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Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$.

What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming convention to distinguish between binary composition and the intermediary terms from composition operations of higher arity?

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Normally, if we have a morphism $f\colon X\rightarrow Z$ such that there exist morphisms $l\colon X\rightarrow Y$ and $r\colon Y\rightarrow Z$, such that $f=r\circ l$ we say that the morphism $f$ factors through $Y$ via $l$ and $r$. Is this the sort of terminology you're looking for?

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  • $\begingroup$ That is the type of terminology I'm looking for. If available, I am also specifically interested in the terminology to indicate: (1) the arity of the composition; (2) the object that is factored through; and, (3) the order of a particular object that is factored through in the sequence of objects that are factored through in context of the composition (if the composition operation has an arity > 2). $\endgroup$ Commented Mar 22, 2013 at 18:19
  • $\begingroup$ I'm not aware of any specific naming conventions for maps with those properties (the 'factoring through' convention is itself not universally used as far as I know). I suppose one would just say "The map $f\colon Y_0\rightarrow Y_{n+1}$ factors through the $n$ objects $Y_1,\ldots, Y_n$ via the $n+1$ factor maps $f_i\colon Y_i\rightarrow Y_{i+1}$". I'm not sure what you mean by the 'order' of an object but I'm also not a category theorist so this is probably just due to my ignorance. $\endgroup$
    – Dan Rust
    Commented Mar 22, 2013 at 18:28
  • $\begingroup$ I am using "order" in the the common usage context. e.g. In a sequence <A,B,C> : the order of "A" is 1, the order of "B" is 2, etc. $\endgroup$ Commented Mar 22, 2013 at 18:30
  • $\begingroup$ To confirm what I got from your last comment, would you call each morphism a "factor map"/"factor"? $\endgroup$ Commented Mar 22, 2013 at 18:35
  • $\begingroup$ Ah I see. I don't believe there's any standard notation beyond indexing your objects and referring to the index, but once again I'm not a category theorist so there may well be. As to your second comment, I think if you called the intermediary maps factors I don't think anyone would object, as it is analogous to the concept of factorisation in ring theory. As just one example, the definition of a model category (en.wikipedia.org/wiki/Model_category) on Wikipedia specifically makes mention of factorisation. $\endgroup$
    – Dan Rust
    Commented Mar 22, 2013 at 18:39

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