Pure maps are strict monomorphisms

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.