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Add effective topos #353

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@dschepler

An idea I've been toying around with is to see what it would take to add the effective topos to the database: https://ncatlab.org/nlab/show/effective+topos

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".)

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  1. dschepler commented on Sep 5, 2026

    @dschepler
    ContributorAuthor

    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

  2. ScriptRaccoon commented on Sep 5, 2026

    @ScriptRaccoon
    Owner

    Looks good! Let's add it.
    That example has been on my list for a while (because of MSE/649829).

  3. linked a pull request that will close this issueAdd the effective topos #362on Sep 8, 2026
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