Date: Mon, 19 Jul 1999 11:09:43 -0400 (EDT)
From: Nikita Danilov <nikitadanilov@yahoo.com>
Subject: categories: universal property of tangent bundle

Dear category people,

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.



Date: Wed, 21 Jul 1999 17:58:09 +0200
From: Andree Ehresmann <ehres@u-picardie.fr>
Subject: categories: Answer to Davilov

Davilov writes:

>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?

Several authors have been interested in this problem many years ago, that
has led to the study of the functors from Man to the category of vector
bundles. Such functors are completely characterized in

        Epstein, "Natural vector bundles", Lecture Notes in Math. 99,
Springer 1969,         p. 171-195.

>From his theorems, it results in particular that Ehresmann's first order
velocity functors Tn which associate to a manifold M the vector bundle of
the 1-jets from R^n to M (in particular T for n=1) are the only such product
preserving functors.

This result is generalized in an abstract setting to characterize
connections in:
        Bowshell, "Abstract velocity functors", Cahiers de Topologie et
Geom. Diff.         XII-1 (1971), 57-82.

Later on, related (or more general) problems are considered in:

        Palais & Terng, "Natural bundles have finte order", Topology 16
(1977),       271-277,
        Epstein & Thurston, "Transformation groups and natural bundles",
Proc.  London Math. Soc. 38 (1979), 219-236.
        Eck, "Product preserving functors on smooth manifolds, J. Pure &
App. Algebra 42 (1986), 133-140.
        Kolar, An abstract characterization of the jet spaces, Cahiers de
Topologie et Geom. Diff. XXXIV-2 (1993), 121-125.
        Dupovec & Kolar, On the jets of fibered manifold morphisms, Cahiers
de Topologie et Geom. Diff. XXXIV-2 (1993), 121-125.
        
        Hoping these old references can be of some interest
                                        Best regards


                                                Andree C. Ehresmann        



Date: Mon, 2 Aug 1999 11:22:56 +0100 (BST)
From: Kirill Mackenzie <K.Mackenzie@sheffield.ac.uk>
Subject: categories: Re: universal property of tangent bundle


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.
> 
> 
> 
>