description

Let $\Gamma$ be a group. A description is a surjective $\Gamma$-map $d\colon X\to Y$ between $\Gamma$-sets; when a quotient $\varphi\colon\Gamma\to G$ acts on $Y$, $\Gamma$ acts on $Y$ through $\varphi$. The kernel of $d$ at a point $x_0$ is $K_d(x_0)=\{\gamma\in\Gamma:\ d(\gamma x_0)=d(x_0)\}=\Gamma_{d(x_0)}$, the stabilizer of the image of $x_0$. What $d$ forgets at $x_0$ is the fibre $d^{-1}(d(x_0))$; if $X$ is transitive, it is the orbit $K_d(x_0)\,x_0$.

A seam identifies two incarnations and forgets nothing. Most maps between theories are not seams: they go one way and forget something. Reduction modulo a prime, the passage from a double cover to its quotient, and the passage from a group to one of its orbits all lose information, and in each case the loss is a subgroup. Here the group may be infinite and the sets need not be transitive.

The kernel contains $\Gamma_{x_0}$, and it changes by conjugation when $x_0$ moves in its orbit. A description forgets nothing exactly when $K_d(x)=\Gamma_x$ for every $x$, that is, when $d$ is a bijection: a description that forgets nothing is a seam.

For transitive $G$-sets $X$ and $Y$ with fixed markings, a description $X\to Y$ exists if and only if a stabilizer of $X$ is contained in a stabilizer of $Y$. The bridge between $X$ and $Y$ is built if and only if some description $X\to Y$ forgets nothing; if $|X|=|Y|$, it is refuted if and only if there is no description from $X$ to $Y$ at all.