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Measurable cardinals in category theory

Andrew Brooke-Taylor

University of Leeds

Reflections on Set Theoretic Reflection Joan Bagaria’s 60th birthday Catalonia, November 2018

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 1 / 26

Joint work in progress with Joan Bagaria and Jiˇ r´ ı Rosick´ y, and I’ve been talking about closely related things with Will Boney.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 2 / 26

Category theory preliminaries

Recall

A category C consists of a class of objects, and for every pair of objects A and B of C, a set HomC(A, B) of morphisms from A to B, with identity morphisms and a composition (partial) function of morphisms, satisfying suitable axioms.

E.g.s

Set is the category with sets as objects and functions as morphisms. Gp is the category with groups as objects and group homomorphisms as morphisms.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 3 / 26

Limits and colimits

We think of a diagram as being a set of objects and morphisms between them. The limit of a diagram D is an object L along with a cone δ of projection maps to the objects of D (such that the triangles formed with the morphisms

• f D commute) such that any other such cone from an object of C factors

uniquely through δ. The colimit of a diagram is the same in reverse.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 4 / 26

E.g.

In Set, every diagram D has a limit and a colimit: The limit is the subset of the product of the sets in D consisting of all element whose coordinates “cohere” under the functions of the diagram. The colimit is the disjoint union of the sets in D, modulo identifying elements with their images under the functions in D.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 5 / 26

E.g.

In Set, every diagram D has a limit and a colimit: The limit is the subset of the product of the sets in D consisting of all element whose coordinates “cohere” under the functions of the diagram. The colimit is the disjoint union of the sets in D, modulo identifying elements with their images under the functions in D. Gp has all limits & colimits too: limits are the same as in Set, and colimits are free products modulo identifications.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 5 / 26

Given a set A of objects in a category C and an object C of C, the canonical diagram of C with respect to A is the diagram with for every object A in A and every morphism f : A → C, a copy of A, which we shall denote by Af , as morphisms, all morphisms h: Af → Bg such that g ◦ h = f .

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 6 / 26

Note that the morphisms f : Af → C form a cocone to C. If this cocone makes C the colimit of its canonical diagram with respect to A, we say that C is a canonical colimit of objects from A.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 7 / 26

Note that the morphisms f : Af → C form a cocone to C. If this cocone makes C the colimit of its canonical diagram with respect to A, we say that C is a canonical colimit of objects from A. If every object is a canonical colimit of

• bjects from from A, we say that A is dense.

E.g.s

ω is dense in Set: every set is the colimit of the diagram of all of its finite subsets, which are the images of functions from finite sets. Any set of representatives of all the isomorphism classes of finite groups is dense in Gp: every group is the colimit of the diagram of all of its finite subgroups.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 7 / 26

Note that being a canonical colimit of objects from A is stronger in general than just being a colimit of some diagram of objects from A.

E.g.

Let VectR be the category of real vector spaces, with linear transformations as the

• morphisms. Consider the set A = {R}. Then every object of VectR is a colimit of
• bjects from A, but A is not dense. Indeed, consider a function ϕ: R2 → R2

respecting scalar multiplication but not addition. Then there is a cocone mapping each Rf to R2 by ϕ ◦ f , but it doesn’t factor through the canonical cocone by any linear map.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 8 / 26

Given a category C, Cop is the category with the same objects as C, and the same morphisms but in the opposite direction. Identity functions remain identity functions, and compositions of morphisms remain compositions of morphisms, just in the opposite order.

E.g.

Setop is the category with sets as objects, and functions as morphisms, with any f : X → Y in the usual sense being considered as going from Y to X.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 9 / 26

Given a category C, Cop is the category with the same objects as C, and the same morphisms but in the opposite direction. Identity functions remain identity functions, and compositions of morphisms remain compositions of morphisms, just in the opposite order.

E.g.

