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Embedding theorems for (regular) Mal'tsev categories

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Mal’tsev characterised [1] varieties of universal algebras for which the composition of congruences is commutative as varieties whose theory contains a ternary operation p(x,y,z) satisfying p(x,y,y)=x and p(x,x,y)=y. This property of commutativity of the composition of equivalence relations can be studied in the more general context of regular categories and gives rise to the notion of a regular Mal’tsev category [2]. Moreover, a regular category is a Mal’tsev category if and only if every reflexive relation is an equivalence relation [2]. This last property makes sense in any finitely complete category and enables us to extend the concept of Mal’tsev categories to this context [3]. Using the theory of approximate Mal’tsev operations from [4], we present in this talk an essentially algebraic category M which is regular Mal’tsev and such that each small regular Mal’tsev category admits a conservative regular embedding into a power of M [5]. This gives a way to translate varietal proofs using elements and Mal’tsev operations into categorical proofs for regular Mal’tsev categories in general. Afterwards, we also give an embedding theorem for weakly Mal’tsev and Mal’tsev categories in the category of partial Mal’tsev algebras [6].

[1] A. I. Mal’tsev, On the general theory of algebraic systems, Mat. Sbornik, N.S. 35 (77) (1954), 3 -20”

[2] A. Carboni, J. Lambek and M.C. Pedicchio, Diagram chasing in Mal’cev categories, J. Pure Appl. Algebra 69 (1990), 271- 284.

[3] A. Carboni, M.C. Pedicchio and N. Pirovano, Internal graphs and internal groupoids in Mal’tsev categories, Canadian Math. Soc. Conf. Proc. 13 (1992), 97 -109.

[4] D. Bourn and Z. Janelidze, Approximate Mal’tsev operations, Theory and Appl. of Categ. 21 No. 8 (2008), 152 -171.

[5] P.-A. Jacqmin, An embedding theorem for regular Mal’tsev categories, submitted to J. Pure Appl. Algebra.

[6] P.-A. Jacqmin, Partial Mal’tsev algebras and an embedding theorem for (weakly) Mal’tsev categories, submitted to Cah. Topol. Géom. Diff ér. Catég.

This talk is part of the Category Theory Seminar series.

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