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algebraic-geometry--vakil--foundations-of-algebraic-geometry.tex
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algebraic-geometry--vakil--foundations-of-algebraic-geometry.tex
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Picture a set $A$ on the left and a set $B$ on the right.
A {\defn function} or {\defn map} $f: A \to B$ attaches to every element of $A$ exactly one directed edge, pointing to an element
of $B$.
$f$ \defn{factors through} a set $C$ if there exists $g: A \to C$ and $h: C \to B$ such that $f = h \circ g$.
\begin{claim}
$f: A \to B$ factors through $C$ if and only if $|C| \geq \min\{ ~|A|, ~|B|~ \}$.
\end{claim}
\begin{proof}
Cases:
\begin{enumerate}
\item $|A| < |B|$
If $|C| < |A|$ then some $c \in C$ will be hit by more than one $a \in A$.
\end{enumerate}
\end{proof}
\section{Product}
product map is given by intersection (which necessarily has one element) of preimages of projection maps?
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