Let $S$ be an inverse semigroup. Define a category $C(S)$ as follows:

- the objects of $C(S)$ are the elements of $S$
- for any $a,b,e\in S$ let $e\colon a\to b$ be a morphism of $C(S)$ iff $aa^{-1}eb^{-1}b=e$
- if $e\colon a\to b$ and $f\colon b\to c$ are morphisms, let $f\circ e\equiv eb^{-1}f\colon a\to c$ (this is a valid morphism because $eb^{-1}f=(aa^{-1}eb^{-1}b)b^{-1}(bb^{-1}fc^{-1}c)=aa^{-1}(eb^{-1}f)c^{-1}c$)

(I would much prefer to write $e\circ f$ for the composition but am bowing to convention.) Easy to see that composition is associative. Identities are $a\colon a\to a$. So it's a category. (I'm tempted to call it the *overlap category*.) This construction occurred to me a few days ago and I'm puzzled because I don't remember coming across it in the literature. But I'm not well versed with inverse semigroups. Anyone seen this before?