nLab
Moore complex

Idea

The Moore complex of a simplicial group – also known in its normalized version as the complex of normalized chains – is a chain complex whose differential is built from the face maps of the simplicial group.

The operation of forming the Moore complex of chains of a simplicial group is one part of the Dold-Kan correspondence that relates simplicial (abelian) groups and chain complexes.

Recall that a simplicial group $G$, being in particular a Kan complex, may be thought of, in the sense of the homotopy hypothesis, as a combinatorial space equipped with a group structure. The Moore complex of $G$ is a chain complex

• whose $n$-cells are the ”$n$-disks with basepoint on their boundary” in this space, with the basepoint sitting on the identity element of the space;

• the boundary map on which acts literally like a boundary map should: it sends an $n$-disk to its boundary, read as an $(n-1)$-disk whose entire boundary is concentrated at the identity point.

This is entirely analogous to how a crossed complex is obtained from a strict ω-groupoid. In fact it is a special case of that, as discussed at Dold-Kan correspondence in the section on the nonabelian version.

Definition

On general simplicial groups

Definition

Given a simplicial group $G$, the $\mathbb{N}$-graded chain complex complex $((NG{)}_{•},\partial )$ of (possibly nonabelian) groups is

• in degree $n$ the joint kernel

  $$(NG{)}_n=\bigcap _{i=1}^n\ker d_i^n$$(N G)_n=\bigcap_{i=1}^{n}ker\,d_i^n

  of all face maps except the 0-face

• with differential given by the remaining 0-face

  $${\partial }_n:=d_0^n{\mid }_{(NG{)}_n}:(NG{)}_n\to (NG{)}_{n-1}$$\partial_n := d_0^n|_{(N G)_n} : (N G)_n \rightarrow (N G)_{n-1}

Remark

Equivalently one can take the joint kernel of all but the $n$-face map and take that remaining face map, $d_n^n$, to be the differential.

It is important to note, and simple to prove, that $NG$ is a normal complex of groups, so that it is easy to take the homology of the complex, even though the groups involved may be non-abelian.

Remark

We may think of the elements of the complex $NG$ in degree $k$ as being $k$-dimensional disks in $G$:

• an element in degree 1 element $g\in N{G}_1$ is a 1-disk

  $$1\stackrel{g}{\to }\partial g,$$1 \stackrel{g}{\to} \partial g \,,

• an element $h\in N{G}_2$ is a 2-disk

  $$\begin{array}{ccc}& & 1\\ & {}^1\nearrow & {\Downarrow }^h& {\searrow }^{\partial h}\\ 1& & \stackrel{1}{\to }& & 1\end{array},$$\array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,

• a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-sphere

  $$\begin{array}{ccc}& & 1\\ & {}^1\nearrow & {\Downarrow }^h& {\searrow }^{\partial h=1}\\ 1& & \stackrel{1}{\to }& & 1\end{array},$$\array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,

etc.

Lemma

For every simplicial group $G$ the complex $(NG{)}_{•}$ is a normal complex of groups.

On abelian simplicial groups

Let here $A$ be a simplicial abelian group. Then $(NA{)}_{•}\in {\mathrm{Ch}}_{•}^{+}$ is an ordinary connective chain complex in the abelian category Ab.

There are two other chain complexes naturally associated with $A$:

Definition

The alternating face map complex $CA$ of $A$ is

• in degree $n$ given by the group $A_n$ itself

  $$(CA{)}_n:=A_n$$(C A)_n := A_n

• with the differential given by the alternating sum of face maps (using the abelian group structure on $A$)

  $${\partial }_n:=\sum _{i=0}^n(-1{)}^i{d}_i:(CA{)}_n\to (CA{)}_{n-1}.$$\partial_n := \sum_{i = 0}^n (-1)^i d_i : (C A)_n \to (C A)_{n-1} \,.

Definition

The complex modulo degeneracies, $(CA)/D(A)$ is the complex

• which in degree $n$ is given by the quotient group obtained by dividing out the group

  $$D{A}_n:=⟨{\cup }_i{\sigma }_i(A_{n-1})⟩$$D A_n := \langle \cup_i \sigma_i(A_{n-1}) \rangle

  generated by the degenerate elements in $A_n$

  $$((CA)/D(A){)}_n:=A_n/D(A_n)$$((C A)/D(A))_n := A_n / D(A_n)

• with differential being the induced action of the alternating sum of faces on the quotient.

