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Moore complex

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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 with chain complexes.

Recall that a simplicial group GG, 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 GG is a chain complex

  • whose nn-cells are the “nn-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 nn-disk to its boundary, read as an (n1)(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

For general simplicial groups

Definition

Given a simplicial group GG, its normalized chain complex or Moore complex is the \mathbb{N}-graded chain complex ((NG) ,)((N G)_\bullet,\partial ) of (possibly nonabelian) groups which

  • is in degree nn the joint kernel

    (NG) n= i=1 nkerd 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 map

    n:=d 0 n| (NG) n:(NG) n(NG) n1. \partial_n := d_0^n|_{(N G)_n} : (N G)_n \rightarrow (N G)_{n-1} \,.
Remark

In def. 1 one may equivalently take the joint kernel of all but the nn-face map and take that remaining face map, d n nd_n^n, to be the differential.

Remark

We may think of the elements of the complex NGN G, def. 1, in degree kk as being kk-dimensional disks in GG all whose boundary is captured by a single face:

  • an element gNG 1g \in N G_1 in degree 1 is a 1-disk

    1gg, 1 \stackrel{g}{\to} \partial g \,,
  • an element hNG 2h \in N G_2 is a 2-disk

    1 1 h h 1 1 1, \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

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

etc.

Proposition

For every simplicial group GG the normalized chain complex (NG) (N G)_\bullet in def. 1 is a normal complex of groups,

Remark

This means that is easy to take the homology of the complex, even though the groups involved may be non-abelian.

For simplicial abelian groups

Let now AA be a simplicial abelian group. Then its normalized chain complex (NA) Ch +(N A)_\bullet \in Ch_\bullet^+ of def. 1 is an ordinary connective chain complex in the abelian category Ab.

In this abelian cases are two other chain complexes naturally associated with AA:

Definition

For AA a simplicial abelian group its alternating face map complex (CA) (C A)_\bullet of AA is the chain complex which

  • in degree nn is given by the group A nA_n itself

    (CA) n:=A n (C A)_n := A_n
  • with differential given by the alternating sum of face maps (using the abelian group structure on AA)

    n i=0 n(1) id i:(CA) n(CA) n1. \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,.

    (see lemma 1).

Lemma

The differential in def. 2 is well-defined in that it indeed squares to 0.

Proof

Using the simplicial identity d id j=d j1d id_i \circ d_j = d_{j-1} \circ d_i for i<ji \lt j one finds:

n n+1 = i,j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd j1d i = ij(1) i+jd id j ik(1) i+kd kd i =0. \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} \,.
Definition

Given a simplicial group AA (or in fact any simplicial set), then an element aA n+1a \in A_{n+1} is called degenerate (or thin) if it is in the image of one of the simplicial degeneracy maps s i:A nA n+1s_i \colon A_n \to A_{n+1}. All elements of A 0A_0 are regarded a non-degenerate. Write

D(A n+1) is i(A n)A n+1 D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1}

for the subgroup of A n+1A_{n+1} which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements).

Definition

For AA a simplicial abelian group its alternating face maps chain complex modulo degeneracies, (CA)/(DA)(C A)/(D A) is the chain complex

  • which in degree 0 equals is just ((CA)/D(A)) 0A 0((C A)/D(A))_0 \coloneqq A_0;

  • which in degree n+1n+1 is the quotient group obtained by dividing out the group the degenerate elements, def. 3:

    ((CA)/D(A)) n+1:=A n+1/D(A n+1) ((C A)/D(A))_{n+1} := A_{n+1} / D(A_{n+1})
  • whose differential is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma 2).

Lemma

Def. 4 is indeed well defined in that the alternating face map differential respects the degenerate subcomplex.

Proof

Using the mixed simplicial identities we find that for s j(a)A ns_j(a) \in A_n a degenerate element, its boundary is

i(1) id is j(a) = i<j(1) is j1d i(a)+ i=j,j+1(1) ia+ i>j+1(1) is jd i1(a) = i<j(1) is j1d i(a)+ i>j+1(1) is jd i1(a) \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

Proposition

Given a simplicial abelian group AA, the evident composite of natural morphisms

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

from the normalized chain complex, def. 1, into the alternating face map complex modulo degeneracies, def. 4, (inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.

e.g. (Goerss-Jardine, theorem III 2.1).

Corollary

For AA a simplicial abelian group, there is a splitting

C (A)N (A)D (A) C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)

of the alternating face map complex, def. 2 as a direct sum, where the first direct summand is naturally isomorphic to the normalized chain complex of def. 1 and the second is the degenerate cells from def. 4.

Proof

By prop. 2 there is an inverse to the diagonal composite in

CA p (CA)/(DA) i NA. \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,.

This hence exhibits a splitting of the short exact sequence given by the quotient by DAD A.

0 DA CA p (CA)/(DA) 0 i iso NA. \array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,.
Theorem (Eilenberg-MacLane)

Given a simplicial abelian group AA, then the inclusion

i:NACA i \colon N A \hookrightarrow C A

of the normalized chain complex, def. 1 into the full alternating face map complex, def. 2, is a natural quasi-isomorphism and in fact a natural chain homotopy equivalence, i.e. the complex D (X)D_\bullet(X) is null-homotopic.

