Re: pushouts in toposes

[categories <cat-dist@mta.ca>]

Date: Sun, 22 Jun 1997 11:54:47 -0300
From: RJ Wood <rjwood@cs.dal.ca>

A belated, somewhat tangential comment, on the distributivity
condition 
A v /\Bi  =  /\(A v Bi)
that Peter mentioned in his posts. The subobject classifier,
Omega, satisfies this condition (internally) if and only if the
topos is boolean. See the proof of Theorem 10 in Constructive
Complete Distributivity II, Math Proc Cam Phil Soc, (1991) 110,
245-249, by Rosebrugh and Wood, which shows that if Omega^op
is Heyting then Omega is Boolean. This result was discovered
independently by Richard Squire in his thesis.

It has always struck me as somewhat surprising but in the Rosebrugh/
Wood proof it is an immediate consequence of the corollary of the
following result which I believe is due to Benabou and which seems
to be not well known:

LEMMA If f <= id:Omega--->Omega then f(u) = u/\f(true).

COROLLARY If f <= id:Omega--->Omega and f(true) = true then f = id.

(If Omega^op is Heyting, apply the corollary to --, where - is the
negation for Omega^op.)
RJ