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I want to distinguish between two disjoint classes of morphisms in a category: (1) those morphisms that are composed of other morphisms (other than identities) and could conceivably be factored into a sequence of other morphisms; and, (2) those morphisms that cannot be factored further.

For the time being, I am referring to the first class of morphisms as "composed" and the second class as "atomic". What is the correct terminology for this distinction?

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In a general category this doesn't seem like what you want; there will be no morphisms in the second class as long as there are nontrivial isomorphisms running around. You want to consider factorizations that don't contain isomorphisms, and then "irreducible" seems like a reasonable word for the second class. –  Qiaochu Yuan Jul 15 '13 at 19:32
    
(But this still isn't great: for example, in any category with biproducts any morphism $f : A \to B$ trivially admits factorizations of the form $A \to B \oplus C \to B$. What sort of categories do you have in mind where you'd like to apply this notion? The only reasonable candidates I can think of are either posets, in which case you recover a version of the Hasse diagram, or free categories on graphs, in which case you recover the underlying graph.) –  Qiaochu Yuan Jul 15 '13 at 19:34

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As far as I know there is no fixed or common terminology for these concepts. Atomic is a good name, as would indecomposable or irreducible would be. I wouldn't use 'composed' for the non-atomic ones, but rather 'composite'.

Of course few categories in nature will have interesting behaviour with respect to these concepts. For instance, isomorphisms may be atomic but then they decompose an identity. A category with reasonable structure will tend to have lots and lots of morphisms, making atomic ones very rare. You might want to consider morphisms that are atomic with respect to a fixed class of morphisms that you may call 'redundant'. These would be morphisms that for some reason you wish to ignore for the purposes of reducibility. It all depends on what you want to achieve.

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