Addendum to Elephant A4.5.8

Addendum to Elephant A4.5.8

[ptj]

Dear category-theorists,

Recently I happened to ask myself when, given a local operator j in a
topos, the j-dense subobject classifier J is injective. The answer
turns out to be equivalent to a bunch of conditions that I've known about
ever since 1973, when Jim Lambek asked me (at the first Aarhus Open House)
what could be said about those j's for which the associated sheaf functor
preserves Omega. Those conditions appeared in the Elephant as Proposition
A4.5.8 (I think, though I don't have my copy to hand, that they also
appeared as an exercise for the reader in `Topos Theory'), and injectivity
of J would certainly have been included among the conditions of A4.5.8
if I'd known about it at the time. The proof of equivalence is sketched
below; the purpose of this e-mail is to ask whether anyone knew it before
now, and if so whether it's written down anywhere.

Here's the proof: among the conditions of A4.5.8 is the condition that
the inclusion Omega_j >--> Omega is j-dense. If this holds, let
r: Omega --> J be its classifying map. To show that ri is the identity,
where i is the inclusion J >--> Omega, it's enough to show that

         1 >-----> Omega_j
         v            v
         |            |
         |            |
         v    i       v
         J >-----> Omega

is a pullback; but this is obvious, since the intersection of J and Omega_j
is a classifier for subobjects which are both closed and dense, i.e. for
isomorphisms. So J is a retract of Omega, and hence injective.

The converse is similar: if J is injective, let r: Omega --> J be a
retraction for the inclusion, and let A >-> Omega be the dense subobject
it classifies. We have a pullback square as before (but with A instead
of Omega_j in the top right corner); so 1 >--> A is j-closed, since i
is the j-closure of \top. So we have factored the generic mono
\top: 1 >--> Omega as a closed mono followed by a dense mono; hence every
mono in E has such a factorization, which is another of the conditions of
A4.5.8.

Peter Johnstone



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