Re: Dualities arising via pairs of schizophrenic objects

[Vaughan Pratt <pratt@cs.stanford.edu>]

There's something odd about when this term is used (under whatever
name).  The implication is that it's two manifestations of the "same"
object, one in each of a dual pair C, C' of categories, d in C and d' in C'.

When C is equivalent to C' (self-duality), as with FinVect, CSLat,
Chu(V,k), etc., this point of view seems mathematically appropriate.

But when not, as with the Stone duality of Boolean algebras, the duality
D: C --> C' doesn't even carry d to d', and moreover C(d,d) and
C'(d',d') typically don't even have the same number of endomorphisms.
Typically d and d' cogenerate and their respective images D(d) and
D'(d') generate.

In this case it would seem preferable to call D(d) the counterpart (up
to isomorphism) in C' of d in C, and conversely for D'(d') (writing D'
for the adjoint to D making it a duality).

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.

(I made the same point in the previous flurry on this topic a year or so
ago, hopefully more clearly this time around.)

Vaughan


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