homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
and
nonabelian homological algebra
A crossed complex (of groupoids) is a nonabelian and many object generalization of a chain complex of abelian groups.
Crossed complexes are an equivalent way to encode the information contained in strict ω-groupoids: the groups appearing in the crossed complex in degree $n \geq 2$ are the source-fibers of the collection of $n$-morphisms of the $\omega$-groupoid.
See also homotopy n-type.
One way to think of a crossed complex is as a chain complex in which the bottom part is a crossed module and the rest is a chain complex of modules over the fundamental group of the crossed complex (that is its cokernel). This is easy to think of in the case where there is a single object (crossed complex of groups), and it is a simple step to extend to the many object case.
Later on we will look in a bit more detail at the fundamental crossed complex of a filtered space, and that is a good example to keep in mind. For simplicity assume we have a CW-complex, or similar space, together with a filtration by some nice subspaces. We have the fundamental groupoid, $\Pi_1(X_1,X_0)$, of the ‘1-skeleton’ based at the vertices. For any vertex $x$, we then have $\pi_2(X_2,X_1,x)$, the relative homotopy group of the 2-dimensional stuff relative to the 1-dimensional stuff, based at $x$. Varying $x$ we get a family of groups which we think of as a groupoid having just vertex groups without any arrows joining distinct vertices. In the next dimension we have $\pi_3(X_3,X_2,x)$, which is the relative homotopy group taking account of the 3-cells modulo the 2-cells, (which is abelian), and so on. Change of base point gives an action of $\Pi_1(X_1,X_0)$ on all of these. It was studying these groups , actions etc. that gave the abstract definition that follows.
A crossed complex (“of groupoids”) $C$ is
a groupoid $C_1 \stackrel{\overset{\delta_t}{\to}}{\underset{\delta_s} {\to}} C_0$
together with a sequence of skeletal groupoids $(C_k)_{k = 2}^\infty$ over $C_0$, i.e. of bundles $C_k = \coprod_{x \in C_0} (C_k)_x$ of groups over $C_0$, abelian for $k \geq 3$, sitting in a diagram
such that
the morphisms $\delta_k$ for $k \geq 2$ are morphisms of groupoids over $C_0$, compatible with the action by $C_1$
$im(\delta_2) \subset C_1$ acts by conjugation on $C_2$ and trivially on $C_k$ for $k \geq 3$
$\delta_{k-1} \circ \delta_k = 0$ for $k \geq 3$.
There is an obvious notion of morphisms $f : C \to D$ of crossed complexes, being sequences of maps $(f_k : C_k \to D_k)$ preserving all the above structure. The resulting category is often denoted $Crs$ or $CrsCpx$.
While the above definition of a crossed complex may seem slightly ‘baroque’, it can naturally be understood as being precisely the data obtained from a globular strict ∞-groupoid by retaining for $k \geq 2$ precisely only those k-morphisms whose source is a an identity $k-1$-morphisms on an object.
(crossed complex associated to a strict $\infty$-groupoid)
For $\mathcal{G}$ a globular strict ∞-groupoid
the corresponding crossed complex $[\mathcal{G}]$ is defined as follows:
the groupoid $[\mathcal{G}]_1 \stackrel{\to}{\to} [\mathcal{G}]_0$ is just the groupoid $\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0$; underlying $\mathcal{G}$ by forgetting all k-morphisms for $k \geq 2$
for $k \geq 2$ the bundle of groups $[\mathcal{G}]_k$ is over $x \in \mathcal{G}_0$ the group of k-morphisms of $\mathcal{G}$ whose source is the the identity on $x$:
where the group operation is given by the horizontal composition of k-morphisms (along objects). By the Eckmann-Hilton argument this is indeed an abelian group structure for $k \geq 3$.
The action of $[\mathcal{G}]_1$ on $[\mathcal{G}]_k$ is given by whiskering/conjugation of k-morphisms by 1-morphisms in $\mathcal{G}$.
The boundary maps $\delta := t : [\mathcal{G}]_{k} \to [\mathcal{G}]_{k-1}$ are the restrictions of the target maps $t : \mathcal{G}_k \to \mathcal{G}_{k-1}$, sending a $k$-morphisms with source an identity on an object to its target $k-1$-morphism.
