Definable Orthogonality Classes inAccessible Categories are Small

Carles Casacuberta, Universitat de Barcelona

Joint work with J. Bagaria, A. R. D. Mathias and J. Rosický

Mülheim an der Ruhr, May 31, 2011

Herzliche Glückwünsche und vielen Dank

an Professor Rüdiger Göbel!

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:I Π0-formulas are the same as Σ0-formulas, namely those

in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:

I Π0-formulas are the same as Σ0-formulas, namely thosein which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:I Π0-formulas are the same as Σ0-formulas, namely those

in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a).

Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:I Π0-formulas are the same as Σ0-formulas, namely those

in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:I Π0-formulas are the same as Σ0-formulas, namely those

in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.

I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Summary

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

Complexity

Complexity is meant in the sense of the Lévy hierarchy:I Π0-formulas are the same as Σ0-formulas, namely those

in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Suchformulas are absolute for transitive models of set theory.

I Σn+1-formulas are those of the form ∃x ϕ where ϕ is Πn.I Πn+1-formulas are those of the form ∀x ψ where ψ is Σn.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.

A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible.

Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Accessible and Locally Presentable Categories

Let λ denote a regular cardinal and let C be a category.

I An object X in C is λ-presentable if the functor C(X ,−)preserves λ-filtered colimits.

I C is λ-accessible if λ-filtered colimits exist in C and thereis a set X of λ-presentable objects in C such that everyobject of C is a λ-filtered colimit of objects from X .

A category is accessible if it is λ-accessible for some λ.A cocomplete accessible category is locally presentable.

For every theory T with any signature Σ, the category ofmodels of T (i.e., Σ-structures satisfying all sentences of T )is accessible. Conversely, every accessible category is acategory of models of some theory [Adámek–Rosický, 1994].

Accessible categories are definable with absolute formulas.

Orthogonality Classes

For a classM of morphisms in a category C, theorthogonal classM⊥ consists of those objects X such thatC(f ,X ) : C(B,X ) ∼= C(A,X ) for all f : A→ B inM.

A∀f∈M //

∀g ��???

????

B

∃!h��X

We say that a class X of objects is

I an orthogonality class if X =M⊥ for some class ofmorphismsM;

I a small-orthogonality class if X =M⊥ for some setof morphismsM.

Orthogonality Classes

For a classM of morphisms in a category C, theorthogonal classM⊥ consists of those objects X such thatC(f ,X ) : C(B,X ) ∼= C(A,X ) for all f : A→ B inM.

A∀f∈M //

∀g ��???

????

B

∃!h��X

We say that a class X of objects is

I an orthogonality class if X =M⊥ for some class ofmorphismsM;

I a small-orthogonality class if X =M⊥ for some setof morphismsM.

Orthogonality Classes

For a classM of morphisms in a category C, theorthogonal classM⊥ consists of those objects X such thatC(f ,X ) : C(B,X ) ∼= C(A,X ) for all f : A→ B inM.

A∀f∈M //

∀g ��???

????

B

∃!h��X

We say that a class X of objects is

I an orthogonality class if X =M⊥ for some class ofmorphismsM;

I a small-orthogonality class if X =M⊥ for some setof morphismsM.

Orthogonality Subcategory Problem

Under which assumptions on a category C and a classof morphismsM does it follow that the orthogonalclassM⊥ is reflective? [Freyd–Kelly, 1972]

A full subcategory L in C is reflective if the inclusion J : L ↪→ Chas a left adjoint K : C → L. Then the composite L = J ◦ K iscalled a reflection or a localization onto L.

I If C is locally presentable, then every small-orthogonalityclass is reflective in C.

Vopenka’s Principle is equivalent to the statement that allorthogonality classes in locally presentable categories aresmall-orthogonality classes. (This cannot be proved in ZFC.)

Orthogonality Subcategory Problem

Under which assumptions on a category C and a classof morphismsM does it follow that the orthogonalclassM⊥ is reflective? [Freyd–Kelly, 1972]

A full subcategory L in C is reflective if the inclusion J : L ↪→ Chas a left adjoint K : C → L.

Then the composite L = J ◦ K iscalled a reflection or a localization onto L.

I If C is locally presentable, then every small-orthogonalityclass is reflective in C.

Vopenka’s Principle is equivalent to the statement that allorthogonality classes in locally presentable categories aresmall-orthogonality classes. (This cannot be proved in ZFC.)

