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Yesterday I was talking to somebody from the Haskell community. Late in the night we found ourselves discussing possible topoi.

Lets order topoi (up to equivalence, ...) by number of objects/morphisms

Questions:

  • Are there topoi "in between" FinSet & Set? (There are more Set-like topos, limiting by cardinality of the set, see nlab.) Are there more toposes "in between" FinSet & Set?
  • Is each topos with countable many objects automatically equivalent to FinSet? (take the minimum over all equivalent topoi ...)

Edits:

  • added "definitions" of Set & FinSet (via linking nlab)
  • I noted that the nlab entry already answers the first questions, so I made the question more precise
  • added a rough definition of smallnes
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    $\begingroup$ Depends on what you mean by $Set$. But if you mean a category whose objects are 'sets' in the sense of modeling ZFC, then you should know that the replacement axiom allows for sets like $\bigcup_n P^n(\mathbb{N})$ which are inaccessible from the topos axioms alone. (I may make this an answer later if someone else doesn't do it first.) $\endgroup$
    – Todd Trimble
    Commented Sep 21, 2018 at 20:58
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    $\begingroup$ You have a lot of toposes similar to finite sets. Like the category of finite presheaves on a finite category. $\endgroup$
    – Simon Henry
    Commented Sep 21, 2018 at 21:01
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    $\begingroup$ What do you mean by countably many? What about the toposes of sets of a countable transitive model of ZFC in ZFC+ε? Not to mention FinSet is not absolute, since one can have non-standard models... $\endgroup$
    – David Roberts ♦
    Commented Sep 22, 2018 at 0:58
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    $\begingroup$ Also, 'size' is not a meaningful option, since FinSet coming from ZFC has a proper class of objects. Perhaps better are structural notions, eg the free topos, which is initial for logical functors. $\endgroup$
    – David Roberts ♦
    Commented Sep 22, 2018 at 1:00
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    $\begingroup$ If $\lambda$ is an inaccessible cardinal the category $\text{Set}_{\lambda}$ of sets of cardinality less then $\lambda$ and functions between them is an elementary topos. Observe that FinSet is $\text{Set}_{\omega}$. $\endgroup$
    – Ivan Di Liberti
    Commented Sep 22, 2018 at 12:03

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Here are some examples, that should show you that there is a lot of countable toposes and lot of things in between finite sets and sets, too much to actually hope to list or classifies.

  • First as I said a lots of usual construction of Grothendieck toposes have a finistic version, which produces elementary toposes (in fact, bounded toposes over FinSet most of the time). For example given a finite category $C$ the category of finite presheaves on $C$ is a topos, in fact you can even take a topology and look at finite sheaves. You can have a look to chapter 5.2 in the third volume of Borceux's handbook of categorical algebra for some details on these.

  • If you have an inaccesible cardinal $\lambda$, then the category of sets of cardinality less than $\lambda$ is an essentially small elementary topos with a natural number objects.

  • As mentioned by Todd Trimble, elementary toposes are a bit weaker than ZF because they don't allow for unbounded replacement. concretely it means that you can construct essentially small full subcategory of the category of sets that are elementary topos without invoking an inaccessible cardinal. If you take any set $X$, the full subcategory of sets that appears as element of $\mathcal{P}^{n}(X)= \mathcal{P} (\mathcal{P}( \dots \mathcal{P}(X)) \dots )$ for some $n$ ($\mathcal{P}(X)$ being the set of subsets of $X$) is an elementary topos. If $X$ is finite, this is justs the category of finite sets, but if $X$ is infinite it will have a natural number object, but it will be uncountable. This produces a lot of concrete example of full subcategory of the category of sets that are small toposes.

  • Every time you have a model of ZF, its "category of sets" is a topos. Because of Löwenheim–Skolem there exists countable model of ZF, so this produces example of countable toposes admiting natural number objects.

  • There exists a "free topos" which admit a unique logical morphisms to any topos, and a "free topos with a natural number object" which admit a unique logical morphisms preserving the natural number object to any topos with a natural number object, see there as well as the reference given there for more details. It admit a syntactical description which shows that it is countable (informally: its objects can be describe as certain formula involving the operation that you have in a topos, like taking power objects and finite limits, and there is only a countable number of these, same for the maps).

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    $\begingroup$ Instead of applying Löwenheim-Skolem to ZF, you could apply it directly to topos theory: Every small topos has a countable elementary submodel. (And you can omit "small" if you work in a theory where satisfaction relations can be defined for proper-class models.) So, as far as first-order expressible (in the language of categories) properties are concerned, countable topoi can do everything that uncountable ones can. $\endgroup$
    – Andreas Blass
    Commented Sep 22, 2018 at 12:32

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