Pure maps are strict monomorphisms

TL;DR

This paper proves that under certain conditions in accessible categories, pure maps are equivalent to strict monomorphisms, with results depending on logical axiomatizability and large cardinal assumptions.

Contribution

It establishes conditions under which pure maps in accessible categories are strict monomorphisms, linking logical axiomatizability and large cardinal hypotheses.

Findings

Pure maps are strict monomorphisms in certain axiomatizable accessible categories.

Under large cardinal assumptions, pure maps become strict monomorphisms for some larger regular cardinal.

Results connect category-theoretic purity with logical and set-theoretic principles.

Abstract

We prove that $i)$ if $\mathcal{A}$ is $\lambda $-accessible and it is axiomatizable in (finitary) coherent logic then $\lambda $-pure maps are strict monomorphisms and $ii)$ if there is a proper class of strongly compact cardinals and $\mathcal{A}$ is $\lambda $-accessible then for some $\mu \vartriangleright \lambda $ every $\mu $-pure map is a strict monomorphism.