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Split Short Five Lemma for Clots and Subtractive Categories

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Abstract

It is well-known that a pointed variety is classically ideal determined (or “BIT speciale”) if and only if it satisfies the split short five lemma (i.e. if and only if it is a protomodular category in the sense of D. Bourn). A much weaker property than being classically ideal determined is “subtractivity”, defined as follows: a variety with a constant 0 is said to be subtractive if its theory contains a binary term s satisfying s(x,x) = 0 and s(x,0) = x. In the case of a pointed variety (i.e. when 0 is the unique constant), this condition can be reformulated purely categorically (as many other similar term conditions), which gives rise to the notion of a subtractive category. In the present paper we show that in a certain general categorical context subtractivity is equivalent to a special restriction of the split short five lemma to the class of clots, i.e. monomorphisms that are pullbacks of reflexive relations R→Y×Y along product injections (1 Y ,0): Y→Y×Y.

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Authors and Affiliations

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

    Zurab Janelidze

  2. Dipartimento di Scienze Matematiche ed Informatiche, Universita’ di Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Aldo Ursini

Authors
  1. Zurab Janelidze
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  2. Aldo Ursini
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Correspondence to Zurab Janelidze.

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Dedicated to the memory of Gregory Maxwell Kelly

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Janelidze, Z., Ursini, A. Split Short Five Lemma for Clots and Subtractive Categories. Appl Categor Struct 19, 233–255 (2011). https://doi.org/10.1007/s10485-009-9192-5

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  • DOI: https://doi.org/10.1007/s10485-009-9192-5

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