2 Aug 12:22 1999

## Re: universal property of tangent bundle

Kirill Mackenzie <K.Mackenzie <at> sheffield.ac.uk>

1999-08-02 10:22:56 GMT

1999-08-02 10:22:56 GMT

In addition to Madame Ehresmann's references, there is in Spivak's Comprehensive Introduction... an abstract characterization of the tangent bundle ( removed from the main text in the second edition `due to the pressure of public distaste') Kirill Mackenzie > > Given an object M in the ``normal'' category of finitely dimensional > smooth manifolds Man (not in SDG sense), what it the universal property > of the tangent bundle TM? > > So far, I found only the following: > > For every manifold M there is a functor F:I -> Man0, where Man0 is > category of open areas in R^n and smooth mapping, such that M=Colim F, > F corresponding to the atlas on M and M is represented as a result of > gluing instances of R^n in the atlas. This functor can be trivially > modified (by multiplying its values on objects on R^n and modifying > morphisms appropriately) to get functor TF:I -> Man0, such that > TM=Colim TF. > > But this doesn't seem satisfactory because: > > 1. Construction of TF follows one particular construction of TM as a > set of triples (x,(U,f),h) where x \in U, (U,f) is in atlas and h \in > R^n with appropriate points identified. > > 2. I hope there should be universal construction with \pi: TM -> M as > universal arrow. > > 3. As tangent bundle is so ubiquitous there should be nice universal > property for it. > > With regards, > N. Danilov. > > > >