# An abstract elementary class non-axiomatizable in $L_{(\infty,\kappa)}$

## TL;DR

This paper proves that certain large categories, including those of sets with large cardinalities and monomorphisms, cannot be represented as models of any $L_{(
fty,eta)}$-theory, highlighting limitations of infinitary logic in categorizing these structures.

## Contribution

It introduces a categorified Scott topology and demonstrates that these large categories are not the points of any topos, thus not models of certain infinitary logical theories.

## Key findings

- Large categories of sets with uncountable cardinalities are not $L_{(
infty,eta)}$-theories.
- The categorified Scott topology acts as a left adjoint to the points functor for topoi.
- Techniques also apply to categories of vector spaces and algebraically closed fields.

## Abstract

We show that for any uncountable cardinal $\lambda$, the category of sets of cardinality at least $\lambda$ and monomorphisms between them cannot appear as the category of point of a topos, in particular is not the category of models of a $L_{(\infty,\omega)}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of $\kappa$-points of a $\kappa$-topos, in particular, not the category of models of a $L_{(\infty,\kappa)}$-theory. The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least $\lambda$ and monomorphisms between them. The same techniques also applies to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.