Write $Str \infty Grpd$ for the 1-category of globular strict ∞-groupoids. The above construction defines an evident functor
The idea of the proof is that a strict ∞-groupoid may completely be reconstructed from its objects, 1-morphisms and those $(k \geq 2)$-morphisms that start at an identity by using the action of the 1-morphisms on the higher morphisms induced by conjugation.
For instance a 2-morphism
is, by the exchange law, equal to the horizontal composite of the 2-morphism
(whose source is the identity on $x$) with the 1-morphisms $f$.
A detailed proof is in
Notice that this article says ”$\infty$-groupoid” for strict globular $\infty$-groupoid and ”$\omega$-groupoid” for strict cubical $\infty$-groupoid .
This is a nonabelian and globular version of the Dold-Kan correspondence.
We describe a functorial construction of a crossed complex starting with a chain complex of modules over a groupoid $(A_n, \mathcal{H})$. As a special case it in particular gives an functor sending ordinary chain complexes of abelian groups into the category of crossed complexes, and hence into strict ω-groupoids. See also Nonabelian Algebraic Topology.
Recall the definition of the semidirect product groupoid $\mathcal{H} \ltimes A_n$.
(crossed complex from a chain complex)
For $A$ a chain complex of modules over a groupoid $\mathcal{H}$, let $\Theta A \in Crs$ be the crossed complex
where
and where
is the canonical covering morphism from above.
Here $\mathcal{H} \ltimes A_1$ acts on $A_n$ for $n \geq 2$ via the projection $\mathcal{H} \ltimes A_1 \to \mathcal{H}$, i.e. $A_1$ acts trivially. (…)
Finally set $\Theta(A)_0 := A_0$.
We spell out what this boils down to explicitly.
Explicit description
Let $A_\bullet$ be a chain complex of modules over the groupoid $\mathcal{H}$. Then the crossed complex $\Theta(A)$ is the following.
Its set of objects is $\Theta(A)_0 = A_0$.
Remember that $A_0$ itself is a module over $\matcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0)$, so that $A_0 = \corpdod_{p \in \mathcal{H}_0} (A_0)_p$.
For $x \in (A_0)_p$ and $y \in (A_0)_q$ a morphism in $\Theta(A)_1$ from $x$ to $y$ is labeled by $h \in \mathcal{H}_1$ and $a \in (A_1)_q$
where $\rho$ denotes the action of $\mathcal{H}$ on $A_0$.
The composition law is given by
For $k \geq 2$ the family of groups $\Theta(A)_k$ is over $x \in (A_0)_p$ the group $(A_k)_q$
The boundary maps and actions are the obvious ones
(ordinary abelian chain complex as crossed complex)
Let $C_\bullet$ be an ordinary chain complex of abelian groups, i.e. a chain complex of modules over the trivial groupoid.
Then $(\Theta C)_1$ is the groupoid with objects $C_0$ and morphisms $\{x \stackrel{b}{\to} (x + \partial b)\}$. And for $n \geq 2$ we have that $(\Theta C)_n$ is $\coprod_{x \in C_0} C_n$.
These form a pair of adjoint functors
where
This is proposition 7.4.29.
(…)
Say that a crossed complex $C$ is $n$-truncated if $C_k$ is trivial for $k \gt k$.
Then
0-truncated crossed complexes are canonically equivalent to sets, equivalent to homotopy 0-types.
1-truncated crossed complexes are canonically equivalently to groupoids, equivalent to homotopy 1-types).
2-truncated crossed complexes are equivalent to strict 2-groupoids equivalent to homotopy 2-types.
2-truncated and 0-connected crossed complex, i.e. a 2-truncated one for which $C_0 = *$ is the point is the same as a crossed module of groups. The equivalence of these to strict 2-groups is due to
R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Kon. Ned. Akad. v. Wet, 79, (1976), 296 – 302.)
A discussion of the kind of homotopy types generally modelled by crossed complexes, namely a linear model is in homotopy n-type.
The notion of crossed complex generalizes the notion of chain complex of abelian groups. Clearly in degree $k \geq 3$ a crossed complex with $C_0 = *$ is a chain complex of abelian groups. To regard the first 2 degrees $A_1 \stackrel{\delta}{\to} A_0$ of a chain complex of abelian groups as a crossed module, form the groupoid
and take the action of this groupoid on all $C_k$ to be trivial. This yields a functor
that embeds chain complexes of abelian groups into crossed complexes.
