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Is there a term to describe a morphism in a category that cannot be written as a composition of two (non-identity) morphisms? Naively Googling around I haven't found anyone talking about this, so I don't know if there is an established word for this. I really hope it's the word indecomposable, since it describes an morphism that is not a composite. But the adjectives prime, primitive, and irreducible could also work. I mostly want to make sure I'm don't define such morphisms opposing an established precedent.

Furthermore, is there a name for a category where every morphism can be written as a composite of these "indecomposable" morphisms? Something like a Krull-Schmidt category but for the morphisms instead of the objects. This question is motivated by looking at the functor that takes a quiver and gives you the free category on that quiver. I want the category-theoretic term that describes the morphisms in that free category that came precisely from the arrows of the quiver.

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    $\begingroup$ I don't know of a common name for them, but I'm pretty sure "indecomposable" would be immediately understood. $\endgroup$ – Malice Vidrine Nov 27 '18 at 22:58
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    $\begingroup$ Every morphism can be uniquely written as a composite of irreducibles if and only if your category is free. I'm not immediately sure what kind of category has every morphism just some composite of irreducibles...not sure it's a useful concept. Reedy categories are a class of examples, but certainly aren't exhaustive. Note that such a category will always be skeletal, so you might rather ask for a morphism which isn't a composite of two non-isomorphisms. Or maybe not! $\endgroup$ – Kevin Carlson Nov 27 '18 at 23:57
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    $\begingroup$ For the specific case of the indecomposable arrows in a free category, I would probably call them "generators" (or possibly "generating arrows" if I was afraid of conflicting with some other terminology). They literally are generators in the usual algebraic sense if you view a (small) category as a multi-sorted essentially algebraic theory. $\endgroup$ – Derek Elkins Nov 28 '18 at 3:33
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Since the answer does not appear to be an immediate Yes, I figure this CW answer should be made to collect everyone's suggestions and comments.

  • Calling such morphisms indecomposable is alright for the reason mentioned in the question.

  • As Derek Elkins says in the comments, one could call such morphisms generators, or generating morphisms, since they literally are generators for the rest of the morphisms in the usual algebraic sense, when viewing your (small) category as a multi-sorted essential algebraic theory.

  • You might be inspired to call them primitive morphisms, after the primitive elements of a coalgebra, the elements $x$ such that $\Delta(x) = x\otimes 1 + 1 \otimes x$. Calling them this isn't ideal though. I think that you can't quite regard the path algebra of a quiver (paths in a category) as a coalgebra, so the analogy has some holes.

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