Joachim Lambek: The Interplay of Mathematics, Logic, and Linguistics

Mal'tsev categories turned out to be a central concept in categorical algebra. On one hand, the simplicity and beauty of notion is revealed through lot characterizations different flavour. Depending on context, can define as those for which `any reflexive relation an equivalence'; difunctional'; `the composition equivalence relations same object commutative'; `each fibre fibration points unital' or forgetful functor from internal groupoids graphs saturated subobjects'. For variety universal algebras, these are also equivalent existence its algebraic theory operation, i.e. ternary operation $p(x,y,z)$ satisfying axioms $p(x,x,y)=y$ $p(x,y,y)=x$. other have been shown right context develop centrality relations, Baer sums extensions, some homological lemmas such denormalized $3 \times 3$ Lemma, whose validity regular category weaker `Goursat property', has wide interest.