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Proof. Let us check the claim is the case of stacks. 15. Let us describe the presheaf Isom???? (????1 , ????2 ).

We refer to the exposition [39] for a complete discussion and proofs. But what kind of object is an algebraic space? 51 a Zariski cover by an ´etale cover and keep only the functorial way of thinking. Algebraic spaces were introduced by Artin, precisely for that purpose. Our presentation will be very sketchy since algebraic spaces are indeed algebraic stacks, precisely Deligne-Mumford stacks. The definition of algebraic spaces modelled on the categorical definition of (separated) schemes is as follows16 .

Indeed let ???? = ???? ???????? . Then by ii) ???? = ???? ×???? ???? is a scheme equipped with two projections ???? ????0 ????1 GG ???? . 51 ???? ⊂ ???? × ???? is the graph of a closed equivalence relation. But now the projections are not local immersions, they are only ´etale. ∐ This is readily seen from ii), indeed the restriction of ????1 to the open set ????−1 (???? ) = ???? 1 ???? ???????? ×???? ???????? is ´etale over ???????? . One can conclude that the sheaf ???? is the quotient, in a categorical sense, ???? = ????/????, meaning that the ´etale sheaf ???? is a coequalizer G ????.

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