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Stream: community: general

Topic: Use of large cardinals

Reuben Stern (they/them) (May 03 2020 at 19:34):

Super vague question: anyone have any fun examples of using cardinals larger than $\aleph_0$ in category theory? Like specific examples of $\kappa$-filtered colimits or something. Not really sure what I'm looking for here

Fabrizio Genovese (May 03 2020 at 19:40):

I think you are more or less asking if big cardinals are useful outside of set theory :P

Fabrizio Genovese (May 03 2020 at 19:41):

Anyway an obvious place to look I guess would be foundations of category theory, where people discuss about which constructions/things are possible and which ones aren't :slight_smile:

Reuben Stern (they/them) (May 03 2020 at 19:43):

I guess that is what I'm asking, but I think "useful" can be interpreted very broadly haha

Mike Shulman (May 03 2020 at 20:46):

Maybe any place that one applies the machinery of locally presentable categories to categories that are not locally finitely presentable?

(=_=) (May 04 2020 at 01:02):

Reuben Stern said:

Super vague question: anyone have any fun examples of using cardinals larger than $\aleph_0$ in category theory? Like specific examples of $\kappa$-filtered colimits or something. Not really sure what I'm looking for here

You can have $\kappa$-ary extensive categories, for example.

@David Michael Roberts, would you like to comment further?

David Michael Roberts (May 04 2020 at 01:09):

The category of Banach spaces and short maps is aleph_1-presentable, for example.

David Michael Roberts (May 04 2020 at 01:12):

Or consider the setup for condensed sets, where one takes a fibred topos over the partially ordered class of uncountable strong limit cardinals.

David Michael Roberts (May 04 2020 at 01:12):

In each fibre one really does use that cardinality bound to ensure one has a Grothendieck topos.

David Michael Roberts (May 04 2020 at 01:14):

David Michael Roberts said:

The category of Banach spaces and short maps is aleph_1-presentable, for example.

Good discussion here on some consequences/details of this: https://math.stackexchange.com/questions/1424777/is-the-category-of-banach-spaces-and-bounded-linear-maps-accessible

(=_=) (May 04 2020 at 01:18):

David Michael Roberts said:

Or consider the setup for condensed sets, where one takes a fibred topos over the partially ordered class of uncountable strong limit cardinals.

Here's a quick backgrounder for condensed sets.

David Michael Roberts (May 04 2020 at 01:26):

Not that Clausen and Scholze use this terminology, but that is what they are doing (man, I wish people would read SGA4 more! Including me :-P)

Mike Shulman (May 04 2020 at 03:27):

I'm not sure whether this counts as "in category theory", but when interpreting dependent type theory into a category we generally need inaccessible cardinals to construct universe objects closed under the type constructors.

Matt Feller (May 04 2020 at 14:08):

Using Vopěnka's principle, Cisinski showed that every "A-localizer" is accessible. Combined with the rest of his theory, what that means is: in the category of presheaves on some small category A, if you have a class of morphisms W which looks like it should be the weak equivalences in a model structure where the cofibrations are the monomorphisms (i.e., W is an A-localizer), then in fact such a model structure exists and the acyclic cofibrations are generated by a set of morphisms (i.e., the model structure is cofibrantly generated).

Matt Feller (May 04 2020 at 14:14):

See section 1.4 of his '06 book (in French). Looking more closely at it now, I see that he also cites this paper by Casacuberta, Scevenels, and Smith.

John Baez (May 04 2020 at 15:15):

Vopěnka's principle is nice because it's a large cardinal axiom that can be stated in a way category theorists can like:

Every discrete full subcategory of a locally presentable category is small.

Here's another way to state it:

No full subcategory of the category of graphs is equivalent to the category of ordinals.

Mike Shulman (May 04 2020 at 19:19):

Here's a way to state Vopěnka's principle that I think many category theorists would like even better:

Every full subcategory of a locally presentable category that's closed under colimits is coreflective.

Similarly, we can state "weak Vopěnka's principle" as

Every full subcategory of a locally presentable category that's closed under limits is reflective.

Gershom (May 04 2020 at 23:22):

Mike Shulman said:

I'm not sure whether this counts as "in category theory", but when interpreting dependent type theory into a category we generally need inaccessible cardinals to construct universe objects closed under the type constructors.

Relatedly, induction-recursion is related to the existence of an external Mahlo cardinal as in http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf

Cody Roux (May 05 2020 at 01:32):

Gershom said:

Mike Shulman said:

I'm not sure whether this counts as "in category theory", but when interpreting dependent type theory into a category we generally need inaccessible cardinals to construct universe objects closed under the type constructors.

Relatedly, induction-recursion is related to the existence of an external Mahlo cardinal as in http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf

Even in a predicative theory? Or is this a "predicative Mahlo cardinal"?

