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Is there any example showing that the composition of morphisms is not necessarily associative?

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Composition of morphisms is always associative. – Qiaochu Yuan Nov 14 '12 at 22:28
In fact, it is an axiom of category that composition of morphisms is associative. – tomasz Nov 14 '12 at 22:50
If you would like to know examples of "non-associative categories", specializing to one object these are sets with a non-associative binary operation and a unit element, which include non-associative algebras. There is a list on wikipedia: – Martin Brandenburg Nov 15 '12 at 11:46
If you meant morphisms in the sense of category theory, then (category-theory) tag would be good for this question. – Martin Sleziak Jul 23 '13 at 18:29

I don't know how much category theory you know, but I guess you can look up any terms you don't recognize.

In a cateogry, composition is associative per definition. However, when generalizing categories to higher categories ($n$-categories), it is sometimes useful not to demand associativity, but only "weak associativity". Weak associativity is the same as associativity up to isomorphism in the layer above. That is, for any triple $f,g,h$ of morphisms where $gf$ and $hg$ are defined, there is an "isomorphism of morphisms" $F: (hg)f \rightarrow h(gf)$.

For example, if we define a path in a topological space $X$ to be a continuous function $\alpha:[0,1]\rightarrow X$, there is a well defined operation of "composition" of paths which is weakly associative, but not associative. In this case we get a 2-category where composition is associative up to homotopy.

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Of course, we have to take homotopy classes of homotopies in order to have an associative composition for those... – Zhen Lin Nov 14 '12 at 22:44
Yes, this chain goes on to infinity, and is the basis for the homotopy hypothesis, that $\infty$-groupoids are essentially the same as topological spaces. (See for example – Espen Nielsen Nov 14 '12 at 22:49

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