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More precisely, consider a Segal groupoid $X_*$ in an infinity category of derived schemes : dSch

In Toen's note, 'Derived Algebraic Geometry', he defines a 1-Artin stacks as a homotopy colimits of a smooth Segal groupoid $X_*$.

smooth means that morphism $X_1 \to X_0$ from groupoid structure are all smooth.

Then I'm curious about if we replace smooth morphism by open inclusion of derived scheme, then the colimit of the Segal groupoid $X_*$ again lies in dSch or not?

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  • $\begingroup$ You need to take Zariski local isomorphism rather than open inclusion, or the groupoid will be trivial. Then you need to impose constraints on the matching objects to avoid stacky structure - see for instance Prop 4.1 and Rem 1.27 of "Presenting higher stacks as simplicial schemes". $\endgroup$ – Jon Pridham Feb 16 '18 at 9:13

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