Conjugation in a group satisfies a nice extendibility property: namely, if $a\in G$ and $f: G\rightarrow H$, then the map $$Conj_a[f]="x\mapsto f(a)xf(a)^{-1}"$$ is again an automorphism of $H$. In addition, conjugation satisfies a nice definability property: the conjugation map given by an element $a$ is defined by a term with $a$ as a parameter.
Moreover, definability and extendibility go together nicely: in order to "push conjugation along" a homomorphism $f$, we move the parameter $a$ along $f$ and then use the same term as in the original group $G$. Roughly speaking, as long as a group contains $a$, then "conjugation by $a$" is an endomorphism; in particular, "conjugation by $a$" almost defines an endomorphism on arbitrary groups.
I'm interested in generalizing this picture to arbitrary varieties, in the following way (incidentally, see this article of Bergman for a different - and indescribably more serious and mathematically interesting - take on the extendability of conjugation):
Given a variety $V$, a virtual endomorphism in $V$ is a tuple $\nu=(m, t(x, \overline{y}), \overline{a}, A)$ where
$A$ is an algebra in $V$ (the starting domain),
$\overline{a}\in A$ is an $m$-tuple,
$t$ is a term in the language of $V$ in $m+1$ variables (note: no parameters from $A$ are allowed in $t$), such that
for any $B\in V$ and any homomorphism $f: A\rightarrow B$, the term $t(x, f(\overline{a}))$ defines an endomorphism of $B$. I'll denote "$t(x, f(\overline{a}))$" by "$\nu[f]$".
For example, if the language of $V$ contains a constant symbol $c$, then the term "$c$" corresponds to a virtual endomorphism; we don't distinguish between "versions" of $c$ with different "dummy information" (e.g. the algebra $A$ or the parameters $\overline{a}\in A$, which don't actually play any role in evaluating the constant map $x\mapsto c$). Even more generally, the term "$x$" always corresponds to the identity endomorphism in any algebra in any variety.
Less trivially, for any element $g$ of any group $G$, "conjugation by $g$" is a virtual endomorphism in the variety of groups.
This seems like an interesting notion, but it is annoyingly difficult (for me at least) to work with. Right off the bat, this raises the first question:
Question 1. What is the right notion of "equality" between virtual endomorphisms?
One reasonable first guess is to consider the relation $\sim$ defined as follows: $\mu=(m, t(x, \overline{y}), \overline{a}, A)\sim \nu=(n, s(x, \overline{z}), \overline{b}, B)$ iff for every $C\in V$ and every pair of homomorphisms $f:A\rightarrow C, g: B\rightarrow C$ which agree on $A\cap B$, we have $\mu[f]=\nu[g]$.
However - besides being somewhat evil in the categorical sense - $\sim$ isn't obviously an equivalence relation! I earlier claimed that it was, but Keith Kearnes pointed out in a commment that there's no reason to believe it is - and I now suspect that in fact it is not.
If we restrict attention to virtual endomorphisms with the same starting domain, then $\sim$ is definitely an equivalence relation - but it seems pretty weak to me: whether it is correct hinges hinges, for me, on the question of whether "conjugation by $a$" (in $G$) should be the same as "conjugation by $a$" (in $H$) for groups $a\in G\subseteq H$.
Leaving that aside for the moment, my main (overly broad) question is:
Question 2. What can we tell about a variety $V$, from its collection of virtual endomorphisms?
Well, that's way too broad; can we make it more precise?
It's always a good idea to try to give a collection of objects a structure beyond that of a pure set. Virtual endomorphisms are "like" arrows, but not quite: they define an arrow for any algebra in $V$, but they don't really constitute arrows on their own (what's the target of a virtual endomorphism?). So we can view them as objects in their own right, and see if there is a reasonable notion of morphism between virtual endomorphisms which will turn this collection into a meaningful category.
The most natural approach here that I see is the following: if $\alpha=(m, t, \overline{a}, A),\beta=(n, s,\overline{b}, B)$ are virtual endomorphisms, a morphism $m: \alpha\rightarrow\beta$ is a homomorphism $h: B\rightarrow A$ such that for every homomorphism $g: A\rightarrow C$ (with $C\in V$), we have $\beta[gh]=\alpha[g]$. That is, a morphism from $\alpha$ to $\beta$ is a way of viewing each instance of $\alpha$ as an instance of $\beta$.
My first question is:
Question 3. Is this a natural structure to put on the class of virtual endomorphisms of $V$? If not, what's a better one?
Assuming the answer to the above question is yes - or, more weakly, that there is some meaningful way to view the class of virtual endomorphisms as yielding a category - we may then ask:
Question 4. What do we know about varieties $V, W$ if their "categories of virtual endomorphisms" (whatever exactly that means) are equivalent?
(That is, what "Morita-like" equivalence between varieties do we get from virtual endomorphisms? (Yes, I realize this is completely unrelated to actual Morita equivalence.))