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Stream: learning: questions

Topic: Basic question on topoi (toposes?).

Keith Elliott Peterson (Jan 30 2024 at 18:08):

There's the topoi (toposes?) $Set$, $Set_{fin}$, $Ord$, and $Ord_{fin}$, are there any topoi (toposes?) inbetween these two extremes? Is there a countable topos of sets and/or ordinals? It doesn't seem too out of place that every ordinal $<\omega_1$ is closed under the required operations needed for a topos, but I could also be missing some subtler points (heh).

Reid Barton (Jan 30 2024 at 18:09):

What kind of topoi (toposes?) are you talking about, and what is the category Ord?

Kevin Arlin (Jan 30 2024 at 18:25):

If $\kappa$ is any strongly inaccessible cardinal then the category of sets smaller than $\kappa$ is closed in $\mathsf{Set}$ under all the topos operations. But whether this gives more examples is independent of ZFC. $\mathsf{Set}_{\mathrm{fin}}$ itself is equivalent to a countable topos.

Kevin Arlin (Jan 30 2024 at 18:26):

Along with Reid I'm a little unclear on what you're saying/thinking about ordinals here but note that the powerset of a countable set is uncountable, which makes it hard to see how the thing that doesn't seem too out of place for you could be interpreted as something true.

Mike Shulman (Jan 30 2024 at 19:09):

Actually it's enough for $\kappa$ to be a strong limit cardinal; the (elementary) topos axioms don't contain an analogue of replacement. So even in ZFC there are lots of elementary toposes in between $\rm FinSet$ and $\rm Set$.

Mike Shulman (Jan 30 2024 at 19:10):

If by $\rm Ord$ you mean the well-ordered set of ordinals considered as a poset and hence a category, then that is not a topos. If you mean the category of ordinals and all functions between them, then that category is (assuming the axiom of choice) equivalent to $\rm Set$.

Kevin Arlin (Jan 30 2024 at 21:15):

Oh right, that's the condition for the $\kappa$-sized sets to be a model of ZFC, not just a topos.

Mike Shulman (Jan 30 2024 at 23:11):

Not quite that either. Inaccessibility of $\kappa$ implies that the set $V_\kappa$ of sets of rank $<\kappa$ satisfies the "second-order replacement axiom", which is quite a bit stronger than the first-order replacement axiom schema of ZFC. In fact, if I remember correctly, then if $\kappa$ is inaccessible, then the set of cardinal numbers $\lambda <\kappa$ such that $V_\lambda$ is a model of ZFC has cardinality $\kappa$.

Kevin Arlin (Jan 31 2024 at 00:16):

Ah, I see, thanks~

Keith Elliott Peterson (Jan 31 2024 at 02:26):

Mike Shulman said:

If by $\rm Ord$ you mean the well-ordered set of ordinals considered as a poset and hence a category, then that is not a topos. If you mean the category of ordinals and all functions between them, then that category is (assuming the axiom of choice) equivalent to $\rm Set$.

The latter. $Ord$ as a skeletal category.

Keith Elliott Peterson (Jan 31 2024 at 02:30):

I'm just curious if there is something like a topos of countable ordinals since if I do recall, countable ordinals are closed under limits, colimits, and exponentiation, much like finite ordinals/sets.

Mike Shulman (Jan 31 2024 at 04:34):

No, countable sets are not closed under exponentiation: $2^\omega$ is the power set of the naturals, which is uncountable. (Countable ordinals are, I think, closed under ordinal exponentiation, but that's not the same as set/cardinal exponentation.)

Keith Elliott Peterson (Jan 31 2024 at 05:17):

Mike Shulman said:

No, countable sets are not closed under exponentiation: $2^\omega$ is the power set of the naturals, which is uncountable. (Countable ordinals are, I think, closed under ordinal exponentiation, but that's not the same as set/cardinal exponentation.)

Why does ordinal exponentiation not count, if I may ask?

Mike Shulman (Jan 31 2024 at 05:19):

Ordinal exponentiation is just defined differently than cardinal exponentiation. Cardinal exponentiation is defined to be the cardinal of the function-set, which is the internal-hom in the category of sets. Ordinal exponentiation is defined to be "continuous" in the exponent, so for instance ordinal $2^\omega = \lim_{n<\omega} 2^n = \omega$.

Keith Elliott Peterson (Jan 31 2024 at 10:09):

Mike Shulman said:

Ordinal exponentiation is just defined differently than cardinal exponentiation. Cardinal exponentiation is defined to be the cardinal of the function-set, which is the internal-hom in the category of sets. Ordinal exponentiation is defined to be "continuous" in the exponent, so for instance ordinal $2^\omega = \lim_{n<\omega} 2^n = \omega$.

I assume this means ordinal exponentiation does not satisfy something like the hom-product adjunction, or something else required of topoi.

David Michael Roberts (Jan 31 2024 at 10:30):

What are the maps of ordinal you want? And what is the subobject classifier?

David Michael Roberts (Jan 31 2024 at 10:31):

I mean what would the subobject classifier have to be, in a putative topos of countable ordinal?

Mike Shulman (Jan 31 2024 at 15:27):

Right, cardinal exponentiation satisfies the hom-product adjunction, so ordinal exponentiation doesn't. (Assuming you take all functions between ordinals to make your category, as you suggested earlier. Although note this is not skeletal either, since for instance $\omega \cong \omega+1$.)

John Baez (Jan 31 2024 at 23:38):

Keith seems to be using $\rm Ord$ as a full subcategory of $\rm Set$: ordinals, thought of as sets, and all functions between them.

Given this, there's no point in talking about $\rm Ord$ as a topos: since it's equivalent to $\rm Set$, all the topos-theoretic things you can do with $\rm Ord$ can also be done with $\rm Set$, and it's less confusing to use $\rm Set$.

The idea of "ordinal exponentiation" only shows up when you use a different category: the poset with ordinals as objects and a single morphism from $x$ to $y$ when $x \le y$, no morphisms otherwise. This poset is very interesting, but it's not a topos.

John Baez (Jan 31 2024 at 23:41):

In short, Keith: don't talk about the topos $\rm Ord$.

Mike Shulman (Feb 01 2024 at 01:04):

In particular, ordinal exponentiation is not, as far as I know, even a functor on the subcategory of Set consisting of ordinals.