Setop is the category with sets as objects, and functions as morphisms, with any f : X → Y in the usual sense being considered as going from Y to X.

Question

Is there a dense set in Setop?

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 9 / 26

For any cardinal κ and any set X, consider the canonical diagram D in Setop of X with respect to κ.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 10 / 26

For any cardinal κ and any set X, consider the canonical diagram D in Setop of X with respect to κ. Since morphisms are reversed, this is the diagram with an

• bject for every function from X to an ordinal less than κ, with a function h from

αf to βg if h ◦ f = g.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 10 / 26

For any cardinal κ and any set X, consider the canonical diagram D in Setop of X with respect to κ. Since morphisms are reversed, this is the diagram with an

• bject for every function from X to an ordinal less than κ, with a function h from

αf to βg if h ◦ f = g. We can think about such functions f : X → α in terms of the partitions {f −1{γ} | γ ∈ α} that they define. In this context, the functions in the diagram represent coarsening maps.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 10 / 26

The elements of the limit are elements u = (uf )αf ∈D of the product of the

• rdinals αf in D — in the αf coordinate, the element uf of αf is chosen.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 11 / 26

The elements of the limit are elements u = (uf )αf ∈D of the product of the

• rdinals αf in D — in the αf coordinate, the element uf of αf is chosen.

This corresponds to the choice of a piece from each of the partitions (f −1{uf } in the partition corresponding to f : X → α), in a way that the coarsening maps respect — we can think of this as choosing a “big” piece from each partition.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 11 / 26

The elements of the limit are elements u = (uf )αf ∈D of the product of the

• rdinals αf in D — in the αf coordinate, the element uf of αf is chosen.

This corresponds to the choice of a piece from each of the partitions (f −1{uf } in the partition corresponding to f : X → α), in a way that the coarsening maps respect — we can think of this as choosing a “big” piece from each partition. These choices form a κ-complete ultrafilter on X!

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 11 / 26

The elements of the limit are elements u = (uf )αf ∈D of the product of the

• rdinals αf in D — in the αf coordinate, the element uf of αf is chosen.

This corresponds to the choice of a piece from each of the partitions (f −1{uf } in the partition corresponding to f : X → α), in a way that the coarsening maps respect — we can think of this as choosing a “big” piece from each partition. These choices form a κ-complete ultrafilter on X! Indeed by coarsening, if Y is chosen in any partition, it is chosen in the partition {Y , X Y }, from which it can be seen that Y is chosen in every partition containing it.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 11 / 26

The elements of the limit are elements u = (uf )αf ∈D of the product of the

• rdinals αf in D — in the αf coordinate, the element uf of αf is chosen.

This corresponds to the choice of a piece from each of the partitions (f −1{uf } in the partition corresponding to f : X → α), in a way that the coarsening maps respect — we can think of this as choosing a “big” piece from each partition. These choices form a κ-complete ultrafilter on X! Indeed by coarsening, if Y is chosen in any partition, it is chosen in the partition {Y , X Y }, from which it can be seen that Y is chosen in every partition containing it. So let U be the set of Y ⊆ X such that Y is chosen in some (any) partitition in which it appears as a piece (i.e., if Y = f −1(uf )).

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 11 / 26

U is a κ-complete ultrafilter

Let χY : X → 2 be the characteristic function of Y , χY (x) = 1 ↔ x ∈ Y . Then U = {Y ⊆ X | ∃α < κ∃f : X → α(Y = f −1{uf })} = {Y ⊆ X | uχY = 1}.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 12 / 26

U is a κ-complete ultrafilter

Let χY : X → 2 be the characteristic function of Y , χY (x) = 1 ↔ x ∈ Y . Then U = {Y ⊆ X | ∃α < κ∃f : X → α(Y = f −1{uf })} = {Y ⊆ X | uχY = 1}. If Y ∈ U and Y ⊆ Z ⊆ X, then {Y , Z Y , X Z} coarsens to {Z, X Z}, so Z ∈ U.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 12 / 26

