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For example, I think this is an example of an elementary topos with natural numbers object which does not have countable copowers (and in fact the natural numbers object is not a countable copower of the terminal object).
I'm not sure how well I'd be able to assign other properties, though. I think I've just barely gotten to the point where I can grasp the basic definitions of what an object is and what a morphism is, and I can start to reason about the category.
(Anyway, my immediate plans would be to finish off the quasitopos project by adding a property for "has effective regular congruences" - the grouping is "has effective (regular congruences)" rather than "has (effective and regular) congruences". Then add the Grothendieck quasitopos property, and then maybe add the category SepPsh(X) for X a topological space. And then possibly a few of the other examples we've discussed, such as "category of sets equipped with a binary relation", "category of subsequential spaces", "category of pseudotopological spaces" (that one I think would be an example of a complete and cocomplete quasitopos which is not a Grothendieck quasitopos because it doesn't have a small extremal generating set), "category of simple graphs".)
along with the general result that an elementary topos with countable powers has countable copowers (since the power object functor is monadic and therefore creates limits)...
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dschepler commentedon Sep 5, 2026
Actually, I just did a quick experiment. It turned out that with these easy or well-known assignments:
is locally small
is an elementary topos
has a natural numbers object
is semi-strongly connected
is not skeletal
does not have countable copowers
is not essentially small
is not locally finite
along with the general result that an elementary topos with countable powers has countable copowers (since the power object functor is monadic and therefore creates limits)...
Then there were just 14 unknown properties, mostly related to accessibility or coaccessibility:
is accessible
is coaccessible
has a cogenerating set
has a cogenerator
is concretizable
has an extremal cogenerating set
has an extremal cogenerator
has an extremal generating set
has an extremal generator
has a generating set
has a generator
is ℵ₁-accessible
has ℵ₁-cofiltered limits
has ℵ₁-filtered colimits
ScriptRaccoon commentedon Sep 5, 2026
Looks good! Let's add it.
That example has been on my list for a while (because of MSE/649829).