2
$\begingroup$

Zero morphism $0_{XY}$ is defined by the formulas $a\circ 0_{XY}=b\circ 0_{XY}$ and $0_{XY}\circ c= 0_{XY}\circ d$ for every morphisms $a$, $b$, $c$, $d$ of suitable sources and destinations.

I define a partially ordered category as a category with a partial order on each of its Hom-sets, such that

$$f_1\le f_2\wedge g_1\le g_2 \Rightarrow g_1\circ f_1\le g_2\circ f_2.$$

My question: Is there any connection between zero morphisms and minimal morphisms of a Hom-set? Do they imply each other? Do they imply each other under some additional conditions? Maybe an implication in one or the other direction works?

So: Under which conditions zero morphism and minimal morphism of a Hom-set are the same?

$\endgroup$
  • $\begingroup$ That these are not equivalent in the general case is clear changing the order of say category $\mathbf{\operatorname{Rel}}$ to the dual order. But they may be equivalent under some additional conditions. $\endgroup$ – porton Oct 19 '12 at 16:26
  • $\begingroup$ No, I don't see any reason why bi-monotone composition should imply that the bottom morphism (assuming it even exists!) is preserved. $\endgroup$ – Zhen Lin Oct 19 '12 at 17:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.