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 $\mu ⊳\lambda$ every $\mu$-pure map is a strict monomorphism.