APPROXIMATE MAL ’ TSEV OPERATIONS Dedicated to Walter Tholen on the occasion of his 60 th birthday

Let X and A be sets and α : X → A a map between them. We call a map μ : X ×X ×X → A an approximate Mal’tsev operation with approximation α, if it satisfies μ(x, y, y) = α(x) = μ(y, y, x) for all x, y ∈ X. Note that if A = X and the approximation α is an identity map, then μ becomes an ordinary Mal’tsev operation. We prove the following two characterization theorems: a category X is a Mal’tsev category if and only if in the functor category SetX op×X there exists an internal approximate Mal’tsev operation homX×homX×homX → A whose approximation α satisfies a suitable condition; a regular category X with finite coproducts is a Mal’tsev category, if and only if in the functor category XX there exists an internal approximate Mal’tsev cooperation A → 1X + 1X + 1X whose approximation α is a natural transformation with every component a regular epimorphism in X. Note that in both of these characterization theorems, if require further the approximation α to be an identity morphism, then the conditions there involving α become equivalent to X being a naturally Mal’tsev category.