A complete classification of categoricity spectra of accessible categories with directed colimits
Résumé
We provide a complete classification of all the possible categoricity spectra, in terms of internal size, that can appear in a large accessible category with directed colimits, assuming the Singular Cardinal Hypothesis (SCH), and providing as well explicit threshold cardinals for eventual categoricity. This includes as a particular case the first complete classification of categoricity spectra of abstract elementary classes (AEC's) entirely in ZF C. More specifically, we have: Theorem. Let K be a large κ-accessible category with directed colimits. Assume the Singular Cardinal Hypothesis SCH (only if the restriction to monomorphisms is not an AEC). Then the categoricity spectrum Cat(K) = {λ ≥ κ : K is λ-categorical} is one of the following: 1. Cat(K) = ∅. 2. Cat(K) = [α, β] for some α, β ∈ [κ, ℶ ω (κ)). 3. Cat(K) = [χ, ∞) for some χ ∈ [κ, ℶ (2 κ) +). This solves in particular Shelah categoricity conjecture for AEC's. There are examples of each of the three cases of the classification, showing that they indeed occur.
Domaines
Mathématiques [math]
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