and
nonabelian homological algebra
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.
Given a simplicial group $G$, the $\mathbb{N}$-graded chain complex complex $((N G)_\bullet,\partial )$ of (possibly nonabelian) groups is
in degree $n$ the joint kernel
of all face maps except the 0-face
with differential given by the remaining 0-face
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 $N G$ 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.
We may think of the elements of the complex $N G$ 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
an element $h \in N G_2$ is a 2-disk
a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-sphere
etc.
For every simplicial group $G$ the complex $(N G)_\bullet$ is a normal complex of groups.
Let here $A$ be a simplicial abelian group. Then $(N A)_\bullet \in Ch_\bullet^+$ is an ordinary connective chain complex in the abelian category Ab.
There are two other chain complexes naturally associated with $A$:
The alternating face map complex $C A$ of $A$ is
in degree $n$ given by the group $A_n$ itself
with the differential given by the alternating sum of face maps (using the abelian group structure on $A$)
The complex modulo degeneracies, $(C A)/D(A)$ is the complex
which in degree $n$ is given by the quotient group obtained by dividing out the group
generated by the degenerate elements in $A_n$
with differential being the induced action of the alternating sum of faces on the quotient.
This is indeed well defined in that the alternating face boundary map satisfies $\partial \circ \partial = 0$ in $C_\bullet(A)$ and restricts to a boundary map on the degenerate subcomplex $\partial : A_n|_{s(A_{n-1})} \to A_{n-1}|_{s(A_{n-2})}$.
For the first statement one checks
using the simplicial identity $d_i \circ d_j = d_{j-1} \circ d_i$ for $i \lt j$.
Similarly, using the mixed simplicial identities we find that for $s_j(a) \in A_n$ a degenerate element, its boundary is
which is again a combination of elements in the image of the degeneracy maps.
Let $A$ be a simplicial abelian group.
There is a splitting
where the first summand is naturally isomorphic to the Moore complex as defined above.
Explicitly,
The evident composite of natural morphisms
(inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.
This appears as theorem 2.1 in (GoerssJardine).
The inclusion
is a natural quasi-isomorphism and in fact a natural chain homotopy equivalence, i.e. the complex $D_\bullet(X)$ is null-homotopic.
Following the proof of theorem 2.1 in (GoerssJardine) we look for each $n \in \mathbb{N}$ and each $j \lt n$ at the groups
and similarly at
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
is an isomorphism of all $j \lt n$. For $j = n-1$ this is then the desired result.
(…)
The functor $N : sAb \to Ch_\bullet^+(A)$ is an equivalence of categories.
This is the statement of the Dold-Kan correspondence. See there for details.
Notice that the simplicial set underlying any simplicial group $G$ (as described there) is a Kan complex. Write
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.
For $A$ a simplicial abelian group there are natural isomorphisms
between the simplicial homotopy groups and the chain homology groups of the unnormalized and of the normalized chain complexes.
The first isomorphism follows with the Eckmann-Hilton argument. The second directly from the Eilenberg-MacLane theorem above.
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.
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 $c\in N G_1$,
Suppose next that $N G_k = 1$ for $k \gt 2$, then the Moore complex is a 2-crossed module.
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
A proof by Cartan is in
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