A complete classification of categoricity spectra of accessible categories with directed colimits
Christian Espindola

TL;DR
This paper classifies all possible categoricity spectra in large accessible categories with directed colimits under SCH, including a complete classification for AECs in ZFC, and provides explicit thresholds for eventual categoricity.
Contribution
It offers a complete classification of categoricity spectra in accessible categories with directed colimits, including the first ZFC classification for AECs, and confirms the Shelah categoricity conjecture.
Findings
Categoricity spectra are either empty, an interval [α, β], or [χ, ∞).
Provides explicit thresholds for categoricity in accessible categories.
Includes examples for each case of the classification.
Abstract
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 (), 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 . More specifically, we have the following theorem: Let be a large -accessible category with directed colimits. Assume the Singular Cardinal Hypothesis (only if the restriction to monomorphisms is not an AEC). Then the categoricity spectrum \mathcal{C}at(\mathcal{K})=\{\lambda\geq \kappa: \mathcal{K} \text{ is \lambda-categorical}\} is one of the following: 1) . 2)…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
