Re: Dualities arising via pairs of schizophrenic objects

[David Roberts <david.roberts@adelaide.edu.au>]

Vaughan wrote:

> What's odd is that the term seems to be used precisely when it is
> mathematically inappropriate in the above sense (quite apart from
> medical or sensitivity issues).
>
> The real manifestation of the "same" entity is not with objects at all
> but with homsets, namely the homset C(D'(d'), d) in C and the homset
> C'(D(d), d') in C', which *do* have the same number of morphisms.
>
> If anything deserves the epithet in question it is that homset in each
> category.  The two homsets are in bijection, but their targets don't
> correspond, having only in common that they are the dualizers in the
> respective categories.

Yes, I always thought it odd that even when one wants to accept
category-theoretic
foundations (e.g. ETCS or similar), then suddenly something like this
comes along,
where people start saying there is a thing which is an object of two
different categories.
Such a property isn't even expressible in type theory-style
foundations, where even
elements of two different sets aren't comparable...

But since both of the categories in each pair Vaughan mentioned are
enriched over
Set, we *are* allowed to compare hom-sets, at least using some sort of roughly
canonical isomorphism. Using the formalism suggested (hom-sets), it
seems much easier
to set down a definition of these slippery objects.

And (more whimsically) regarding terminology: if someone wanted to use
the analogy
of a door, why not a window? We can have fenestral objects, by which
one can 'see'
from one category to another. And unfortunately, 'liminal' gives rise
to subliminal, which
might be a natural prefix extension mathematically but is even more
confusing than
the existing inaccurate term. :-)

David


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