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 $\mathbf{B}^n A$ 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 $\mathbf{H}$ be an (∞,1)-topos.

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

• a pointed object $* \to X \in \mathbf{H}$

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

$$\pi_n : \mathbf{H}_* \to Disc(\mathbf{H}_*)$$

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 $Disc(\mathbf{H})$; moreover, the restriction $\pi_0:Disc(\mathbf{H}_*)\to Disc(\mathbf{H}_*)$ 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 $Disc(\mathbf{H})$.

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

Proof

This is HTT, prop. 7.2.2.12.

Definition

For $\mathbf{H}$ 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 $A \in Disc(\mathbf{H})$ an (abelian) group object we say that

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

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

Proposition

We have that

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

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 $\mathbf{H}$ over ∞Grpd

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

For fixed ∞-groupoid $K$ this

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

preserves $(\infty,1)$-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 $Top$

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

In $(\infty,1)$-sheaf $(\infty,1)$-toposes

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

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

Under the Dold?Kan correspondence?

$$N : sAb \stackrel{\leftarrow}{\to} Ch_+ : \Gamma$$

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

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

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

Under the equivalence

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

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

Cohomology

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

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

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

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

• Rick Jardine, Fields Lectures: Simplicial Presheaves (pdf)

Discussion in equivariant homotopy theory (see also at Bredon cohomology) is in

• {Lewis92} L. G. Lewis, Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems, Topology Appl., 48 (1992), no. 1, pp. 25–61.

Formalization in homotopy type theory is in

• Dan Licata, Eric Finster, Eilenberg-MacLane spaces in homotopy type theory, LICS 2014 (pdf text, Agda HoTT code, web discussion)