Harris' definition of quotient of a variety, what does he mean with "doing something stupid" I am currently reading "Algebraic Geometry" by J. Harris (http://userpage.fu-berlin.de/aconstant/Alg2/Bib/Harris_AlgebraicGeometry.pdf). On page 123, he defines the quotient of a variety $X$, that gets acted on by an algebraic group $G$:
A quotient is a variety $Y$ with regular surjective map $\pi: X \rightarrow Y$ s.t.:

*

*$\pi(x) = \pi(y) \iff \exists g\in G$ s.t. $x = g(y)$, for $x,y \in X$

*If $Z$ is a variety, and $\varphi:X \rightarrow Z$ a regular map, then: $\varphi$ factors through $\pi$ $\iff \varphi(x) = \varphi(g(x)) $ $\forall x\in X, \forall g\in G$
Then, he wrote:
"This prevents us from  doing something stupid like composing the map $\pi$  with  a  map $\rho: Y \rightarrow Y'$  that is  one to
one but not an isomorphism."
I don't quite follow. What's so "stupid" about this composition? Could someone explain in to me, please? Is this maybe referring to the "if and only if" in 2.?
 A: A quotient could naively be defined as an object $Y$ and a morphism $\pi : X\rightarrow Y$ so that $\pi(x) = \pi(x')$ if and only if $x, x'$ are in the same $G$-orbit. Why do we need the second condition?
Well, imagine we take some variety $Y'$ and a morphism $\rho : Y\rightarrow Y',$ where $Y$ is the quotient of $X$ by $G$, and $\rho$ is a bijection but not an isomorphism. Then since $\rho$ is a bijection, $\rho\circ\pi : X\rightarrow Y'$ still obeys the naive definition of quotient: $\rho\circ\pi$ is surjective and only fails to be injective along $G$-orbits. But then $Y, Y'$ are non-isomorphic, and so the naive definition fails to pin down a quotient uniquely. Adding in condition two makes this a true well-defined construction (you can think of it as question 1 is really lifting the `wrong' notion of quotient--it is modeled on the definition of quotient in the category of sets, where surjectivity and injectivity are very useful, but in the category of varieties because bijectivity is no longer equivalent to isomorphism, then that second condition is a much better definition).