U is a κ-complete ultrafilter

Let χY : X → 2 be the characteristic function of Y , χY (x) = 1 ↔ x ∈ Y . Then U = {Y ⊆ X | ∃α < κ∃f : X → α(Y = f −1{uf })} = {Y ⊆ X | uχY = 1}. If Y ∈ U and Y ⊆ Z ⊆ X, then {Y , Z Y , X Z} coarsens to {Z, X Z}, so Z ∈ U. If Y ∈ U and {Zγ | γ < α} is a partition of Y into fewer than κ many pieces, then since {X Y } ∪ {Zγ | γ < α} coarsens to {Y , X Y }, one of the Zγ is in U, so U is κ-complete.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 12 / 26

U is a κ-complete ultrafilter

Let χY : X → 2 be the characteristic function of Y , χY (x) = 1 ↔ x ∈ Y . Then U = {Y ⊆ X | ∃α < κ∃f : X → α(Y = f −1{uf })} = {Y ⊆ X | uχY = 1}. If Y ∈ U and Y ⊆ Z ⊆ X, then {Y , Z Y , X Z} coarsens to {Z, X Z}, so Z ∈ U. If Y ∈ U and {Zγ | γ < α} is a partition of Y into fewer than κ many pieces, then since {X Y } ∪ {Zγ | γ < α} coarsens to {Y , X Y }, one of the Zγ is in U, so U is κ-complete. For any Y ⊆ X, Y ∈ U if uχY = 1 and X Y ∈ U if uχY = 0, so U is ultra.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 12 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x).

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x). Note that the χ{x} component of ux is 1, so {x} is in the corresponding ultrafilter — it is the principle ultrafilter defined by x.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x). Note that the χ{x} component of ux is 1, so {x} is in the corresponding ultrafilter — it is the principle ultrafilter defined by x. So there is a non-principal κ-complete ultrafilter on X if and only if this map X → lim D is not a bijection

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x). Note that the χ{x} component of ux is 1, so {x} is in the corresponding ultrafilter — it is the principle ultrafilter defined by x. So there is a non-principal κ-complete ultrafilter on X if and only if this map X → lim D is not a bijection i.e. not an isomorphism in Set

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x). Note that the χ{x} component of ux is 1, so {x} is in the corresponding ultrafilter — it is the principle ultrafilter defined by x. So there is a non-principal κ-complete ultrafilter on X if and only if this map X → lim D is not a bijection i.e. not an isomorphism in Set i.e. X is not the limit of D. Note that by definition, there is a dense set in Setop if and only if for some κ, every X is the limit of its canonical diagram with respect to κ,

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

The canonical cone from X to D factors through the limit cone by the map x → ux, where the αf component of ux is f (x). Note that the χ{x} component of ux is 1, so {x} is in the corresponding ultrafilter — it is the principle ultrafilter defined by x. So there is a non-principal κ-complete ultrafilter on X if and only if this map X → lim D is not a bijection i.e. not an isomorphism in Set i.e. X is not the limit of D. Note that by definition, there is a dense set in Setop if and only if for some κ, every X is the limit of its canonical diagram with respect to κ, if and only if there are no non-principal κ-complete ultrafilters on any set.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 13 / 26

So we have shown

Theorem (Isbell, 1960)

There is a dense set in Setop if and only if there are only boundedly many measurable cardinals.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 14 / 26

We have characterised κ-complete ultrafilters category-theoretically.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 15 / 26

We have characterised κ-complete ultrafilters category-theoretically.

Question (Joan)

Can we characterise normal ultrafilters on κ?

