Groupoids in schemes, relations and etale equivalence relations

Fix a base scheme $S$ (Schemes and morphisms over a base). A groupoid in $S$-schemes is a tuple $(U,R,s,t,c,e,i)$ consisting of $S$-schemes $U$ and $R$ and morphisms of $S$-schemes $s,t\text{\hspace{0.17em}⁣}:R\to U$ (source and target), $c\text{\hspace{0.17em}⁣}:R{\times }_{s,U,t}R\to R$ (composition), $e\text{\hspace{0.17em}⁣}:U\to R$ (identity) and $i\text{\hspace{0.17em}⁣}:R\to R$ (inverse), subject to the usual identities of a small groupoid. Here $c$ is defined on composable pairs $(r_1,r_2)$ with $s(r_1)=t(r_2)$ and $c(r_1,r_2)$ is the composite "$r_2$ first, then $r_1$", with $s(c(r_1,r_2))=s(r_2)$ and $t(c(r_1,r_2))=t(r_1)$; the identities are $s\circ e=t\circ e={\mathrm{i}\mathrm{d}}_U,s\circ c=s\circ {\mathrm{p}\mathrm{r}}_2,t\circ c=t\circ {\mathrm{p}\mathrm{r}}_1,$ $c\circ (c\times {\mathrm{i}\mathrm{d}}_R)=c\circ ({\mathrm{i}\mathrm{d}}_R\times c),c\circ ({\mathrm{i}\mathrm{d}}_R,e\circ s)=c\circ (e\circ t,{\mathrm{i}\mathrm{d}}_R)={\mathrm{i}\mathrm{d}}_R,$ $s\circ i=t,t\circ i=s,c\circ ({\mathrm{i}\mathrm{d}}_R,i)=e\circ t,c\circ (i,{\mathrm{i}\mathrm{d}}_R)=e\circ s,$ where the fibre products and projections are those of Fibre product of schemes and all morphisms are morphisms of $S$-schemes (Morphisms of schemes); associativity is stated on the triple fibre product $R{\times }_{s,U,t}R{\times }_{s,U,t}R$, where $c\times {\mathrm{i}\mathrm{d}}_R$ and ${\mathrm{i}\mathrm{d}}_R\times c$ are formed using the source/target identifications. A groupoid in $S$-schemes is precisely a groupoid object in the category of $S$-schemes in the sense of these diagrams, and its functor of points on the category of $S$-schemes is a groupoid-valued functor.

With $j=(t,s)\text{\hspace{0.17em}⁣}:R\to U{\times }_{S}U$, the groupoid is a relation when $j$ is a monomorphism (Monomorphism and epimorphism by left and right cancellation); then $j$ presents $R$ as a subobject of $U{\times }_{S}U$, and the groupoid axioms exhibit a reflexive ($e$), symmetric ($i$) and transitive ($c$) set-theoretic relation on the points of $U$ in the sense of Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$. The groupoid is an equivalence relation on $U$ over $S$ when it is a relation, and an etale equivalence relation when in addition $s$ and $t$ are etale (Étale morphism of schemes).

Restriction is well defined as follows. Let $g\text{\hspace{0.17em}⁣}:U^{\mathrm{\prime }}\to U$ be a morphism of $S$-schemes and form $R^{\mathrm{\prime }}=R{\times }_{U{\times }_{S}U}(U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}),$ the fibre product along $j$ and $g\times g$, with its two projections ${\mathrm{p}\mathrm{r}}_R$ and ${\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$; set $t^{\mathrm{\prime }}={\mathrm{p}\mathrm{r}}_1\circ {\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$ and $s^{\mathrm{\prime }}={\mathrm{p}\mathrm{r}}_2\circ {\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$. The map $e^{\mathrm{\prime }}$ sends $u^{\mathrm{\prime }}$ to $(e(g(u^{\mathrm{\prime }})),(u^{\mathrm{\prime }},u^{\mathrm{\prime }}))$, where $(u^{\mathrm{\prime }},u^{\mathrm{\prime }})$ is the diagonal $U^{\mathrm{\prime }}\to U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}$; the map $i^{\mathrm{\prime }}$ sends $(r,(u_1^{\mathrm{\prime }},u_2^{\mathrm{\prime }}))$ to $(i(r),(u_2^{\mathrm{\prime }},u_1^{\mathrm{\prime }}))$; and $c^{\mathrm{\prime }}$ sends a composable pair $((r_1,(u_1^{\mathrm{\prime }},u_2^{\mathrm{\prime }})),(r_2,(u_2^{\mathrm{\prime }},u_3^{\mathrm{\prime }})))$ to $(c(r_1,r_2),(u_1^{\mathrm{\prime }},u_3^{\mathrm{\prime }}))$. All three are defined by the universal property of the relevant fibre products, and the groupoid identities for $(U^{\mathrm{\prime }},R^{\mathrm{\prime }},s^{\mathrm{\prime }},t^{\mathrm{\prime }},c^{\mathrm{\prime }},e^{\mathrm{\prime }},i^{\mathrm{\prime }})$ follow from those for $(U,R,s,t,c,e,i)$ after applying the universal property; this tuple is the restriction $R{\mathrm{\mid }}_{U^{\mathrm{\prime }}}$ of the groupoid along $g$. If $j$ is a monomorphism, then so is $j^{\mathrm{\prime }}=(t^{\mathrm{\prime }},s^{\mathrm{\prime }})$, because a monomorphism is stable under base change in any category with fibre products: given two morphisms into the fibre product with equal composites to $U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}$ and to $R$, the universal property of the fibre product makes them equal. Hence restricting an equivalence relation along an arbitrary morphism of $S$-schemes yields an equivalence relation. Restriction of the etale property needs $g$ etale and is recorded separately in Restriction of an etale equivalence relation: its local flatness, finite-presentation and fibre arguments establish the required stability without a choice assumption.