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It is pointed out in Geometry from the spectral point of view the following: If one considers a discrete space, say, the two-point space $\{1,2\}$, after identifying its points $X=\{1,2\}/\sim$, the algebra of functions $A=C(\{1,2\}/\sim,\mathbb{C})$ is the algebra of matrices $M_2(\mathbb{C})$ with usual matrix product. According to the author, the noncommutativity of this algebra is a result of the relation between the "points".

(If one takes the quotient one has $X=\{*\}$, whose algebra is $\mathbb{C}$. I still have no problem with this apparent ambiguity, i.e. $M_2(\mathbb{C})$ is Morita equivalent to $\mathbb{C}$, so it will have the same "noncommutative topology", so to say). However, concerning Connes' statement,

Question: where does the usual matrix product comes from?

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It comes from composition of isomorphisms. One version of the "algebra of functions" on, say, a finite groupoid $G$ is its groupoid algebra $\mathbb{C}[G]$, which is a direct generalization of the group algebra: take the free vector space on the morphisms in $G$ with multiplication given by composition (or $0$ if there is no composition). If $G$ is an equivalence relation, then $\mathbb{C}[G]$ is a finite direct product of matrix algebras.

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    Ah, I see, thanks, Qiaochu!2012-11-01