Dear David,
[...]
> For this reason, I don't think CategoryWithBasis (or more
> generally CategoryWith[Free]Generators) is a natural
> category -- unless the generators are structures intended
> to be preserved, as is the case with a category of pointed
> sets.
As Nicolas already explained, FreeModuleWithBasis (and all their friend) is
*not* an (abstract) category in the mathematical sense of the term. It is the
category FreeModule + a little assumption on how the object of the category
are implemented (The object are stored as expanded on a particular
basis). Sorry to use this kind of argument but we spend two year designing
this stuff in MuPAD and six more using it in a *very* large scale. I'm more
than convinced that this is a good design choice...
Here we are not just doing mathematics. We are speaking implementation and
generic algorithms. It is not because you know that RR is a free QQ-module
that you know an algorithm to compute the QQ-rank of a bunch of real (think
about [sqrt(i) for i in range(100)]... However if you give me a basis and a
way to expand any real on this basis, I can implement easily gaussian
elimination...
The goal of the category theory is to provide generic reasoning and
constructions. Here we have to be a little more concrete, because we want to
provide generic implementations.
Cheers,
Florent
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