MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Isn't this just the left adjoint of the inclusion from $B$ into $A$? See also; 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

From Hodges, model theory, section 9.3 Word-constructions pag. 431:

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.

share|cite|improve this answer

protected by Todd Trimble Aug 9 '15 at 23:52

Thank you for your interest in this question. Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).

Would you like to answer one of these unanswered questions instead?

Not the answer you're looking for? Browse other questions tagged or ask your own question.