Morita Equivalence (of Lie groupoids) - work in progress edit
In differential geometry, Morita equivalence is an equivalence relation between Lie groupoids that preserves many properties. Two Lie groupoids are Morita equivalent if and only if there is a principal bibundles between them or, alternatively, if their categories of principal bundles are equivalent.
It was introduced by Ping Xu in 1991,[1] inspired by the similar notions of Morita equivalence for rings in abstract algebra,[2] and for C* algebras,[3][4] and building up on Morita equivalence between topological groupoids.[5]
Definition edit
There are several equivalent definitions.
Definition with bibundles edit
Two Lie groupoids are Morita equivalent if there is a manifold together with two surjective submersions and together with a left -action and a right -action,
- is a principal -bundle;
- is a principal -bundle;
- the two actions commute.
One says also that is a principal bibundle.
Definition with groupoid morphisms edit
A Lie groupoid map between two Lie groupoids and is called Morita if it is
- fully faithful: blabla
- essentially surjective: blabla
Sometimes called a weak equivalence or essential equivalence[6]
Two Lie groupoids and are Morita equivalent if and only if there exists a third Lie groupoid together with two Morita maps from G to K and from H to K.
Equivalence between the definitions edit
A generalised equivalence is blabla
Then there is a 1-1 correspondence between Morita maps and generalised equivalence?[7]
Properties and objects preserved by Morita equivalence edit
The following properties are Morita invariant
- being proper (i.e. the source map is proper)
- being Hausdorff
- being transitive (actually much more holds; see below)
- being ?
Other properties, i.e. being étale (see below), are not Morita invariant.
Moreover, a Morita equivalence between and preserves their transverse geometry, i.e. it induces:
- a homeomorphism between the orbit spaces and , where the orbit at corresponds to that at if blabla;
- an isomorphism between the isotropy groups at corresponding points and ;
- an isomorphism between the normal representations of the isotropy groups at corresponding points and .
Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.[8]
Examples edit
- Isomorphic Lie groupoids are trivially Morita equivalent.
- Two Lie groups are Morita equivalent if and only if they are isomorphic as Lie groups.
- Two unit groupoids are Morita equivalent if and only if the base manifolds are diffeomorphic.
- Any transitive Lie groupoid is Morita equivalent to its isotropy groups.
- Given a Lie groupoid and a surjective submersion , the pullback groupoid is Morita equivalent to .
- Given a free and proper Lie group action of on (therefore the quotient is a manifold), the action groupoid is Morita equivalent to the unit groupoid .
- A Lie groupoid is Morita equivalent to an étale groupoid if and only if all isotropy groups of are discrete.[9]
A concrete instance of the last example goes as follows. Let M be a smooth manifold and an open cover of . Its Čech groupoid is defined by the disjoint unions and , where . The source and target map are defined as the embeddings and , and the multiplication is the obvious one if we read the as subsets of M (compatible points in and actually are the same in and also lie in ). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion , of the unit groupoid . As such, Čech groupoids associated to different open covers of are Morita equivalent.
Relations with differentiable stacks edit
Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a smooth stack as a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence: an example is the Lie groupoid cohomology.
Since the notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks. Other classes of examples include orbifolds, which are (equivalence classes of) proper étale Lie groupoids, and orbit spaces of foliations.
Symplectic Morita equivalence and Poisson dual pairs edit
A symplectic Morita equivalence between symplectic groupoids is blabla[1]
This is the same thing as a Poisson dual pair between Poisson manifolds, i.e. a couple of
Extra structure, e.g. diffeological Morita equivalence
Morita equivalence is an equivalence relation only between integrable Poisson manifolds[10]
References edit
- ^ a b Xu, Ping (1991). Dazord, Pierre; Weinstein, Alan (eds.). "Morita Equivalent Symplectic Groupoids". Symplectic Geometry, Groupoids, and Integrable Systems. New York, NY: Springer US: 291–311. doi:10.1007/978-1-4613-9719-9_20. ISBN 978-1-4613-9719-9.
- ^ Morita, Kiiti (1958). "Duality for Modules and its Applications to the Theory of Rings with Minimum Condition". Science Reports of the Tokyo Kyoiku Daigaku, Section A. 6 (150): 83–142. ISSN 0371-3539.
- ^ Rieffel, Marc A (1974-06-01). "Induced representations of C∗-algebras". Advances in Mathematics. 13 (2): 176–257. doi:10.1016/0001-8708(74)90068-1. ISSN 0001-8708.
- ^ Rieffel, Marc A. (1974-08-01). "Morita equivalence for c∗-algebras and w∗-algebras". Journal of Pure and Applied Algebra. 5 (1): 51–96. doi:10.1016/0022-4049(74)90003-6. ISSN 0022-4049.
- ^ Muhly, Paul S.; Renault, Jean N.; Williams, Dana P. (1987). "Equivalence and isomorphism for groupoid C*-algebras". Journal of Operator Theory. 17 (1): 3–22. ISSN 0379-4024.
- ^ Moerdijk, I.; Mrcun, J. (2003). Introduction to Foliations and Lie Groupoids. Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511615450. ISBN 978-0-521-83197-0.
- ^ Hoyo, Matias L. del (2013-08-02). "Lie groupoids and their orbispaces". Portugaliae Mathematica. 70 (2): 161–209. doi:10.4171/pm/1930. ISSN 0032-5155.
- ^ Crainic, Marius (2003-12-31). "Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes". Commentarii Mathematici Helvetici. 78 (4): 681–721. doi:10.1007/s00014-001-0766-9. ISSN 0010-2571.
- ^ Crainic, Marius; Moerdijk, Ieke (2001-02-10). "Foliation Groupoids and Their Cyclic Homology". Advances in Mathematics. 157 (2): 177–197. doi:10.1006/aima.2000.1944. ISSN 0001-8708.
- ^ Xu, Ping (1991-12-01). "Morita equivalence of Poisson manifolds". Communications in Mathematical Physics. 142 (3): 493–509. doi:10.1007/BF02099098. ISSN 1432-0916.