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I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in https://arxiv.org/pdf/1408.1615.pdf

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 https://link.springer.com/article/10.1007/s00233-014-9657-1

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)

I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in https://arxiv.org/pdf/1408.1615.pdf

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.

The journal version is here https://link.springer.com/article/10.1007/s00233-014-9657-1

I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in https://arxiv.org/pdf/1408.1615.pdf

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 https://link.springer.com/article/10.1007/s00233-014-9657-1

I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in https://arxiv.org/pdf/1408.1615.pdf

We define it for semigroups in general but it should boil down to what you wrote for inverse semigroups.

The journal version is here https://link.springer.com/article/10.1007/s00233-014-9657-1

I think this is isomorphic to what Alfredo and I call the Schutzenberger category of a semigroup in https://arxiv.org/pdf/1408.1615.pdf

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.

The journal version is here https://link.springer.com/article/10.1007/s00233-014-9657-1