In this introductory lecture on non-commutative geometry, Connes gives an elementary example of how to identify two points.
We take a two-point space $S:=\{a,b\}$ and we wish to identify (i.e. quotient) these two points into one. One trivial example is to consider the algebra of functions on $S$ of functions $A:=\{f:S\to\mathbb{C}\}$, and then consider the subalgebra where the functions take the same values at the two points $A_s:=\{f:S\to\mathbb{C} \ \ | \ \ f(a)=f(b) \}$.
Another way to do it is to put these two functions in a matrix \begin{pmatrix} f_{aa} & 0 \\ 0 & f_{bb} \end{pmatrix} The idea is then to add off-diagonal terms that will identify two points (if anyone can elucidate this as well that would be nice). That means we consider the algebra \begin{pmatrix} f_{aa} & f_{ab} \\ f_{ba} & f_{bb} \end{pmatrix} Call this algebra $K$. He then claims that the categories of modules of these two algebras (considered as Rings in this context I suppose) are the same $M_K\cong M_{A_s}$ i.e. that they are Morita Equivalent.
How would one go about proving such a thing in such a (seemingly elementary) example? I'm not so well versed in homological algebra, in general, please bear with me if this seems to be trivial.