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1. • CommentRowNumber1.
   • CommentAuthorZhen Huan
   • CommentTimeJun 29th 2022
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   Author: Zhen Huan
   Format: TextI have two questions about plus construction. 1) In Nikolaus and Schweigert's paper "Equivariance in Higher Geometry", the plus construction is defined on presheaves over the category of manifolds and is given explicitly. Is it a special case of the plus construction given in Lurie's Higher Topos theory, which is on presheaves over any small site with a Grothendieck topology? 2) Can the plus construction in NS' paper be defined on presheaves over the category of Lie groupoids? Somebody said yes. But this point is not proved explicitly in NS' paper. Thanks.

   I have two questions about plus construction.
   
   1) In Nikolaus and Schweigert's paper "Equivariance in Higher Geometry", the plus construction is defined on presheaves over the category of manifolds and is given explicitly. Is it a special case of the plus construction given in Lurie's Higher Topos theory, which is on presheaves over any small site with a Grothendieck topology?
   
   2) Can the plus construction in NS' paper be defined on presheaves over the category of Lie groupoids? Somebody said yes. But this point is not proved explicitly in NS' paper.
   
   Thanks.
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 29th 2022
   • (edited Jun 29th 2022)
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   Author: Urs
   Format: MarkdownItexThe following is not exactly an answer to your questions, but maybe relevant nonetheless: On 1): Without looking into details of the formulas given, one can say that the iteration of the plus constructions must agree, up to equivalence, or else there would be something wrong in their definitions. Namely, the iterated plus construction is meant to model the $\infty$-stackification (the 2-stackification in the case of NS's article) in either case. On 2): Just to remark that passing from the site of smooth manifolds to that of action groupoids by finite groups gives the "singular-smooth" homotopy theory which we discuss in *[[schreiber:Proper Orbifold Cohomology|Proper Orbifold Cohomology]]* and in *[[schreiber:Equivariant principal infinity-bundles|Equivariant principal $\infty$-bundles]]*. There we discuss $\infty$-stackification not via the plus-construction but via special-purpose tools that apply thanks to the special nature of the site of smooth manifolds. (Essentially: Once a simplicial presheaf satisfies descent over just Cartesian spaces ,which is a rather more simple condition, then its stackification over all manifolds may be computed by evaluating it on Čech nerves of good open covers, which is typically exactly what one wants to compute anyways. The analogous statement holds also for the equivariant version over a site of finite action groupoids.)

   The following is not exactly an answer to your questions, but maybe relevant nonetheless:

   On 1): Without looking into details of the formulas given, one can say that the iteration of the plus constructions must agree, up to equivalence, or else there would be something wrong in their definitions. Namely, the iterated plus construction is meant to model the $\infty$-stackification (the 2-stackification in the case of NS’s article) in either case.

   On 2): Just to remark that passing from the site of smooth manifolds to that of action groupoids by finite groups gives the “singular-smooth” homotopy theory which we discuss in Proper Orbifold Cohomology and in Equivariant principal $\infty$-bundles. There we discuss $\infty$-stackification not via the plus-construction but via special-purpose tools that apply thanks to the special nature of the site of smooth manifolds. (Essentially: Once a simplicial presheaf satisfies descent over just Cartesian spaces ,which is a rather more simple condition, then its stackification over all manifolds may be computed by evaluating it on Čech nerves of good open covers, which is typically exactly what one wants to compute anyways. The analogous statement holds also for the equivariant version over a site of finite action groupoids.)

3. • CommentRowNumber3.
   • CommentAuthorZhen Huan
   • CommentTimeJun 30th 2022
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   Author: Zhen Huan
   Format: TextThanks a lot for the helpful answers and the latest paper.

   Thanks a lot for the helpful answers and the latest paper.