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Studying posets I encountered the notation $a\prec b$. It means that $a<b$ and no $c$ exists with $a<c<b$. If $a\prec b$ then in words $a$ is covered by $b$. Looking at a poset $P$ as a category you could say that for the arrow $f$ in $P\left(a,b\right)$ we have: $$f=g\circ h\Rightarrow g=1\vee h=1$$ It reminds me of elements in a ring that are irreducible.

Suppose that more generally in a category $\mathcal{C}$ there is an arrow $f$ - which is not an isomorphism itself - satisfying: $$f=g\circ h\Rightarrow g\text{ is isomorphism}\vee h\text{ is isomorphism}$$ Is there a name (p.e. irreducible) for arrows having that property?

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Good question. Notice that your notion of irreducibility coincides with the one for monoids (regarded as categories with one object). But one should also require that they are no isomorphisms themselves (à la: irreducibles are no units). – Martin Brandenburg Jan 22 '14 at 22:14
@MartinBrandenburg. Yes, I see. I will add that in an edit. – drhab Jan 23 '14 at 9:31
up vote 4 down vote accepted

Auslander and Reiten (Representation theory of artin algebras) introduced the notion of an irreducible morphism, but it differs from your notion: A morphism $g$ is irreducible if $g$ is neither a split mono nor a split epi, and if, for any factorization $g=fh$, either $f$ is a split mono or $h$ is a split epi.

For monoids (regarded as categories with one object), this reads: $g$ is irreducible, if $g$ is neither left nor right invertible, and if, for any factorization $g=fh$, either $f$ is left invertible or $h$ is right invertible. But your notion becomes the usual notion of irreducibility (no unit, and in every factorization a unit occurs). In the commutatice case, there is no difference.

I think it really depends on the application which notion one uses.

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