From: Nikita Danilov <nikitadanilov@yahoo.com>
To: categories@mta.ca
Subject: universal property of tangent bundle
Date: Mon, 19 Jul 1999 11:09:43 -0400 (EDT) [thread overview]
Message-ID: <19990719150943.3947.rocketmail@web105.yahoomail.com> (raw)
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.
next reply other threads:[~1999-07-19 15:09 UTC|newest]
Thread overview: 2+ messages / expand[flat|nested] mbox.gz Atom feed top
1999-07-19 15:09 Nikita Danilov [this message]
1999-08-02 10:22 Kirill Mackenzie
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