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 15 / 26

Taking a step back

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different. In particular, different elements of sets are distinguishable. Indeed, if we’re talking about a cardinal, the elements even have a canonical well-ordering.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different. In particular, different elements of sets are distinguishable. Indeed, if we’re talking about a cardinal, the elements even have a canonical well-ordering. In category theory, the elements of sets are considered to really be indistinguishable.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different. In particular, different elements of sets are distinguishable. Indeed, if we’re talking about a cardinal, the elements even have a canonical well-ordering. In category theory, the elements of sets are considered to really be

• indistinguishable. In particular, things defined category-theoretically in Set should

probably be permutation-invariant.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different. In particular, different elements of sets are distinguishable. Indeed, if we’re talking about a cardinal, the elements even have a canonical well-ordering. In category theory, the elements of sets are considered to really be

• indistinguishable. In particular, things defined category-theoretically in Set should

probably be permutation-invariant. However, normality is not permutation-invariant: given an ultrafilter U on κ and a permutation π of κ, we have an ultrafilter πU on κ defined by Y ∈ πU iff π−1Y ∈ U.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Taking a step back

In set theory, different sets are different. In particular, different elements of sets are distinguishable. Indeed, if we’re talking about a cardinal, the elements even have a canonical well-ordering. In category theory, the elements of sets are considered to really be

• indistinguishable. In particular, things defined category-theoretically in Set should

probably be permutation-invariant. However, normality is not permutation-invariant: given an ultrafilter U on κ and a permutation π of κ, we have an ultrafilter πU on κ defined by Y ∈ πU iff π−1Y ∈ U. Consider the permutation π swapping every limit ordinal with its successor. Then if U is normal, the ultrafilter πU is not normal — it contains the set of successor

• rdinals.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 16 / 26

Changing the question

A more suitable notion in this context is selectivity:

Definition

An ultrafilter U on κ is selective if for every partition of κ into κ many sets Yα none of which is in U, there is some set X ∈ U such that |X ∩ Yα| ≤ 1 for every α.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 17 / 26

Changing the question

A more suitable notion in this context is selectivity:

Definition

An ultrafilter U on κ is selective if for every partition of κ into κ many sets Yα none of which is in U, there is some set X ∈ U such that |X ∩ Yα| ≤ 1 for every α. Clearly selectivity is permutation-invariant.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 17 / 26

Changing the question

A more suitable notion in this context is selectivity:

Definition

An ultrafilter U on κ is selective if for every partition of κ into κ many sets Yα none of which is in U, there is some set X ∈ U such that |X ∩ Yα| ≤ 1 for every α. Clearly selectivity is permutation-invariant. Note that any normal ultrafilter is selective: suppose {Yα | α < κ} is a partition

• f κ into pieces not in U, and assume WLOG that min(Yα) ≥ α for all α. Then

△

α<α κ Yα

is a set in U as required.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 17 / 26

The partial converse was pointed out to me by Jonathan Verner:

Proposition

If a nonprincipal κ-complete ultrafilter U on κ is selective then there is a permutation π of κ such that πU is normal.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 18 / 26

The partial converse was pointed out to me by Jonathan Verner:

Proposition

If a nonprincipal κ-complete ultrafilter U on κ is selective then there is a permutation π of κ such that πU is normal.

Proof

Let jU : V → M be the ultrapower embedding from U. Let f : κ → κ be such that [f ]U = κ. By κ-completeness, f −1{α} / ∈ U for each α ∈ κ (otherwise we would have [f ]U = j(α) = α), so by selectivity there is some X ∈ U such that f ↾X is injective.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 18 / 26

The partial converse was pointed out to me by Jonathan Verner:

Proposition

If a nonprincipal κ-complete ultrafilter U on κ is selective then there is a permutation π of κ such that πU is normal.

