Definable orthogonality classes in accessible categories are small

We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Levy hierarchy. For example, the statement that, for a class S of morphisms in a locally presentable category C of structures, the orthogonal class of objects is a small-orthogonality class (hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from the existence of a proper class of supercompact cardinals if S is \Sigma_2, and from the existence of a proper class of what we call C(n)-extendible cardinals if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a new hierarchy, and we show that Vopenka's principle is equivalent to the existence of C(n)-extendible cardinals for all n. As a consequence, we prove that the existence of cohomological localizations of simplicial sets, a long-standing open problem in algebraic topology, is implied by the existence of arbitrarily large supercompact cardinals. This result follows from the fact that cohomology equivalences are \Sigma_2. In contrast with this fact, homology equivalences are \Sigma_1, from which it follows (as is well known) that the existence of homological localizations is provable in ZFC.


Summary
The assumptions needed to infer reflectivity or smallness of orthogonality classes in accessible categories may depend on the complexity of the formulas in the language of set theory defining those classes.

Summary
The assumptions needed to infer reflectivity or smallness of orthogonality classes in accessible categories may depend on the complexity of the formulas in the language of set theory defining those classes.

Summary
The assumptions needed to infer reflectivity or smallness of orthogonality classes in accessible categories may depend on the complexity of the formulas in the language of set theory defining those classes.

Complexity
Complexity is meant in the sense of the Lévy hierarchy: Π 0 -formulas are the same as Σ 0 -formulas, namely those in which all quantifiers are bounded (∀x ∈ a, ∃x ∈ a). Such formulas are absolute for transitive models of set theory. preserves λ-filtered colimits.
C is λ-accessible if λ-filtered colimits exist in C and there is a set X of λ-presentable objects in C such that every object 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 of models of T (i.e., Σ-structures satisfying all sentences of T ) is accessible. Conversely, every accessible category is a category of models of some theory [Adámek-Rosický, 1994].
Accessible categories are definable with absolute formulas.

Orthogonality Classes
For a class M of morphisms in a category C, the orthogonal class M ⊥ consists of those objects X such that

Orthogonality Classes
For a class M of morphisms in a category C, the orthogonal class M ⊥ consists of those objects X such that We say that a class X of objects is an orthogonality class if X = M ⊥ for some class of morphisms M; For a class M of morphisms in a category C, the orthogonal class M ⊥ consists of those objects X such that

Orthogonality Subcategory Problem
Under which assumptions on a category C and a class of morphisms M does it follow that the orthogonal class M ⊥ is reflective? [Freyd-Kelly, 1972]

Orthogonality Subcategory Problem
Under which assumptions on a category C and a class of morphisms M does it follow that the orthogonal class M ⊥ is reflective? [Freyd-Kelly, 1972] A full subcategory L in C is reflective if the inclusion J : L → C has a left adjoint K : C → L.

Orthogonality Subcategory Problem
Under which assumptions on a category C and a class of morphisms M does it follow that the orthogonal class M ⊥ is reflective? [Freyd-Kelly, 1972] A full subcategory L in C is reflective if the inclusion J : L → C has a left adjoint K : C → L. Then the composite L = J • K is called a reflection or a localization onto L.

Orthogonality Subcategory Problem
Under which assumptions on a category C and a class of morphisms M does it follow that the orthogonal class M ⊥ is reflective? [Freyd-Kelly, 1972] A full subcategory L in C is reflective if the inclusion J : L → C has a left adjoint K : C → L. Then the composite L = J • K is called a reflection or a localization onto L.
If C is locally presentable, then every small-orthogonality class is reflective in C.

Orthogonality Subcategory Problem
Under which assumptions on a category C and a class of morphisms M does it follow that the orthogonal class M ⊥ is reflective? [Freyd-Kelly, 1972] A full subcategory L in C is reflective if the inclusion J : L → C has a left adjoint K : C → L. Then the composite L = J • K is called a reflection or a localization onto L.
If C is locally presentable, then every small-orthogonality class is reflective in C.
Vopěnka's Principle is equivalent to the statement that all orthogonality classes in locally presentable categories are small-orthogonality classes. (This cannot be proved in ZFC.) If VP holds, then for every homotopical localization L on simplicial sets there is a map f such that L L f . [C-Scevenels-Smith, 2005] If VP holds, then cohomological localizations of simplicial sets exist. [C-Scevenels-Smith, 2005] If VP holds, then every localizing subcategory of a triangulated category with combinatorial models is coreflective, and every colocalizing subcategory is reflective. [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 smallness of orthogonality classes in accessible categories may depend on the complexity of the formulas in the language of set theory defining those classes.