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Stream: learning: questions

Topic: irreductible/indecomposable arrow

Moana Jubert (Feb 17 2023 at 10:55):

Hello everyone, I was wondering if there is a standard terminology for "indecomposable arrows", i.e. given an arrow:
$c \longrightarrow d$
if it factors through another object e:
$c \longrightarrow e \longrightarrow d$
then e must be equal to either c or d, and in that case the corresponding arrow is the identity.

Also, I am actually working with posets so maybe it has a specific name (e.g. "non-dense") in that case?

Graham Manuell (Feb 17 2023 at 16:12):

If you mean the category itself is a poset, instead of that you are working in the category of all posets, then it does have a name. In order theory we say that $d$ covers $c$.

Moana Jubert (Feb 17 2023 at 16:36):

Thank you! This is exactly what I was looking for.