Re: Not invariant but good

[Thomas Streicher <streicher@mathematik.tu-darmstadt.de>]

Dear Michael,

> Does this clarify what I meant?

Yes, indeed that has been very helpful. 

> Perhaps "not well-behaved" was a poor choice of words since it implies
> a value judgement, but I think it is correct to say that in the
> absence of choice, category theory becomes very unfamiliar unless we
> replace functors with anafunctors.  For instance, if we insist on
> using only functors, then a category with finite products does not
> necessarily become a monoidal category, as there is no "product"
> functor from A×A to A.  Also, without choice the 2-category Cat using
> functors is not even a regular 2-category, let alone a 2-topos.  Since
> the defining characteristics of Set include that it is a well-pointed
> 1-topos, it seems unlikely to me that one will be able to get much of
> anywhere with a version of Cat that is not a 2-topos.

Well, but then we can work with categories with a chosen structure, e.g.
chosen products; that's what is recommended in the Elephant and it looks to
me as close to practice; it's very unlikely to have a nonconstructive proof
of existence of products.

Thomas



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