Abstract
We prove a structure theorem for compact inverse categories. The Ehresmann-Schein-Nambooripad theorem gives a structure theorem for inverse monoids: they are inductive groupoids. A particularly nice case due to Clifford is that commutative inverse monoids become semilattices of abelian groups. It has also been categorified by Hoehnke and DeWolf-Pronk to a structure theorem for inverse categories as locally complete inductive groupoids. We show that in the case of compact inverse categories, this takes the particularly nice form of a semilattice of compact groupoids. Moreover, one-object compact inverse categories are exactly commutative inverse monoids. Compact groupoids, in turn, are determined in particularly simple terms of 3-cocycles by Baez-Lauda.
Date: January 13, 2023. Supported by EPSRC Fellowship EP/R044759/1. We thank Peter Hines for pointing out that the proof of Proposition 9 could be simplified, Martti Karvonen for the idea of the proof of Lemma 23, and Phil Scott for pointing out Theorem 5.
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Notes
- 1.
Robin, in particular, would like to acknowledge Samson’s influence which stretches back to well before the mutual interests discussed in this article. He recalls fondly a “hike” with Samson in Lake Louise, during a Higher Order Banff workshop meeting in the early 1990s. Samson, oblivious to the spectacular scenery, spent the hike explaining his ideas on interaction categories (Abramsky, 1993).
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Cockett, R., Heunen, C. (2023). Compact Inverse Categories. In: Palmigiano, A., Sadrzadeh, M. (eds) Samson Abramsky on Logic and Structure in Computer Science and Beyond. Outstanding Contributions to Logic, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-031-24117-8_22
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