Orthogonality Subcategory Problem

Under which assumptions on a category C and a classof morphismsM does it follow that the orthogonalclassM⊥ is reflective? [Freyd–Kelly, 1972]

A full subcategory L in C is reflective if the inclusion J : L ↪→ Chas a left adjoint K : C → L. Then the composite L = J ◦ K iscalled a reflection or a localization onto L.

I If C is locally presentable, then every small-orthogonalityclass is reflective in C.

Vopenka’s Principle is equivalent to the statement that allorthogonality classes in locally presentable categories aresmall-orthogonality classes. (This cannot be proved in ZFC.)

Orthogonality Subcategory Problem

Under which assumptions on a category C and a classof morphismsM does it follow that the orthogonalclassM⊥ is reflective? [Freyd–Kelly, 1972]

A full subcategory L in C is reflective if the inclusion J : L ↪→ Chas a left adjoint K : C → L. Then the composite L = J ◦ K iscalled a reflection or a localization onto L.

I If C is locally presentable, then every small-orthogonalityclass is reflective in C.

Vopenka’s Principle is equivalent to the statement that allorthogonality classes in locally presentable categories aresmall-orthogonality classes. (This cannot be proved in ZFC.)

Orthogonality Subcategory Problem

Under which assumptions on a category C and a classof morphismsM does it follow that the orthogonalclassM⊥ is reflective? [Freyd–Kelly, 1972]

A full subcategory L in C is reflective if the inclusion J : L ↪→ Chas a left adjoint K : C → L. Then the composite L = J ◦ K iscalled a reflection or a localization onto L.

I If C is locally presentable, then every small-orthogonalityclass is reflective in C.

Vopenka’s Principle is equivalent to the statement that allorthogonality classes in locally presentable categories aresmall-orthogonality classes. (This cannot be proved in ZFC.)

Vopenka’s Principle

Vopenka’s Principle states that, for every proper class ofstructures with the same signature, there exists a nontrivialelementary embedding between two of them.

This is in fact an axiom schema, since it is a statement aboutclasses. In the language of set theory, it is formulated asfollows, for all formulas ϕ(x , y) of the language of set theorywith two free variables:

∀x [(∀y ∀z (ϕ(x , y) ∧ ϕ(x , z) → y and z are Σ-structures for some Σ)

∧ ∀α ∈ Ord ∃y (rank(y) > α ∧ ϕ(x , y))) →

∃y ∃z (ϕ(x , y) ∧ ϕ(x , z) ∧ y 6= z ∧ ∃e (e : y → z is elementary))].

Vopenka’s Principle

Vopenka’s Principle states that, for every proper class ofstructures with the same signature, there exists a nontrivialelementary embedding between two of them.

This is in fact an axiom schema, since it is a statement aboutclasses. In the language of set theory, it is formulated asfollows, for all formulas ϕ(x , y) of the language of set theorywith two free variables:

∀x [(∀y ∀z (ϕ(x , y) ∧ ϕ(x , z) → y and z are Σ-structures for some Σ)

∧ ∀α ∈ Ord ∃y (rank(y) > α ∧ ϕ(x , y))) →

∃y ∃z (ϕ(x , y) ∧ ϕ(x , z) ∧ y 6= z ∧ ∃e (e : y → z is elementary))].

Some well-known consequences of Vopenka’s Principle (VP):

I If VP holds, then all torsion theories of abelian groupsare singly generated. [Dugas–Göbel, 1985]

I If VP holds, then every full subcategory of anaccessible category has a small dense fullsubcategory. [Adámek–Rosický, 1994]

I If VP holds, then every full limit-closed subcategory of alocally presentable category is reflective and every fullcolimit-closed subcategory is coreflective.[Adámek–Rosický, 1994]

Some well-known consequences of Vopenka’s Principle (VP):

I If VP holds, then all torsion theories of abelian groupsare singly generated. [Dugas–Göbel, 1985]

I If VP holds, then every full subcategory of anaccessible category has a small dense fullsubcategory. [Adámek–Rosický, 1994]

I If VP holds, then every full limit-closed subcategory of alocally presentable category is reflective and every fullcolimit-closed subcategory is coreflective.[Adámek–Rosický, 1994]

Some well-known consequences of Vopenka’s Principle (VP):

I If VP holds, then all torsion theories of abelian groupsare singly generated. [Dugas–Göbel, 1985]

I If VP holds, then every full subcategory of anaccessible category has a small dense fullsubcategory. [Adámek–Rosický, 1994]

I If VP holds, then every full limit-closed subcategory of alocally presentable category is reflective and every fullcolimit-closed subcategory is coreflective.[Adámek–Rosický, 1994]

More consequences of Vopenka’s Principle (VP):

I If VP holds, then for every homotopical localization Lon simplicial sets there is a map f such that L ' Lf .[C–Scevenels–Smith, 2005]

I If VP holds, then cohomological localizations ofsimplicial sets exist. [C–Scevenels–Smith, 2005]

I If VP holds, then every localizing subcategory of atriangulated category with combinatorial models iscoreflective, and every colocalizing subcategory isreflective. [C–Gutiérrez–Rosický, 2011]

Interesting symmetry break: VP implies that localizing subcategories are singly

generated, yet we have been unable to prove this for colocalizing subcategories.

