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 with 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 ∞-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 $G$, its normalized chain complex or Moore complex is the $\mathbb{N}$-graded chain complex $((N G)_\bullet,\partial )$ of (possibly nonabelian) groups which

• is in degree $n$ the joint kernel

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

  of all face maps except the 0-face;

• with differential given by the remaining 0-face map

  $$\partial_n \coloneqq d_0^n|_{(N G)_n} : (N G)_n \rightarrow (N G)_{n-1} \,.$$

Remark

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

Remark

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

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

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

• an element $h \in N G_2$ is a 2-disk

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

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

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

etc.

Proposition

For every simplicial group $G$ the normalized chain complex $(N G)_\bullet$ in def. 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 $A$ be a simplicial abelian group. Then its normalized chain complex $(N A)_\bullet \in Ch_\bullet^+$ of def. is an ordinary connective chain complex in the abelian category Ab.

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

Definition

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

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

  $$(C A)_n \coloneqq A_n$$

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

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

  (see lemma ).

Lemma

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

Proof

Using the simplicial identity $d_i \circ d_j = d_{j-1} \circ d_i$ for $i \lt j$ one finds:

$$\begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned} \,.$$

Definition

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

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

for the subgroup of $A_{n+1}$ which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements). Elements of $D(A)_{n}$ are often called thin $n$-simplices.

Definition

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

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

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

  $$((C A)/D(A))_{n+1} \coloneqq A_{n+1} / D(A_{n+1})$$

• whose differential is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma ).

Lemma

Def. 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 $s_j(a) \in A_n$ a degenerate element, its boundary is

$$\begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned}$$

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

Properties

Normalization

Proposition

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

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

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

(Goerss-Jardine, theorem III 2.1, see also Schwede-Shipley 03, Section 2.1).

Corollary

For $A$ a simplicial abelian group, there is a splitting

$$C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)$$

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

Proof

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

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

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

$$\array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,.$$

Theorem

(Eilenberg-MacLane)
Given a simplicial abelian group $A$, then the inclusion

$$i \colon N A \hookrightarrow C A$$

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

(Goerss-Jardine, theorem III 2.4)

Proof

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

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

and similarly at

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

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

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

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

(…)

Corollary

Given a simplicial abelian group $A$, then the projection chain map

$$(C A) \longrightarrow (C A)/(D A)$$

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

Proof

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

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

By theorem the vertical map is a quasi-isomorphism and by prop. 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. restricts on simplicial abelian groups to an equivalence of categories

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

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 $G$ (as described there) is a Kan complex. Write

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

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

$$\pi_n(A,0) \simeq H_n(N A) \simeq H_n(A)$$

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 $sAb$ as well as $Ch_\bullet^+$ 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 $N$ 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 $N G$, which satisfies $N G_k = 1$ for $k\gt 1$, 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 $c\in N G_1$,

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

• Suppose next that $N G_k = 1$ for $k \gt 2$, then the Moore complex is a 2-crossed module.

Examples

Example

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

This simplicial abelian group starts out as

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

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

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

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

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

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

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

$$\partial_1 \colon (i \to j) \mapsto (i)$$

$$\partial_0 \colon (i \to j) \mapsto (j) \,.$$

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

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

Notice that alternatively we could consider the topological 1-simplex $\Delta^1 = [0,1]$ and its singular simplicial complex $Sing(\Delta^1)$ in place of the smaller $\Delta[1]$, then the free simplicial abelian group $\mathbb{Z}(Sing(\Delta^1))$ of that. The corresponding alternating face map chain complex $C(\mathbb{Z}(Sing(\Delta^1)))$ 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 $n$-simplex in $[0,1]$ 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.

Example

(normalized chain complex of $E\mathbb{Z}_2$)
We write $\mathbb{Z}_2 \,\equiv\, \mathbb{Z}/2\mathbb{Z}$ for the cyclic group of order 2, whose underlying set we denote by $\{0, 1\}$.

Consider the simplicial group (see here)

$$W \mathbb{Z}_2 \;=\; N\big( \mathbf{E} \mathbb{Z}_2 \big) \;\equiv\; N\big( \mathbb{Z}_2 \times \mathbb{Z}_2 \underoverset {pr_2} {(\text{-})\cdot(-)} {\rightrightarrows} \mathbb{Z}_2 \big) \,.$$

Forming degree-wise linear spans gives a simplicial abelian group

$$\mathbb{Z}\big[ W \mathbb{Z}_2 \big] \;\; \in \;\; Ab^{\Delta^{op}} \,.$$

with normalized chain complex denoted

(2)$$N_\bullet \big( \mathbb{Z} [ W \mathbb{Z}_2 ] \big) \;\; \in \;\; Ch_{\geq 0} \,.$$

Now, since the only non-identity morphisms in $\mathbf{E}\mathbb{Z}_2$ are the two morphsism $0 \to 1$ and $1 \to 0,$ the non-degenerate $n$-simplices of (3) are alternating sequences on $n+1$-elements in $\{0,1\}$. By the alternating property, these are fully determined by their first element (in particular), whence there are exactly two non-degenerate $n$-simplices for all $n \in \mathbb{N}$. On these, the differential is given as follows, using (1) and Prop. , according to which only the 0th and $n$th face maps contribute in degree $n$, the latter with sign $(-1)^n$:

