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Stream: learning: questions

Topic: representably order-reflecting maps

Jonas Frey (Dec 27 2020 at 20:41):

A morphism $f:A\to B$ in a category $C$ is called monomorphism/monic, if all induced postcomposition maps $C(X,A)\to C(X,B)$ are injective.

Is there an established term for a map $f:A\to B$ in a locally ordered category $C$ such that all induced postcomposition maps $C(X,A)\to C(X,B)$ are order-reflecting?

I know that such maps can be called "full", but that's not the kind of analogy I'm looking for since "order-reflecting" as well as "full" are terms in concrete "enriching" categories, and I'm looking for a term for the corresponding "representable" notion in enriched categories.

fosco (Jan 01 2021 at 10:42):

Might not be the best terminology, but if $F : {\cal C} \to {\cal D}$ is a functor of $\cal V$-enriched category, and $P$ a property of morphisms in $\cal V$, I tend to call "hom-wise $P$" the $F$'s such that every $F_{XY} : \hom(X,Y) \to \hom(FX, FY)$ is $P$.

Amar Hadzihasanovic (Jan 01 2021 at 12:03):

@Jonas Frey Why not call that “pointwise” or “objectwise” order-reflecting? It seems consistent with the use in e.g. “(co)limits of presheaves are computed pointwise/objectwise”.
You identify a morphism in $C$ with a morphism of (enriched) presheaves on $C$ via the Yoneda embedding; then “doing things pointwise” means “evaluating the presheaves at each object of $C$”. In this case you're saying that $f: A \to B$ seen as the morphism of presheaves $f_*: C(-,A) \to C(-,B)$ is pointwise order-reflecting.