Is there any example showing that the composition of morphisms is not necessarily associative?
Example of non-associative composition of morphisms
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7Composition of morphisms is always associative. – 2012-11-14
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0In fact, it is an *axiom* of category that composition of morphisms is associative. – 2012-11-14
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3If 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: http://en.wikipedia.org/wiki/Non-associative_algebra – 2012-11-15
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0If you meant morphisms in the sense of category theory, then ([tag:category-theory]) tag would be good for this question. – 2013-07-23
1 Answers
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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0Of course, we have to take homotopy classes of homotopies in order to have an associative composition for _those_... – 2012-11-14
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2Yes, 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 http://ncatlab.org/nlab/show/homotopy+hypothesis) – 2012-11-14