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categories: gluing, lifting and partial maps

I have long regarded it as "well known" that
    the partial map classifier for topological spaces or locales
    by "partial" I mean a continuous function defined on an open subset
    the Artin gluing, Freyd cover or scone (Sierpinski cone).

Can anybody point me to a published proof of this, or even tell me who
first proved it?  

The same construction, with frames replaced by the categories of contexts
and substitutions (a.k.a. classifying categories) for theories in other
fragments of logic,  has also been used with spectacular results to prove
consistency, strong normalisation, etc.   I know of plenty of work on
that application itself,   but I wonder whether anybody has investigated
the connection between these two applications of the construction.


PS Thanks to everyone who wrote to me about 1970s calculators.  I will be
writing back and summarising the responses for "categories" after the end
of term.   When the students have sat my exam paper (sometime in May)
I will also post to "categories" the actual question that I composed.