# Internal sizes in μ-abstract elementary classes

This is the title of a talk given at the Harvard Logic Seminar on September 12, 2017. The talk presents the corresponding paper.

## Abstract

The internal size of an object *M* inside a given category is, roughly, the least infinite cardinal λ such that any morphism from M into the colimit of a λ^{+}-directed system factors through one of the components of the system. In the category of set, the internal size of an object is its cardinality. In the category of vector spaces, the internal size is the dimension, and in the category of metric spaces, the internal size is the least cardinality of a dense subset. We will discuss questions around internal sizes in the framework of μ-abstract elementary classes (μ-AECs), which are, up to equivalence of categories, the same as accessible categories with all morphisms monomorphisms. We will in particular examine an example of Shelah — a certain class of sufficiently-closed constructible models of set theory — which shows that the categoricity spectrum can behave very differently depending on whether we look at categoricity in cardinalities or in internal sizes. This is joint work with Michael Lieberman and Jiří Rosický.

## References

Michael Makkai and Robert Paré, *Accessible Categories: The Foundations of Categorical Model Theory*, Contemporary Mathematics, vol. 104, American Mathematical Society, 1989.

Tibor Beke and Jiří Rosický, *Abstract elementary classes and accessible categories*, Annals of Pure and Applied Logic **163** (2012), 2008–2017.

Will Boney, Rami Grossberg, Michael Lieberman, Jiří Rosický, and Sebastien Vasey, *μ-Abstract elementary classes and other generalizations*, Journal of Pure and Applied Algebra **220** (2016), no. 9, 3048–3066.

Michael Lieberman, Jiří Rosický, and Sebastien Vasey, *Internal sizes in μ-abstract elementary classes*, In preparation. Preprint: pdf arXiv.