2 Aug 12:22 1999

## Re: universal property of tangent bundle


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.
>
>
>
>

`

Gmane