Lemma

This is indeed well defined in that the alternating face boundary map satisfies $\partial \circ \partial =0$ in $C_{•}(A)$ and restricts to a boundary map on the degenerate subcomplex $\partial :A_n{\mid }_{s(A_{n-1})}\to A_{n-1}{\mid }_{s(A_{n-2})}$.

Proof

For the first statement one checks

$$\begin{array}{rl}{\partial }_n{\partial }_{n+1}& =\sum _{i,j}(-1{)}^{i+j}{d}_i\circ d_j\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j+\sum _{i<j}(-1{)}^{i+j}{d}_i\circ d_j\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j+\sum _{i<j}(-1{)}^{i+j}{d}_{j-1}\circ d_i\\ & =\sum _{i\ge j}(-1{)}^{i+j}{d}_i\circ d_j-\sum _{i\le k}(-1{)}^{i+k}{d}_k\circ d_i\\ & =0\end{array}$$\begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned}

using the simplicial identity $d_i\circ d_j=d_{j-1}\circ d_i$ for $i<j$.

Similarly, using the mixed simplicial identities we find that for $s_j(a)\in A_n$ a degenerate element, its boundary is

$$\begin{array}{rl}\sum _i(-1{)}^i{d}_i{s}_j(a)& =\sum _{i<j}(-1{)}^i{s}_{j-1}{d}_i(a)+\sum _{i=j,j+1}(-1{)}^{i}a+\sum _{i>j+1}(-1{)}^i{s}_j{d}_{i-1}(a)\\ & =\sum _{i<j}(-1{)}^i{s}_{j-1}{d}_i(a)+\sum _{i>j+1}(-1{)}^i{s}_j{d}_{i-1}(a)\end{array}$$\begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned}

which is again a combination of elements in the image of the degeneracy maps.

Properties

Normalization

Let $A$ be a simplicial abelian group.

Proposition

There is a splitting

$$C_{•}(A)\simeq N_{•}(A)\oplus D_{•}(A)$$C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)

where the first summand is naturally isomorphic to the Moore complex as defined above.

Explicitly,

Proposition

The evident composite of natural morphisms

$$NA\stackrel{i}{\hookrightarrow }A\stackrel{p}{\to }(CA)/D(A)$$N A \stackrel{i}{\hookrightarrow} A \stackrel{p}{\to} (C A)/D(A)

(inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.

This appears as theorem 2.1 in (GoerssJardine).

Theorem (Eilenberg-MacLane)

The inclusion

$$NA\hookrightarrow CA$$N A \hookrightarrow C A

is a natural quasi-isomorphism and in fact a natural chain homotopy equivalence, i.e. the complex $D_{•}(X)$ is null-homotopic.

Proof

Following the proof of theorem 2.1 in (GoerssJardine) we look for each $n\in \mathbb{N}$ and each $j<n$ at the groups

$$N_n(A{)}_j:={\cap }_{i=0}^j\ker (d_i)\subset A_n$$N_n(A)_j := \cap_{i=0}^j ker (d_i) \subset A_n

and similarly at

$$D_n(A{)}_j=\{s_i{\}}_{i\le j}(A_{n-1})\subset A_n,$$D_n(A)_j = \{s_{i}\}_{i \leq j}(A_{n-1}) \subset A_n \,,

the subgroup generated by the first $j$ degeneracies.

For $j=n-1$ these coincide with $N_n(A)$ and with $D_n(A)$, respectively. We show by induction on $j$ that the composite

$$N_n(A{)}_j\hookrightarrow A_n\stackrel{}{\to }{A}_n/D_n(A{)}_j$$N_n(A)_j \hookrightarrow A_n \stackrel{}{\to} A_n/D_n(A)_j

is an isomorphism of all $j<n$. For $j=n-1$ this is then the desired result.

(…)

Equivalence of categories

Proposition

The functor $N:\mathrm{sAb}\to {\mathrm{Ch}}_{•}^{+}(A)$ is an equivalence of categories.

This is the statement of the Dold-Kan correspondence. See there for details.

