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For an algebraic space $Z$ with the action of multiplicative group scheme $G_{m}$ one can define the attractor space $Z^{+}$ as the functor which sends a scheme $S$ to the set $Map(S \times A^{1},Z)^{G_{m}}$, where the action of $G_{m}$ on affine space $A^{1}$ is given by dilations. (see the paper https://arxiv.org/pdf/1308.2604.pdf).

My question is the following: suppose now that $Z$ is not an algebraic space but some stack (one can assume that it is algebraic). Using for example the results of the paper https://perso.univ-rennes1.fr/matthieu.romagny/articles/group_actions.pdf one can define the action of $G_{m}$ on $Z$. Does there exist a reasonable notion of the attractor stack for the corresponding action of $G_{m}$ on $Z$?

Of course one can consider again the functor which sends $S$ to $Map(S \times A^{1},Z)^{G_{m}}$ but does this set can be endowed with a groupoid structure? (and what does $Map(S \times A^{1},Z)^{G_{m}}$ actually mean in the case of stacks)?

Actually I am just asking an easy question about the existence of the attractor stack, not about its properties.

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    $\begingroup$ If $Z$ is a stack then $Map(-,Z)$ is a groupoid-valued functor, since stacks form a 2-category. The definition of $Z^+$ still makes sense in that context, but everything is now happening in a 2-category. In particular $G_m$-equivariance is not a property of a map but an additional structure on it. $\endgroup$ Commented Apr 19, 2017 at 1:01

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