The notion of Eilenberg–Mac Lane object in an (∞,1)-topos or stable (∞,1)-category generalizes the notion of Eilenberg–Mac Lane space from the (∞,1)-topos Top of topological spaces or the stable (∞,1)-category of spectra:
it is an object obtained from an abelian group object by delooping that times.
The standard model for these Eilenberg–Mac Lane objects are nothing but the familiar Eilenberg–Mac Lane sheaves of fame in abelian sheaf cohomology.
For recall that every -topos is presented by a model category of simplicial sheaves.
Under the Dold–Kan correspondence
chain complexes of abelian groups concentrated in degree map into simplicial sets
and these to the corresponding constant simplicial sheaves (on whatever site) that we denote by the same symbol, for convenience.
Regarding the simplicial sheaf as an object of the -topos presented by the given model structure on simplicial sheaves means regarding them as the -stacks that they correspond to under -stackification. These are then the Eilenberg–Mac Lane objects in our (∞,1)-topos.
From the general nonsense at cohomology and comparing with the case Eilenberg–Mac Lane spaces, we see that for our -topos, for its homotopy category and for any object, the “ordinary” degree -cohomology of is
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