Anticommutativity and the triangular lemma

Abstract

For a variety V, it has been recently shown that binary products commute with arbitrary coequalizers locally, i.e., in every fibre of the fibration of points π : Pt(C) → C, if and only if Gumm's shifting lemma holds on pullbacks in V. In this paper, we establish a similar result connecting the so-called triangular lemma in universal algebra with a certain categorical anticommutativity condition. In particular, we show that this anticommutativity and its local version are Mal'tsev conditions, the local version being equivalent to the triangular lemma on pullbacks. As a corollary, every locally anticommutative variety V has directly decomposable congruence classes in the sense of Duda, and the converse holds if V is idempotent.

The matrix taxonomy of finitely complete categories

“…The corresponding matrix class is the left one in the sixth row (from the top) of Figure 5 which, as we can see on that figure, contains the third one in the twelfth row. As it follows from Remark 2.21 in [23], a Mal'tsev variety is locally anticommutative if and only if it is arithmetical. This implies that, in the algebraic case, the matrix class appearing in the third place of the twelfth row of Figure 5 (which is the same as the matrix class from Remark 1.5) matches with the matrix class given by the arithmetical matrix (Example 1.3).…”

“…We conclude with another example showing context-sensitivity of implications of matrix properties. The Mal'tsev condition for 'local-anticommutativity' given in [23] is the Mal'tsev condition for direct decomposability of congruence classes [15] together with some identities linking some of the 'inner' terms involved in the condition. This results in the syntactical refinement of the two Mal'tsev conditions being the same.…”