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A Characterization of the Higgins Commutator

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Abstract

We show that in an ideal-determined unital category the Higgins commutator can be characterized as the largest binary operation C on subobjects (defined on all subobjects of each object) satisfying the following conditions: (a) C is order-preserving; (b) C(H, K) is always less or equal to the meet of normal closures of H and K;  (c) \(C(f(H),f(K))=f(C(H,K))\) for every pair of subobjects H and K of an object X,  and every morphism f whose domain is X.

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Acknowledgements

This work forms part of the author’s Ph.D. thesis at Stellenbosch university, under the supervision of J.R.A. Gray. Funding was provided by Deutscher Akademischer Austauschdienst (Grant No. 91560461).

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Authors and Affiliations

  1. Mathematics Division, Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland, 7602, South Africa

    Vaino Tuhafeni Shaumbwa

  2. Department of Mathematics, University of Namibia, Windhoek, Namibia

    Vaino Tuhafeni Shaumbwa

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  1. Vaino Tuhafeni Shaumbwa
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Correspondence to Vaino Tuhafeni Shaumbwa.

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George Janelidze.

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Shaumbwa, V.T. A Characterization of the Higgins Commutator. Appl Categor Struct 27, 159–162 (2019). https://doi.org/10.1007/s10485-018-9548-9

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  • DOI: https://doi.org/10.1007/s10485-018-9548-9

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