Homology and homotopy groups

Notice that the simplicial set underlying any simplicial group $G$ (as described there) is a Kan complex. Write

$${\pi }_n(G)n\in \mathbb{N}$$\pi_n(G) \;\;\; n \in \mathbb{N}

for the $n$-th simplicial homotopy group of $G$. Notice that due to the group structure of $G$ in this case also ${\pi }_0(G)$ is indeed canonically a group, not just a set.

Proposition

For $A$ a simplicial abelian group there are natural isomorphisms

$${\pi }_n(A,0)\simeq H_n(NA)\simeq H_n(A)$$\pi_n(A,0) \simeq H_n(N A) \simeq H_n(A)

between the simplicial homotopy groups and the chain homology groups of the unnormalized and of the normalized chain complexes.

Proof

The first isomorphism follows with the Eckmann-Hilton argument. The second directly from the Eilenberg-MacLane theorem above.

Remark

Both $\mathrm{sAb}$ as well as ${\mathrm{Ch}}_{•}^{+}$ are naturally categories with weak equivalences given by those morphisms that induce isomorphisms on all simplicial homotopy group and on all chain homology groups, respectively. So the above statement says that the Moore complex functor $N$ respects these weak equivalences.

In fact, it induces an equivalence of categories also on the corresponding homotopy categories. And even better, it induces a Quillen equivalence with respect to the standard model category structures that refine the structures of categories of weak equivalences. All this is discussed at Dold-Kan correspondence.

Hypercrossed complex structure

Proposition

The Moore complex of a simplicial group is naturally a hypercrossed complex.

This has been established in (Carrasco-Cegarra). In fact, the analysis of the Moore complex and what is necessary to rebuild the simplicial group from its Moore complex is the origin of the abstract motion of hypercrossed complex, so our stated proposition is almost a tautology!

Typically one has pairings $N{G}_p\times N{G}_q\to N{G}_{p+q}$. These use the Conduché decomposition theorem, see the discussion at hypercrossed complex.

These Moore complexes are easily understood in low dimensions:

• Suppose that $G$ is a simplicial group with Moore complex $NG$, which satisfies $N{G}_k=1$ for $k>1$, then $(G_1,G_0,d_1,d_0)$ has the structure of a 2-group. The interchange law is satisfied since the corresponding equation in $G_1$ is always the image of an element in $N{G}_2$, and here that must be trivial. If one thinks of the 2-group as being specified by a crossed module $(C,P,\delta ,a)$, then in terms of the original simplicial group, $G$, $N{G}_0=G_0=P$, $N{G}_1\cong C$, $\partial =\delta$ and the action of $P$ on $C$ translates to an action of $N{G}_0$ on $N{G}_1$ using conjugation by $s_0(p)$, i.e., for $p\in G_0$ and $c\in N{G}_1$,

  $$a(p)(c)=s_0(p)c{s}_0(p{)}^{-1}.$$a(p)(c) = s_0(p)c s_0(p)^{-1}.

• Suppose next that $N{G}_k=1$ for $k>2$, then the Moore complex is a 2-crossed module.

References

Original sources are

• John Moore, Homotopie des complexes monoïdaux, I. Séminaire Henri Cartan, 7 no. 2 (1954-1955), Exposé No. 18, 8 p. (numdam)

• John Moore, Semi-simplicial complexes and Postnikov systems , Symposium international de topologia algebraica, Mexico 1958, p. 243].

• John Moore, Semi-simplicial Complexes, seminar notes , Princeton University 1956]

There is also a never published

• Seminar on algebraic homotopy theory. Princeton 1956. Mimeographed notes.

A proof by Cartan is in

• Cartan, Quelques questions de topologies seminar, 1956-57

A standard textbook reference for the abelian version is

• Paul Goerss, Rick Jardine, Simplicial homotopy theory (chapter 3)

Notice that these authors write “normalized chain complex” for the complex that elsewhere in the literature would be called just “Moore complex”, whereas what Goerss–Jardine call “Moore complex” is sometime maybe just called “alternating sum complex”.

A discussion with an emphasis of the generalization to non-abelian simplicial groups is in section 1.3.3 of

• Tim Porter, Crossed Menagerie

The discusson of the hypercrossed complex structure on the Moore complex of a general simplicial group is in

• P. Carrasco, A. M. Cegarra, Group-theoretic Algebraic Models for Homotopy Types , J. Pure Appl. Alg., 75, (1991), 195–235