More consequences of Vopenka’s Principle (VP):

I If VP holds, then for every homotopical localization Lon simplicial sets there is a map f such that L ' Lf .[C–Scevenels–Smith, 2005]

I If VP holds, then cohomological localizations ofsimplicial sets exist. [C–Scevenels–Smith, 2005]

I If VP holds, then every localizing subcategory of atriangulated category with combinatorial models iscoreflective, and every colocalizing subcategory isreflective. [C–Gutiérrez–Rosický, 2011]

Interesting symmetry break: VP implies that localizing subcategories are singly

generated, yet we have been unable to prove this for colocalizing subcategories.

More consequences of Vopenka’s Principle (VP):

I If VP holds, then for every homotopical localization Lon simplicial sets there is a map f such that L ' Lf .[C–Scevenels–Smith, 2005]

I If VP holds, then cohomological localizations ofsimplicial sets exist. [C–Scevenels–Smith, 2005]

I If VP holds, then every localizing subcategory of atriangulated category with combinatorial models iscoreflective, and every colocalizing subcategory isreflective. [C–Gutiérrez–Rosický, 2011]

Interesting symmetry break: VP implies that localizing subcategories are singly

generated, yet we have been unable to prove this for colocalizing subcategories.

More consequences of Vopenka’s Principle (VP):

I If VP holds, then for every homotopical localization Lon simplicial sets there is a map f such that L ' Lf .[C–Scevenels–Smith, 2005]

I If VP holds, then cohomological localizations ofsimplicial sets exist. [C–Scevenels–Smith, 2005]

I If VP holds, then every localizing subcategory of atriangulated category with combinatorial models iscoreflective, and every colocalizing subcategory isreflective. [C–Gutiérrez–Rosický, 2011]

Interesting symmetry break: VP implies that localizing subcategories are singly

generated, yet we have been unable to prove this for colocalizing subcategories.

Recall

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

More precisely, we define, for each n < ω, what we callC(n)-extendible cardinals, with the following properties:

I If there is a proper class of C(n)-extendible cardinals,where n ≥ 1, then each Σn+1 orthogonality class in anaccessible category is a small-orthogonality class.

I Vopenka’s Principle is equivalent to the claim that there isa proper class of C(n)-extendible cardinals for each n < ω.

Recall

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

More precisely, we define, for each n < ω, what we callC(n)-extendible cardinals, with the following properties:

I If there is a proper class of C(n)-extendible cardinals,where n ≥ 1, then each Σn+1 orthogonality class in anaccessible category is a small-orthogonality class.

I Vopenka’s Principle is equivalent to the claim that there isa proper class of C(n)-extendible cardinals for each n < ω.

Recall

The assumptions needed to infer reflectivity or smallnessof orthogonality classes in accessible categoriesmay depend on the complexity of the formulas in thelanguage of set theory defining those classes.

More precisely, we define, for each n < ω, what we callC(n)-extendible cardinals, with the following properties:

I If there is a proper class of C(n)-extendible cardinals,where n ≥ 1, then each Σn+1 orthogonality class in anaccessible category is a small-orthogonality class.

I Vopenka’s Principle is equivalent to the claim that there isa proper class of C(n)-extendible cardinals for each n < ω.

C(n)-Extendible Cardinals

Let C(n) denote the class of cardinals α such that Vα is aΣn-elementary submodel of the universe V . A cardinalκ ∈ C(n) is C(n)-extendible if, for all λ > κ in C(n), there is anelementary embedding j : Vλ → Vµ for some µ ∈ C(n) withcritical point κ, such that j(κ) ∈ C(n) and j(κ) > λ.

A class S is definable with sufficiently low complexity if

I it is Σ1, orI there is a proper class of supercompact cardinals andS is Σ2, or

I there is a proper class of C(n)-extendible cardinals andS is Σn+2, with n ≥ 1.