Dan Doel (May 05 2020 at 01:55):

In a predicative theory it's similar, but the whole theory is weaker, so the notion of inaccessibility changes.

Dan Doel (May 05 2020 at 01:55):

So people talk more about recursive inaccessibility.

Mike Shulman (May 05 2020 at 02:10):

FWIW, I don't think the Mahlo is intrinsic to the notion of induction-recursion, just that it's needed for the commonest choice about the rules for universe indices in inductive-recursive definitions. But there should be other ways to state such rules that don't require a Mahlo.

David Michael Roberts (May 10 2020 at 02:01):

Matt Feller said:

Using Vopěnka's principle, Cisinski showed that every "A-localizer" is accessible. Combined with the rest of his theory, what that means is: in the category of presheaves on some small category A, if you have a class of morphisms W which looks like it should be the weak equivalences in a model structure where the cofibrations are the monomorphisms (i.e., W is an A-localizer), then in fact such a model structure exists and the acyclic cofibrations are generated by a set of morphisms (i.e., the model structure is cofibrantly generated).

In light of Simon Henry's recent work, I wonder if the last result could be weakened to a weak model category and lose the assumption of Vopěnka's principle?

David Michael Roberts (Jun 04 2020 at 05:50):

Matt Feller said:

Using Vopěnka's principle, Cisinski showed that every "A-localizer" is accessible. Combined with the rest of his theory, what that means is: in the category of presheaves on some small category A, if you have a class of morphisms W which looks like it should be the weak equivalences in a model structure where the cofibrations are the monomorphisms (i.e., W is an A-localizer), then in fact such a model structure exists and the acyclic cofibrations are generated by a set of morphisms (i.e., the model structure is cofibrantly generated).

In light of Simon Henry's recent work, might it be possible to weaken the result to be a weak model category, and in the process, drop the Vopenka assumption. (Sorry, wrote this ages ago but it never sent).

Nikolaj Kuntner (Sep 17 2020 at 18:11):

In ZFC, consider $V_\alpha$ for any infinite ordinal $\alpha$ and $X,Y\in V_\alpha$. Are all functions $X\to Y$ in our theory already forming a set on the level $\alpha$, making this Neumann level catesian closed?
I asked because I read $V_{\omega+\omega}$ is a nice topos, but I don't quite know if that means all conceivable functions are already covered in it, nor whether these statements extend to other ordinals.

Jonas Frey (Sep 18 2020 at 02:58):

Nikolaj Kuntner said:

In ZFC, consider $V_\alpha$ for any infinite ordinal $\alpha$ and $X,Y\in V_\alpha$. Are all functions $X\to Y$ in our theory already forming a set on the level $\alpha$, making this Neumann level catesian closed?
I asked because I read $V_{\omega+\omega}$ is a nice topos, but I don't quite know if that means all conceivable functions are already covered in it, nor whether these statements extend to other ordinals.

I think $V_\alpha$ is cartesian closed whenever $\alpha$ is a limit ordinal, like $\omega+\omega$.

Proof: if $X,Y\in V_\alpha$ and $\alpha$ is limit, then there exists a $\beta<\alpha$ with $X,Y\in V_\beta$. Then $Y^X\subseteq P(X\times Y) \in V_{\beta+1}$, thus $Y^X\in V_{\beta+1}$ since $V_\beta$ is closed under subsets, and therefore $Y^X\in V_{\alpha}$. QED

In particular $V_\omega$ is also cartesian closed and thus a topos, but it doesn't have a 'natural numbers object' (NNO), since all elements of $V_\omega$ are finite. Thus, to get an elementary topos with NNO, we have to iterate the powerset functor at least up to the second limit ordinal $\omega+\omega$.

John Baez (Sep 18 2020 at 15:10):

Nice! That sounds right to me.

One interesting thing is that 0, 1 and $\omega$ would count as inaccessible cardinals if the definition didn't explicitly rule them out.

André Beuckelmann (Sep 18 2020 at 16:35):

Adding to @Jonas Frey's comment, it is also necessary for $\alpha$ to be a limit ordinal. In general, if we define $rank(S)$ to be the least ordinal with $S\subseteq V_\alpha$ (i.e. $S$ first appears in $V_{rank(S)+1}$), then we have $rank(S)=sup(rank(x)+1|x\in S)$. From this, one can see that $rank(\gamma)=\gamma$ for any ordinal, and $rank(id_S)\geq rank(S)$. In particular, if $\alpha=\beta+1$, then we have $\beta\in V_\alpha$, and the rank of any set containing $id_\beta$ must be at least $\beta+1$. Thus, $Set(\beta,\beta)$ cannot be an element of $V_\alpha$.