# What's the term to describe a morphism that isn't a composite of morphisms?

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.

• I don't know of a common name for them, but I'm pretty sure "indecomposable" would be immediately understood. – Malice Vidrine Nov 27 '18 at 22:58
• 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! – Kevin Carlson Nov 27 '18 at 23:57
• 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. – Derek Elkins Nov 28 '18 at 3:33

• 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.