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Eilenberg-Mac Lane object

Idea

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 $B^{n}A$ obtained from an abelian group object $A$ by delooping that $n$ 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 $(\mathrm{\infty }\mathrm{,1})$-topos is presented by a model category of simplicial sheaves.

Under the Dold–Kan correspondence

$$N:\mathrm{SimpAb}\overleftarrow{\to }{\mathrm{Ch}}_{+}:\Gamma$$N : SimpAb \stackrel{\leftarrow}{\to} Ch_+ : \Gamma

chain complexes $A[n]$ of abelian groups concentrated in degree $n$ map into simplicial sets

$$K(A,n):=\Gamma (A[n])$$K(A,n) := \Gamma(A[n])

and these to the corresponding constant simplicial sheaves (on whatever site) that we denote by the same symbol, for convenience.

Regarding the simplicial sheaf $K(A,n)$ as an object of the $(\mathrm{\infty }\mathrm{,1})$-topos presented by the given model structure on simplicial sheaves means regarding them as the $\mathrm{\infty }$-stacks that they correspond to under $\mathrm{\infty }$-stackification. These are then the Eilenberg–Mac Lane objects in our (∞,1)-topos.

Cohomology

From the general nonsense at cohomology and comparing with the case Eilenberg–Mac Lane spaces, we see that for $H$ our $(\mathrm{\infty }\mathrm{,1})$-topos, for $H$ its homotopy category and for $X\in H$ any object, the “ordinary” degree $n$-cohomology of $X$ is

$$H^n(X,\mathbb{Z}):=H(X,K(\mathbb{Z},n)).$$H^n(X, \mathbb{Z}) := H(X, K(\mathbb{Z},n)) \,.

References

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• Jardine, Fields Lectures: Simplicial Presheaves (pdf)