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Stream: theory: category theory

Topic: functors inducing poset isomorphisms

Amar Hadzihasanovic (Oct 16 2022 at 14:49):

Let $r: \mathbf{Cat} \to \mathbf{Pos}$ be the "poset reflection" functor, left adjoint to the inclusion of posets into categories.

Of course $r$ maps equivalences of categories to isomorphisms of posets.
Are there known general characterisations of the functors $F$ such that $rF$ is an isomorphism?

Morgan Rogers (he/him) (Oct 16 2022 at 17:26):

Let $F: \mathcal{C} \to \mathcal{D}$. Consider the "category of loops on $F$", whose objects are quadruples $(C \in \mathcal{C}, D \in \mathcal{D},f:FC \to D,g:D \to FC)$, with no conditions on $f$ and $g$, with morphisms being the evident pairs forming commuting squares with the latter two components - these are actually redundant. Then $rF$ is surjective if and only if the projection to $\mathcal{D}$ is essentially surjective (or equivalently surjective on objects), and order-reflecting if and only if $F$ reflects inhabitedness of hom-sets, in the sense that whenever $\exists f: FC \to FD$, it follows that $\exists g: C \to D$. I thought this could be implemented as fullness of the projection from the loop category, but it is not necessary that the latter morphism commute in any square with the former...

Anyway, being surjective and reflecting the order is necessary and sufficient for an order-preserving map between posets to be an isomorphism.