(Goerss-Jardine, theorem III 2.4)

Proof

Following the proof of (Goerss-Jardine, theorem III 2.1) we look for each nn \in \mathbb{N} and each j<nj \lt n at the groups

N n(A) j:= i=0 jker(d i)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} ij(A n1)A n, D_n(A)_j = \{s_{i}\}_{i \leq j}(A_{n-1}) \subset A_n \,,

the subgroup generated by the first jj degeneracies.

For j=n1j= n-1 these coincide with N n(A)N_n(A) and with D n(A)D_n(A), respectively. We show by induction on jj that the composite

N n(A) jA nA 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<nj \lt n. For j=n1j = n-1 this is then the desired result.

(…)

Corollary

Given a simplicial abelian group AA, then the projection chain map

(CA)(CA)/(DA) (C A) \longrightarrow (C A)/(D A)

from its alternating face maps complex, def. 2, to the alternating face map complex modulo degeneracies, def. 4, is a quasi-isomorphism.

Proof

Consider the pre-composition of the map with the inclusion of the normalized chain complex, def. 1.

CA p (CA)/(DA) i NA \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A }

By theorem 1 the vertical map is a quasi-isomorphism and by prop. 2 the composite diagonal map is an isomorphism, hence in particular also a quasi-isomorphism. Since quasi-isomorphisms satisfy the two-out-of-three property, it follows that also the map in question is a quasi-isomorphism.

Equivalence of categories

Proposition

The normalized chain complex functor of def. 1 restricts on simplicial abelian groups to an equivalence of categories

N:sAbCh +(A) N \colon sAb \stackrel{\simeq}{\longrightarrow} Ch_\bullet^+(A)

between sAb and the category of chain complexes in non-negative degree.

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 GG (as described there) is a Kan complex. Write

π n(G)n \pi_n(G) \;\;\; n \in \mathbb{N}

for the nn-th simplicial homotopy group of GG. Notice that due to the group structure of GG in this case also π 0(G)\pi_0(G) is indeed canonically a group, not just a set.

Proposition

For AA a simplicial abelian group there are natural isomorphisms

π n(A,0)H n(NA)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 sAbsAb as well as Ch +Ch_\bullet^+ 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 NN 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 NG p×NG qNG p+qN 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 GG is a simplicial group with Moore complex NGN G, which satisfies NG k=1N G_k = 1 for k>1k\gt 1, then (G 1,G 0,d 1,d 0)(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 1G_1 is always the image of an element in NG 2N G_2, and here that must be trivial. If one thinks of the 2-group as being specified by a crossed module (C,P,δ,a)(C,P,\delta, a), then in terms of the original simplicial group, GG, NG 0=G 0=PN G_0 = G_0 = P, NG 1CN G_1 \cong C, =δ \partial = \delta and the action of PP on CC translates to an action of NG 0N G_0 on NG 1N G_1 using conjugation by s 0(p)s_0(p), i.e., for pG 0p\in G_0 and cNG 1c\in N G_1,

    a(p)(c)=s 0(p)cs 0(p) 1.a(p)(c) = s_0(p)c s_0(p)^{-1}.
  • Suppose next that NG k=1N G_k = 1 for k>2k \gt 2, then the Moore complex is a 2-crossed module.

Examples

Example

Consider the 1-simplex Δ[1]\Delta[1] regarded as a simplicial set, and write [Δ[1]]\mathbb{Z}[\Delta[1]] for the simplicial abelian group which in each degree is the free abelian group on the simplices in Δ[1]\Delta[1].

This simplicial abelian group starts out as

[Δ[1]]=( 4 3 1 0 2) \mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right)

(where we are indicating only the face maps for notational simplicity).

Here the first 2=\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}, the direct sum of two copies of the integers, is the group of 0-chains generated from the two endpoints (0)(0) and (1)(1) of Δ[1]\Delta[1], i.e. the abelian group of formal linear combinations of the form

2{a(0)+b(1)|a,b}. \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,.

The second 3\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z} is the abelian group generated from the three (!) 1-simplicies in Δ[1]\Delta[1], namely the non-degenerate edge (01)(0\to 1) and the two degenerate cells (00)(0 \to 0) and (11)(1 \to 1), hence the abelian group of formal linear combinations of the form

3{a(00)+b(01)+c(11)|a,b,c}. \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,.

The two face maps act on the basis 1-cells as

1:(ij)(i) \partial_1 \colon (i \to j) \mapsto (i)
0:(ij)(j). \partial_0 \colon (i \to j) \mapsto (j) \,.

Now of course most of the (infinitely!) many simplices inside Δ[1]\Delta[1] are degenerate. In fact the only non-degenerate simplices are the two 0-cells (0)(0) and (1)(1) and the 1-cell (01)(0 \to 1). Hence the alternating face maps complex modulo degeneracies, def. 4, of [Δ[1]]\mathbb{Z}[\Delta[1]] is simply this:

(C([Δ[1]]))/D([Δ[1]]))=(00(11) 2). (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,.

Notice that alternatively we could consider the topological 1-simplex Δ 1=[0,1]\Delta^1 = [0,1] and its singular simplicial complex Sing(Δ 1)Sing(\Delta^1) in place of the smaller Δ[1]\Delta[1], then the free simplicial abelian group (Sing(Δ 1))\mathbb{Z}(Sing(\Delta^1)) of that. The corresponding alternating face map chain complex C((Sing(Δ 1)))C(\mathbb{Z}(Sing(\Delta^1))) is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular nn-simplex in [0,1][0,1] is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.

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

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

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

Revised on March 26, 2015 13:50:03 by Urs Schreiber (80.92.246.195)