C(n)-Extendible Cardinals

Let C(n) denote the class of cardinals α such that Vα is aΣn-elementary submodel of the universe V . A cardinalκ ∈ C(n) is C(n)-extendible if, for all λ > κ in C(n), there is anelementary embedding j : Vλ → Vµ for some µ ∈ C(n) withcritical point κ, such that j(κ) ∈ C(n) and j(κ) > λ.

A class S is definable with sufficiently low complexity if

I it is Σ1, orI there is a proper class of supercompact cardinals andS is Σ2, or

I there is a proper class of C(n)-extendible cardinals andS is Σn+2, with n ≥ 1.

Main Results

I Every full subcategory of an accessible category definablewith sufficiently low complexity is bounded.

I If L is a reflection on an accessible category withcoproducts, and the class of L-equivalences is definablewith sufficiently low complexity, then L ∼= Lf for some f .

I In a locally presentable category, every full limit-closedsubcategory definable with sufficiently low complexity isreflective, and every full colimit-closed subcategorydefinable with sufficiently low complexity is coreflective.

Recall: A class S is definable with sufficiently low complexity if

• it is Σ1, or

• there is a proper class of supercompact cardinals and S is Σ2, or

• there is a proper class of C(n)-extendible cardinals and S is Σn+2, with n ≥ 1.

Implications in Algebraic Topology

For a spectrum E , the class of E∗-acyclic simplicial sets is Σ1,while the class of E∗-acyclic simplicial sets is Σ2.

A simplicial set X is E∗-acyclic if and only if the followingformula is true:

X ∈ sSets∗ ∧ (∀n < ω) ∃M [M ∈ sSets∗

∧ (∀k < ω) [(∀f ∈ Mk ) f ∈ sSets∗(X ∧∆[k ]+,En)

∧ ∀g (g ∈ sSets∗(X ∧∆[k ]+,En)→ g ∈ Mk )]

∧ M is weakly contractible].

The formula states that the mapping space Map(X ,En) is weaklycontractible for all n, where the (fibrant) spectrum E = {En}n≥0

is treated as a parameter.

Thus, it follows from our results that the existence ofE∗-localizations can be proved in ZFC —this was knownsince [Bousfield, 1975]— and that the existence ofE∗-localizations follows from the existence of a properclass of supercompact cardinals.

I The latter improves [C–Scevenels–Smith, 2005], where theexistence of E∗-localizations was inferred from VP.

I More generally, if S is any (possibly proper) class of mapsbetween simplicial sets, then an S-localization exists if S isdefinable with sufficiently low complexity.

Recall: A class S is definable with sufficiently low complexity if

• it is Σ1, or

• there is a proper class of supercompact cardinals and S is Σ2, or

• there is a proper class of C(n)-extendible cardinals and S is Σn+2, with n ≥ 1.

Thus, it follows from our results that the existence ofE∗-localizations can be proved in ZFC —this was knownsince [Bousfield, 1975]— and that the existence ofE∗-localizations follows from the existence of a properclass of supercompact cardinals.

I The latter improves [C–Scevenels–Smith, 2005], where theexistence of E∗-localizations was inferred from VP.

I More generally, if S is any (possibly proper) class of mapsbetween simplicial sets, then an S-localization exists if S isdefinable with sufficiently low complexity.

Recall: A class S is definable with sufficiently low complexity if

• it is Σ1, or

• there is a proper class of supercompact cardinals and S is Σ2, or

• there is a proper class of C(n)-extendible cardinals and S is Σn+2, with n ≥ 1.

Thus, it follows from our results that the existence ofE∗-localizations can be proved in ZFC —this was knownsince [Bousfield, 1975]— and that the existence ofE∗-localizations follows from the existence of a properclass of supercompact cardinals.

I The latter improves [C–Scevenels–Smith, 2005], where theexistence of E∗-localizations was inferred from VP.

I More generally, if S is any (possibly proper) class of mapsbetween simplicial sets, then an S-localization exists if S isdefinable with sufficiently low complexity.

Recall: A class S is definable with sufficiently low complexity if

• it is Σ1, or

• there is a proper class of supercompact cardinals and S is Σ2, or

• there is a proper class of C(n)-extendible cardinals and S is Σn+2, with n ≥ 1.

Please come to:

Large-Cardinal Methods in Homotopy

Advanced Course and Workshop

Institut de Matemàtica, Universitat de Barcelona

1–8 September 2011

Organizing Committee:J. Bagaria, C. Casacuberta, R. Göbel,

A. R. D. Mathias, J. L. Rodríguez, J. Rosický

www.imub.ub.es/hocard11