Proof

Let jU : V → M be the ultrapower embedding from U. Let f : κ → κ be such that [f ]U = κ. By κ-completeness, f −1{α} / ∈ U for each α ∈ κ (otherwise we would have [f ]U = j(α) = α), so by selectivity there is some X ∈ U such that f ↾X is injective. Shrinking X if necessary, we may assume that |κ X| = κ and |κ f “X| = κ. But then we may extend f ↾X to a permutation π of κ with π↾X = f ↾X, so [π]U = [f ]U = κ.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 18 / 26

Now consider the normal ultrafilter V defined from jU: X ∈ V ↔ κ ∈ jU(X) ↔ [π]U ∈ [cX]U where cX is the constant function taking value X ↔ {α ∈ κ | π(α) ∈ X} ∈ U ↔ X ∈ πU. So πU is normal. So a nonprincipal κ-complete ultrafilter on κ is selective if and only if some permutation of it is normal.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 19 / 26

Characterising selectivity in category theory

The precise formulation of selectivity we translate is the following:

Proposition

An ultrafilter on κ is selective if and only if for every f : κ → κ, there is a set X in the ultrafilter on which X is either constant or injective.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 20 / 26

Characterising selectivity in category theory

The precise formulation of selectivity we translate is the following:

Proposition

An ultrafilter on κ is selective if and only if for every f : κ → κ, there is a set X in the ultrafilter on which X is either constant or injective. We also need a little more category theory.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 20 / 26

Terminal objects and elements

A terminal object 1 in a category is an object such that for every object C of C there is a unique morphism C → 1.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 21 / 26

Terminal objects and elements

A terminal object 1 in a category is an object such that for every object C of C there is a unique morphism C → 1.

E.g.

In Set, the ordinal 1 is a terminal object.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 21 / 26

Terminal objects and elements

A terminal object 1 in a category is an object such that for every object C of C there is a unique morphism C → 1.

E.g.

In Set, the ordinal 1 is a terminal object. Note that an element x of a set X can be identified with the function from 1 to X taking the unique element of 1 to x. Thus “elements” can be translated into category-theoretic language, as “morphisms from the terminal object.”

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 21 / 26

Subobjects

In category theory we discuss subobjects in terms of their inclusion morphisms.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 22 / 26

Subobjects

In category theory we discuss subobjects in terms of their inclusion morphisms. The translation of a one-to-one function is as a monomorphism: a morphism f such that f ◦ g = f ◦ h ⇒ g = h.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 22 / 26

Subobjects

In category theory we discuss subobjects in terms of their inclusion morphisms. The translation of a one-to-one function is as a monomorphism: a morphism f such that f ◦ g = f ◦ h ⇒ g = h. Thus, a subobject is just considered to mean a monomorphism.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 22 / 26

Definition

In a category C with a terminal object 1, a subobject classifier is an object Ω equipped with a distinguished morphism t : 1 → Ω such that for every monomorphism i : A → X of C, there is a unique morphism χA,i : X → Ω such that A

i

• X

χA,i

• 1

t

• Ω

is the pullback (that is, A is the limit) of the following diagram. (E.g. in Set, Ω = 2.) X

χA,i

• 1

t

• Ω

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 23 / 26

Theorem

Let U : 1 → lim D be (the morphism that picks out) a non-principal κ-complete ultrafilter on κ. Then U is selective if and only if for every morphism f : κ → κ, there is a subobject A of κ, with inclusion morphism i : A ֒ → κ, such that: the subobject classifier map t : 1 → 2χA,i factors as πχA,i ◦ U, and either

1

f ◦ i factors through 1, or

2

f ◦ i is a monomorphism.

A

mono i

• (2) mono
• 1

U

• t
• (1)∃
• κ

f

• χA,i
• κ

lim D

πχA,i

• 2χA,i

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 24 / 26

Back to normality?

If we work with the category of sets with a preferred partial order (with no requirement that functions respect it), the hom-sets inherit a preferred partial

• rder. In this setting, we can express f < id, and translate the Fodor formulation
• f normality to get a category-theoretic version.

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 25 / 26

Congratulations Joan!

Andrew Brooke-Taylor Measurable cardinals in category theory Joan 60 26 / 26