Definition
Let \(G\) be a sheaf in groups over \(S\) a scheme and let \(\mathcal M\) be a \(G\)-stack over \(S\). For any stack \(\mathcal Y\) over \(S\), there is a trivial \(G\)-stack \((\mathcal Y, \pr_2)\), which gives a \(2\)-functor \(\iota:\Stack \to G\text{-}\Stack\).
- A stack of fixed points \(\mathcal M^G\) is a stack that \(2\)-represents the \(2\)-functor \(\mathsf{Stack}^{\op} \to \Cat\) defined by
\[F(\mathcal N) = \mathcal{H}om_{G-\mathsf{Stack}}(\iota(\mathcal N), \mathcal M)\] (the latter is the stack of Definition 2.1 (iii) in Romagny and \(\Cat\) is the 2-category of categories. It is the “stack of 1-morphisms and 2-morphisms between \(\iota(\mathcal N)\) and \(\mathcal M\) as \(G\)-stacks.”) That is, \(\mathcal M^G(X)\) is the set of \(G\)-equivariant maps from \(X\) to \(\mathcal M\) with respect to the trivial \(G\) action on \(X\). Note that a \(G\)-equivariant morphism \(S\to \mathcal M\) is a pair \((f,\sigma)\) consisting of a morphism \(f:S\to \mathcal M\) and a choice of \(2\)-morphism \(\sigma\) making the diagram
\begin{equation*}\(2\)-commute. Here \(\mu\) and \(\nu\) are simply the action maps. More concretely, if we test this against some test scheme \(T\), then for an object \((g,x)\in G(T)\times S(T)\) this \(2\)-commutativity is simply the requirement that \[\sigma_{(g,x)}:f(\mu(g, x)) \xrightarrow{\sim}\nu\big((\id_G\times f)(g,x)\big),\] or more helpfully, \[\sigma_{(g,x)}:f(g\cdot x) \xrightarrow{\sim}g\cdot f(x).\]