# Universal constructions that factor through endomorphisms

If $\cal A$ is a variety of algebras (e.g., all groups) and $\cal B$ is a subvariety defined by some set of identities $X$ (e.g., abelian groups with $X = \{xy \simeq yx\}$), then there is a functor $F_X : {\cal A} \rightarrow {\cal B}$, that maps $A \in \cal A$, to $A / \sim$, where $\sim$ is the congruence on $A$ generated by $X$ (so for groups and abelian groups this functor is abelianization). Moreover, any homomorphism from $A \in A$ to $B \in \cal B$ factors uniquely through $F_X(A)$. Is there a standard name for this construction?

I am looking at a situation where there is a natural embedding of $F_X(A)$ in $A$, i.e., the natural $\cal A$-homomorphism from $A$ to $F_X(A)$ factors as the composite of an endomorphism of $A$ and an isomorphism. My example generalises the case where $\cal A$ is the variety of brouwerian semilattices, $\cal B$ is the subvariety defined by the identity $\lnot\lnot x \simeq x$ and the endomorphism is $x \mapsto \lnot\lnot x$. Has this kind of situation been studied in general, or does anyone know of any other specific examples? Any references are greatly appreciated.

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Isn't this just the left adjoint of the inclusion from $B$ into $A$? See also ncatlab.org/nlab/show/reflective+subcategory; sometimes such left adjoints are called reflectors. –  Qiaochu Yuan Feb 1 '14 at 21:11
Yes; you are looking at the adjoint of the forgetful functor. –  Arturo Magidin Feb 2 '14 at 1:15
Yes, there is an adjoint of a forgetful functor involved here. But that doesn't really answer either of my questions. –  Rob Arthan Feb 2 '14 at 3:15

There is no agreed name for functors $\Gamma$ as defined above. To category theorists they are the left adjoints of forgetful functors between quasivarieties; but this is too heavy. With slight adjustments, some computer theorists know them as parametrised data types - but this is too specialised. Seeing that Mal'tsev [1958a] first described them, he should name them. He calls $T(A)$ the replica of $A$ in $W$, and accordingly I shall refer to these functors $\Gamma$ as replica functors.
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