We only add this here so that we can later say that coarse quotients correspond to coarse moduli spaces (or moduli schemes).
Definition 83.6.1. Let S be a scheme and B an algebraic space over S. Let j : R \to U \times _ B U be a pre-relation. A morphism \phi : U \to X of algebraic spaces over B is called a coarse quotient if
\phi is a categorical quotient, and
\phi is an orbit space.
If S = B, U, R are all schemes, then we say a morphism of schemes \phi : U \to X is a coarse quotient in schemes if
\phi is a categorical quotient in schemes, and
\phi is an orbit space.
In many situations the algebraic spaces R and U are locally of finite type over B and the orbit space condition simply means that
for all algebraically closed fields k. See Lemma 83.5.20. If j is also a (set-theoretic) pre-equivalence relation, then the condition is simply equivalent to U(k)/j(R(k)) \to X(k) being bijective for all algebraically closed fields k.
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