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Stream: learning: questions

Topic: co(ntra)variant functors and tensors

Peiyuan Zhu (Dec 30 2021 at 03:47):

Does covariant functor has anything to do with covariant derivative or covariant tensor?

John Baez (Dec 30 2021 at 05:14):

Yes.

Mike Shulman (Dec 30 2021 at 19:09):

It's backwards, though, isn't it? Contravariant tensors, like tangent vectors, push forwards along a smooth map of manifolds, i.e. they behave covariantly, while covariant tensors, like cotangent vectors, pull backwards, behaving contravariantly.

Mike Shulman (Dec 30 2021 at 19:09):

Possibly this is because originally the words were applied to the components of those tensors in some basis, rather than to the tensors themselves?

Peiyuan Zhu (Dec 31 2021 at 21:48):

Does category theory has anything to do with numerical methods? I remember watching Andrej Bauer's lecture and he mentioned HoTT proofs reveals numerical stability of algorithms, although I wasn't sure exactly what he meant by that.

Peiyuan Zhu (Dec 31 2021 at 21:54):

How to understand Ilya Prigogine's contribution to quantum mechanics? Why is the Liouville space important?

John Baez (Dec 31 2021 at 21:55):

Mike wrote:

Contravariant tensors, like tangent vectors, push forwards along a smooth map of manifolds, i.e. they behave covariantly, while covariant tensors, like cotangent vectors, pull backwards, behaving contravariantly.

That's old-fashioned terminology. These days the terminology is changing. A lot of people in differential geometry call tangent vectors covariant and cotangent vectors contrvariant, and I certainly did in the book I wrote.

John Baez (Dec 31 2021 at 21:57):

In case anyone is wondering, part of the confusion arises from the fact that we often write a tangent vector as something like

$v = v^i \partial_i$

where $v$ is a tangent vector, $\partial_i$ is a basis of tangent vectors (coming from a coordinate system), and $v^i$ are a bunch of numbers, and we sum over $i$.

John Baez (Dec 31 2021 at 22:00):

If we keep $v$ the same and change the basis of tangent vectors $\partial_i$, the components $v^i$ need to change "in the opposite way" to compensate. This led mathematicians to say the numbers $v^i$ transform "contravariantly", and then go further (bad idea) and say "tangent vectors are contravariant".

John Baez (Dec 31 2021 at 22:00):

Mike was talking about something else (yet related), namely how $v$ itself changes under a map between manifolds.

John Baez (Dec 31 2021 at 22:01):

Anyway, the modern way to understand these issues - which can be incredibly confusing, yet are ultimately quite simple - definitely involves covariant and contravariant functors.