a category associated with an inverse semigroup 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?
 A: I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in this paper (arXiv link) . 
We define it for semigroups in general but it should boil down to what you wrote for inverse semigroups. We however would say your arrow e goes from b to a. So I guess maybe it is the opposite category but for an inverse  semigroup it doesn't matter.  
For an inverse semigroup, or more generally a von Neumann regular semigroup, this category is equivalent to the Karoubi envelope (aka idempotent splitting or Cauchy completion). For non-regular semigroups it is more interesting.  
The journal version is here (Springerlink). Reference: (A. Costa and B.Steinberg, The Schützenberger category of a semigroup, Semigroup Forum (2015) 91(3) 543–559)
A: I have not a name for it, but as already indicated in my comment, your categroy $C$ is isomorphic to the following more clear category $D$:
Objects of $D$ are $S$. Morphisms from $a \in S$ to $b \in S$ are $e \in S$ with $a a^{-1} e b b^{-1} = e$. Composition of morphisms is just ordinary product in $S$.
Then $F:C \rightarrow D: F(e:a \rightarrow b ) := (e b^{-1} :a \rightarrow b)$ is the isomorphism between $C$ and $D$. 
