2023/01/30 by Christian Espíndola, Espindola, Christian
Mathematics · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)
paper · pdf · doi:10.48550/arxiv.2301.13167
openalex publication_date 2023/01/30 · openalex created_date 2023/02/01 · openalex updated_date 2026/08/01
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 ZFC. More specifically, we have the following 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, \bethω(κ)). 3) Cat(K)=[χ, ∞) for some χ∈ [κ, \beth(2κ)+). 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.