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Eilenberg-Mac Lane object

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

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 nA\mathbf{B}^n A obtained from an abelian group object AA by delooping that nn times.

An object that is both nn-truncated as well as nn-connected.

Definition

Definition

Let H\mathbf{H} be an (∞,1)-topos.

For nn \in \mathbb{N} an Eilenberg-MacLane object XX of degree nn

This appears as HTT, def. 7.2.2.1

Remark

If one drops the condition that XX 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\mathbf{H} an (∞,1)-topos, H *\mathbf{H}_* its (∞,1)-category of pointed objects, Disc(H)Disc(\mathbf{H}) the full sub-(∞,1)-category on discrete objects (0-truncated objects) and nn \in \mathbb{N}, write

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

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

  • For n=0n = 0 this establishes an equivalence between the full subcategory on degree 0 Eilenberg-MacLane objects and pointed objects of Disc(H)Disc(\mathbf{H}); moreover, the restriction π 0:Disc(H *)Disc(H *)\pi_0:Disc(\mathbf{H}_*)\to Disc(\mathbf{H}_*) is equivalent to the identity.

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

  • For n2n \geq 2 this establishes an equivalence between the full subcategory on degree nn Eilenberg-MacLane objects and the category of commutative group objects in Disc(H)Disc(\mathbf{H}).

Proof

This is HTT, prop. 7.2.2.12.

Definition

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

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

is the degree nn-Eilenberg-MacLane object of AA.

Proposition

We have that

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

is the nn-fold delooping of the discrete group object AA.

Proof

check

The categorical homotopy groups are defined in terms of the canonical powering of H\mathbf{H} over ∞Grpd

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

For fixed ∞-groupoid KK this

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

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

By the above proposition on the equivalence between Eilenberg-MacLane objects and group objects, this identifies ΩK(A,n)K(A,n1)\Omega K(A,n) \simeq K(A,n-1).

Examples

In TopTop

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

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

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

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

Under the Dold–Kan correspondence

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

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

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

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

Under the equivalence

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

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

Cohomology

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

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

H n(X,A):=H(X,K(A,n)):=π 0H(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

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)

Revised on January 16, 2011 08:02:22 by Toby Bartels (173.190.149.57)