Construction Let denote the -topos (or some localization of it, for example the etale topos).
Let be a commutative grouplike monoid. (This is, by toposic Koszul duality, equivalent to the data of a connective spectrum object in , which in turn is a sheaf with coefficients in connective spectra). is an example of such an object. This is spelled in detail under Multiplication on flat G_m.
The Cartier Dual of is given by To see that this again has the structure of a group stack, we can view it in terms of connective spectra
where lowercase denotes mapping spectrum, and now the RHS manifests lifts to an group.
Remark: we further unpack explicitly the Cartier dual in Cartier dual is lattice of strict characters
The equivalence that already appeared above is interpolated by an tower of equivalences
Map_{CGp(\mathfrak X)}(G, \mathbb G_m) \simeq Map_{CGp(\mathfrak X)}(BG, B\mathbb G_m) \simeq \cdots$$ **Why**: This is using that the mapsGp_{E_n} (\mathfrak X)\xrightarrow{B^n} \mathfrak X_*
are fully faithful, and furthermore, we have intermediate fully faithful embeddings $$nGp(\mathfrak X) \xrightarrow{B^{n-k}} kGp (\mathfrak X)$$ We used this above in the case $k = (n-1)$, then there's a tower of functors indexeed by $n\in \mathbb N^{\downarrow}$ , then we take the $n\to \infty$ limit. Now we get, at each delooping level $n$, evaluation maps $$B^nG \times G^\vee \simeq B^nG \times Map(B^nG, B^n\mathbb G_m) \to B^n\mathbb G_m$$ This in turn defines a strict n-Brauer class on $B^nG \times G^\vee$, Fourier-Mukai integration against this class (viewed as a module) then defines a map $$nQCoh(G^\vee) \to nQCoh(B^nG) := nRep(G)$$ **Remark** This looks a bit like a theory of completions: it's saying using the multiplicative structure on $G^\vee$, we can construct an equivalence between sheaves on the underlying scheme $G^\vee$, and representations of the dual group $G$. Hence, the dual group is like a Koszul dual, a group stack of infinitesimal symmetries.