nLab
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.

An object that is both $n$-truncated as well as $n$-connected.

Definition

Definition

Let $H$ be an (∞,1)-topos.

For $n\in \mathbb{N}$ an Eilenberg-MacLane object $X$ of degree $n$

• a pointed object $*\to X\in H$

• which is both $n$-connective as well as $n$-truncated.

This appears as HTT, def. 7.2.2.1

Remark

If one drops the condition that $X$ has a global point, then this is the definition of ∞-gerbes.

Properties

The next proposition asserts that Eilenberg-MacLane objects defined this way are shifted (∞,1)-categorical group objects:

Proposition

For $H$ an (∞,1)-topos, $H_{*}$ its (∞,1)-category of pointed objects, $\mathrm{Disc}(H)$ the full sub-(∞,1)-category on discrete objects (0-truncated objects) and $n\in \mathbb{N}$, write

$${\pi }_n:H_{*}\to \mathrm{Disc}(H_{*})$$\pi_n : \mathbf{H}_* \to Disc(\mathbf{H}_*)

for the (∞,1)-functor that assigns the $n$-th categorical homotopy groups.

• For $n=0$ this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and pointed objects of $\mathrm{Disc}(H)$; moreover, the restriction ${\pi }_0:\mathrm{Disc}(H_{*})\to \mathrm{Disc}(H_{*})$ is equivalent to the identity.

• For $n=1$ this establishes an equivalence between the full subcategory on degree 1 Eilenberg-MacLane objects and the category of group objects in $\mathrm{Disc}(H)$.

• For $n\ge 2$ this establishes an equivalence between the full subcategory on degree $n$ Eilenberg-MacLane objects and the category of commutative group objects in $\mathrm{Disc}(H)$.

Proof

This is HTT, prop. 7.2.2.12.

Definition

For $H$ an (∞,1)-topos and $n\in \mathbb{N}$ write $K(-,n)$ for the homotopy inverse to the equivalence induced by ${\pi }_n$ by the above proposition. For $A\in \mathrm{Disc}(H)$ an (abelian) group object we say that

$$K(A,n)\in H$$K(A,n) \in \mathbf{H}

is the degree $n$-Eilenberg-MacLane object of $A$.

Proposition

We have that

$$K(A,n)\simeq B^{n}A$$K(A,n) \simeq \mathbf{B}^n A

is the $n$-fold delooping of the discrete group object $A$.

Proof

check

The categorical homotopy groups are defined in terms of the canonical powering of $H$ over ∞Grpd

$$(-{)}^{(-)}:\mathrm{\infty }\mathrm{Grpd}\times H\to H.$$(-)^{(-)} : \infty Grpd \times \mathbf{H} \to \mathbf{H} \,.

For fixed ∞-groupoid $K$ this

$$(-{)}^K:H\to H.$$(-)^{K} : \mathbf{H} \to \mathbf{H} \,.

preserves $(\mathrm{\infty },1)$-limits and hence pullbacks. It follows that the categorical homotopy groups of the loop space object $\Omega K(A,n)$ are those of $K(A,n)$, shifted down by one degree.

By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies $\Omega K(A,n)\simeq K(A,n-1)$.

Examples

In $\mathrm{Top}$

In the archetypical (∞,1)-topos Top$\simeq$ ∞Grpd the notion of Eilenberg-MacLane object reduces to the traditional notion of Eilenberg-MacLane space.

In $(\mathrm{\infty },1)$-sheaf $(\mathrm{\infty },1)$-toposes

Recall that an (∞,1)-sheaf/∞-stack (∞,1)-topos $H={\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)$ may be presented by the model structure on simplicial sheaves on $C$.

In terms of this model the Eilenberg-Mac Lane objects $K(A,n)\in H$ (for abelian $A$) are the Eilenberg-MacLane sheaves of abelian sheaf cohomology theory.

Under the Dold–Kan correspondence

$$N:\mathrm{sAb}\overleftarrow{\to }{\mathrm{Ch}}_{+}:\Gamma$$N : sAb \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 the site $C$, that we denote by the same symbol, for convenience.

Under the equivalence

$$H={\mathrm{Sh}}_{(\mathrm{\infty },1)}(C)\simeq (\mathrm{sSh}(C{)}_{\mathrm{loc}}{)}^{\circ }$$\mathbf{H} = Sh_{(\infty,1)}(C) \simeq (sSh(C)_{loc})^\circ

of $H$ with the Kan complex-enriched full subcategory of $\mathrm{sSh}(C)$ on fibrant cofibrant objects, this identifies the fibrant reeplacement – the ∞-stackification – of $\Gamma (A[n])$ with the Eilenberg-MacLane object in $H$.

Cohomology

The notion of cohomology in the (∞,1)-topos $H$ with coefficients in an object $𝒜\in H$ is often taken to be restricted to the case where $𝒜$ is an Eilenberg-MacLane object.

For $A\in \mathrm{Disc}(A)$ an abelian group object, and $n\in \mathbb{N}$, the degree $n$-cohomology of an object $X\in H$ is the cohomology with coefficients in $K(A,n)$:

$$H^n(X,A):=H(X,K(A,n)):={\pi }_{0}H(X,K(A,n)).$$H^n(X, A) := H(X, K(A,n)) := \pi_0 \mathbf{H}(X, K(A,n)) \,.

References

The general discussion of Eilenberg-MacLane objects is in section 7.2.2 of

• Jacob Lurie, Higher Topos Theory

For a discussion of Eilenberg-MacLane objects in the context of the model structure on simplicial presheaves see top of page 4 of

• Jardine, Fields Lectures: Simplicial Presheaves (pdf)