Let $X$ and $A$ be sets and $\alpha:X\to A$ a map between them. We call a map $\mu:X\times X\times X\to A$ an approximate Mal'tsev operation with approximation $\alpha$, if it satisfies $\mu(x,y,y) = \alpha(x) = \mu(y,y,x)$ for all $x,y\in X$. Note that if $A = X$ and the approximation $\alpha$ is an identity map, then $\mu$ becomes an ordinary Mal'tsev operation. We prove the following two characterization theorems: a category $\mathbb{X}$ is a Mal'tsev category if and only if in the functor category $\mathbf{Set}^{\mathbb{X}^\mathrm{op}\times\mathbb{X}}$ there exists an internal approximate Mal'tsev operation $\mathrm{hom}_{\mathbb{X}}\times \mathrm{hom}_{\mathbb{X}}\times \mathrm{hom}_{\mathbb{X}}\rightarrow A$ whose approximation $\alpha$ satisfies a suitable condition; a regular category $\mathbb{X}$ with finite coproducts is a Mal'tsev category, if and only if in the functor category $\mathbb{X}^\mathbb{X}$ there exists an internal approximate Mal'tsev co-operation $A\rightarrow 1_\mathbb{X}+1_\mathbb{X}+1_\mathbb{X}$ whose approximation $\alpha$ is a natural transformation with every component a regular epimorphism in $\mathbb{X}$. Note that in both of these characterization theorems, if require further the approximation $\alpha$ to be an identity morphism, then the conditions there involving $\alpha$ become equivalent to $\mathbb{X}$ being a naturally Mal'tsev category.
Keywords: Mal'tsev category, Mal'tsev operation, unital category, strongly unital category, subtractive category
2000 MSC: 18C99, 08B05
Theory and Applications of Categories,
Vol. 21, 2008,
No. 8, pp 152-171.
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