Lemma 78.25.2. Notation and assumption as in Lemma 78.21.1. The morphism of quotient stacks
[f] : [U/R] \longrightarrow [U'/R']
is an equivalence if and only if
(U, R, s, t, c) is the restriction of (U', R', s', t', c') via f : U \to U', and
the map
\xymatrix{ U \times _{f, U', t'} R' \ar[r]_-{\text{pr}_1} \ar@/^3ex/[rr]^ h & R' \ar[r]_{s'} & U' }
is a surjection of sheaves.
Part (2) holds for example if \{ h : U \times _{f, U', t'} R' \to U'\} is an fppf covering, or if f : U \to U' is a surjection of sheaves, or if \{ f : U \to U'\} is an fppf covering.
Proof.
We already know that part (1) is equivalent to fully faithfulness by Lemma 78.25.1. Hence we may assume that (1) holds and that [f] is fully faithful. Our goal is to show, under these assumptions, that [f] is an equivalence if and only if (2) holds. We may use Stacks, Lemma 8.4.8 which characterizes equivalences.
Assume (2). We will use Stacks, Lemma 8.4.8 to prove [f] is an equivalence. Suppose that T is a scheme and x' \in \mathop{\mathrm{Ob}}\nolimits ([U'/R']_ T). There exists a covering \{ g_ i : T_ i \to T\} such that g_ i^*x' is the image of some element a'_ i \in U'(T_ i), see Lemma 78.24.1. Hence we may assume that x' is the image of a' \in U'(T). By the assumption that h is a surjection of sheaves, we can find an fppf covering \{ \varphi _ i : T_ i \to T\} and morphisms b_ i : T_ i \to U \times _{g, U', t'} R' such that a' \circ \varphi _ i = h \circ b_ i. Denote a_ i = \text{pr}_0 \circ b_ i : T_ i \to U. Then we see that a_ i \in U(T_ i) maps to f \circ a_ i \in U'(T_ i) and that f \circ a_ i \cong _{T_ i} h \circ b_ i = a' \circ \varphi _ i, where \cong _{T_ i} denotes isomorphism in the fibre category [U'/R']_{T_ i}. Namely, the element of R'(T_ i) giving the isomorphism is \text{pr}_1 \circ b_ i. This means that the restriction of x to T_ i is in the essential image of the functor [U/R]_{T_ i} \to [U'/R']_{T_ i} as desired.
Assume [f] is an equivalence. Let \xi ' \in [U'/R']_{U'} denote the object corresponding to the identity morphism of U'. Applying Stacks, Lemma 8.4.8 we see there exists an fppf covering \mathcal{U}' = \{ g'_ i : U'_ i \to U'\} such that (g'_ i)^*\xi ' \cong [f](\xi _ i) for some \xi _ i in [U/R]_{U'_ i}. After refining the covering \mathcal{U}' (using Lemma 78.24.1) we may assume \xi _ i comes from a morphism a_ i : U'_ i \to U. The fact that [f](\xi _ i) \cong (g'_ i)^*\xi ' means that, after possibly refining the covering \mathcal{U}' once more, there exist morphisms r'_ i : U'_ i \to R' with t' \circ r'_ i = f \circ a_ i and s' \circ r'_ i = \text{id}_{U'} \circ g'_ i. Picture
\xymatrix{ U \ar[d]^ f & & U'_ i \ar[ll]^{a_ i} \ar[ld]_{r'_ i} \ar[d]^{g'_ i} \\ U' & R' \ar[l]_{t'} \ar[r]^{s'} & U' }
Thus (a_ i, r'_ i) : U'_ i \to U \times _{g, U', t'} R' are morphisms such that h \circ (a_ i, r'_ i) = g'_ i and we conclude that \{ h : U \times _{g, U', t'} R' \to U'\} can be refined by the fppf covering \mathcal{U}' which means that h induces a surjection of sheaves, see Topologies on Spaces, Lemma 73.7.5.
If \{ h\} is an fppf covering, then it induces a surjection of sheaves, see Topologies on Spaces, Lemma 73.7.5. If U' \to U is surjective, then also h is surjective as s has a section (namely the neutral element e of the groupoid in algebraic spaces).
\square
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