham complex is further discussed in
The results on differentiable Lie group cohomology used above is in
- P. Blanc, Cohomologie différentiable et changement de groupes Astérisque, vol. 124-125 (1985), pp. 113-130.
- Graeme Segal, Cohomology of topological groups , Symposia Mathematica, Vol IV (1970) (1986?) p. 377
The -site of derived infinitesimal points is discussed in