(3)$$N_\bullet \mathbb{Z} \big[ W \mathbb{Z}_2 \big] \;\simeq\; \Bigg[ \cdots \to \xrightarrow{ \left[ \array{ +1 & +1 \\ +1 & +1 } \right] } \mathbb{Z}_{0101} \oplus \mathbb{Z}_{1010} \xrightarrow{ \left[ \array{ -1 & +1 \\ +1 & -1 } \right] } \mathbb{Z}_{010} \oplus \mathbb{Z}_{101} \xrightarrow{ \left[ \array{ +1 & +1 \\ +1 & +1 } \right] } \mathbb{Z}_{01} \oplus \mathbb{Z}_{10} \xrightarrow{ \left[ \array{ -1 & +1 \\ +1 & -1 } \right] } \mathbb{Z}_0 \oplus \mathbb{Z}_1 \Bigg] \,.$$

One may think of the two generators in degree $n+1$ as corresponding to two $n+1$-dimensional hemispheres with common boundary an equator similarly formed by two $n$-dimensional hemispheres, and so on. Thereby this chain complex is seen to be isomorphically that for the cellular homology of the standard CW-complex-structure on the infinite-dimensional sphere $S^\infty$ (cf. also at real projective space the section Relation to $\mathbb{Z}/2$-classifying space).

Indeed, one sees immediately that (3) has vanishing chain homology in all positive degrees and homology $\simeq \mathbb{Z}$ in degree 0, whence the canonical map to the normalized chain complex on the point is a quasi-isomorphism:

$$N_\bullet\,\mathbb{Z}[W \mathbb{Z}_2] \xrightarrow[\;qiso\;]{} N_\bullet\,\mathbb{Z}[\ast] \,.$$

This reflects the contractible homotopy type of both $E \mathbb{Z}_2$ (the total space of the universal principal $\mathbb{Z}_2$-bundle) and of $S^\infty$.

Example

(normalized chain dg-algebra of $E\mathbb{Z}_2$)

On the normalized chain complex of Ex. , the group structure on $W \mathbb{Z}_2$ induces a simplicial ring-structure, the “simplicial group ring”)

$$\mathbb{Z}\big[ W \mathbb{Z}_2 \big] \;\; \in \;\; Ring^{\Delta^{op}} \,.$$

Since the normalized chain complex-functor $N_\bullet \,\colon\, Ab^{\Delta^{op}} \to Ch_{\geq 0}$ is (see here) a lax monoidal functor via the Eilenberg-Zilber map, this induces on $N_\bullet\big( \mathbb{Z}[W \mathbb{Z}_2] \big)$ the structure of a dg-algebra.

To write this out, denote the two generators in each degree by

$$(0), (1) \;\in\; \mathbb{Z}_{0\cdots} \oplus \mathbb{Z}_{1\cdots} \;=\; N_n\big( \mathbb{Z}[W \mathbb{Z}_2] \big) \,.$$

The non-degenerate cells of the tensor product simplicial group are similarly labeled (via this Prop.) by

1. a pair of first elements of $n+1$-sequences in $\{0,1\}$

2. a $(p, q)$-shuffle,

hence:

$$\big( (g, g'), sh \big) \;\in\; \mathbb{Z} \big[ \{0,1\}^2 \times \sqcup_{p+q = n} Sh(p,q) \big] \;\simeq\; N_{n} \Big( \mathbb{Z}\big[ W \mathbb{Z}_2 \times W \mathbb{Z}_2 \big] \Big) \;\simeq\; N_{n} \Big( \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \otimes \mathbb{Z}\big[ W \mathbb{Z}_2 \big] \Big) \,.$$

One immediately finds that on these generators the induced product map is just the group operation, independent of the shuffle:

$$\array{ N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \otimes \mathbb{Z}[W \mathbb{Z}_2] \big) & \xrightarrow{ N_\bullet \big( \mathbb{Z}[(\text{-})\cdot(\text{-})] \big) } & N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \big) \\ \big( g, g', sh \big) &\mapsto& (g \cdot g') \mathrlap{\,.} }$$

Composed with the Eilenberg-Zilber map this gives

$$\array{ \big( N_\bullet \, \mathbb{Z}[W \mathbb{Z}_2] \big) \otimes \big( N_\bullet \, \mathbb{Z}[W \mathbb{Z}_2] \big) &\xrightarrow{\phantom{----}}& N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \otimes \mathbb{Z}[W \mathbb{Z}_2] \big) & \xrightarrow{ N_\bullet \big( \mathbb{Z}[(\text{-})\cdot(\text{-})] \big) } & N_\bullet \big( \mathbb{Z}[W \mathbb{Z}_2] \big) \\ (g)_p \otimes (g')_q &\mapsto& \displaystyle{ \underset{ \mathclap{ sh \in Sh(p,q) } }{\sum} } \; sgn(sh) \, \big( g,g', sh \big) &\mapsto& \Big( \displaystyle{ \underset{ \mathclap{ sh \in Sh(p,q) } }{\sum} } \; sgn(sh) \Big) \, (g \cdot g')_{p + q} \mathrlap{\,.} }$$

References

Original sources:

• John Moore, Homotopie des complexes monoïdaux, I. Séminaire Henri Cartan 7 2 (1954-1955), Exposé 18 (numdam:SHC_1954-1955__7_2_A8_0)

• 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

• John C. Moore, Algebraic homotopy theory. Princeton 1956. Mimeographed notes [pdf]

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, chapter III.2 of Simplicial homotopy theory

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”.

See also

• Stefan Schwede, Brooke Shipley, Section 2.1 of: Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv:math.AT/0209342, euclid:euclid.agt/1513882376)

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 https://doi.org/10.1016/0022-4049(91)90133-M