The embedding of chain complexes of abelian groups into crossed complexes generalizes to an embedding of chain complexs of modules over a groupoid
For details see Nonabelian Algebraic Topology, section 7.4.v.
If $X_*$ is a filtered space, there is a crossed complex $\Pi X_*$ – the fundamental crossed complex which corresponds to a (filtered and) strict ∞-groupoid version of the fundamental ∞-groupoid of $X$. In degree 1 it is the subgroupoid $\Pi_1(X_1,X_0)$ of the fundamental groupoid $\Pi_1(X_1)$ of $X_1$ on objects in $X_0$. In degree $n \gt 1$ it is the family of relative homotopy group?s $\{\pi_n(X_n,X_{n-1},p) : p\in X_0\}$.
This gives a functor $\Pi$ from filtered spaces to crossed complexes, which may be used to construct the generalisation of the Dold-Kan correspondence, which in this case goes between crossed complexes and simplicial T-complexes.
An important special case of the above is when the filtered space is a CW-complex and the filtration is by skeleta. Particularly useful instances of this are the $n$-cubes and $n$-simplices, with their CW-filtration. We obtain $\Pi(I^n)$ and $\Pi(\Delta^n)$. These are used to define cubical and simplicial nerves of a crossed complex and these in turn define the Dold-Kan correspondence mentioned above. For instance if $C$ is a crossed complex, then its simplicial nerve is the simplicial set with $Ner(C)_n = Crs(\Pi(\Delta^n),C)$ in dimension $n$.
(fundamental crossed complex of the $n$-simplex)
The topological $n$-simplex $\Delta^n$ is canonically a filtered space with $(\Delta^n)_k$ being the union of its $k$-faces.
Then we have that $\Pi_1((\Delta^n)_1, (\Delta^n)_0)$ is the groupoid whose objects are the $n+1$ vertices of $\Delta^n$ and which has precisely one morphism $x_i \to x_j$ for each ordered pair $x_i,x_j \in (\Delta^n)_0$ (all of them being isomorphisms)
At any $x_i$ the relative homotopy group $\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i)$ is a group on the set of 2-faces that have $x_i$ as a 0-face: there is a unique homotopy class of disks in $\Delta^n$ that sits in the 2-faces $(\Delta^n)_2$, whose base point is at $x_j$ and whose boundary runs along the boundary of a given 2-face of $\Delta^n$.
So (using the equivalence of crossed complexes with strict $\omega$-groupoids) for instance $\Pi \Delta^2$ is generated from $\Pi_1((\Delta^2)_1,(\Delta^2)_0)$ as above and a 2-cell
under whiskering and composition. For instance whiskering this with $x_1 \to x_2$ yields the 2-morphism
One sees that $\Pi \Delta^2$ is the strict groupoidification of the second oriental.
Generally, $\Pi \Delta^n$ is the $n$-groupoid freely generated from $k$-morphisms for each $k$-face of $\Delta^n$.
Crossed complexes (of groups) correspond to group T-complexes. Any group $T$-complex is a simplicial group and in the entry for them it is mentioned that a simplicial group has a group $T$-complex structure if and only if $\mathcal{N}G\cap D$ is the trivial graded subgroup, where $D = (D_n)_{n\geq 1}$ is the graded subgroup of $G$ generated by the degenerate elements. If $G$ is such a group $T$-complex then its Moore complex has a natural structure of a crossed complex. In general the obstruction to a given simplicial group to have Moore complex which is a crossed complex is exactly that graded subgroup, $\mathcal{N}G\cap D$. (The Whitehead products for $G$ live in this graded subgroup, so this provides one way of showing that the homotopy types representable by crossed complexes have trivial Whitehead products.)
Conversely, for any crossed complex, $C$, there is a simplicial group, $K(C)$, constructed using an analogue of the inverse in the Dold-Kan correspondence, which is a group $T$-complex and whose Moore complex is isomorphic to $C$.
The generalisation to general crossed complexes (of groupoids) and simplicially enriched groupoids if quite easy to do. We will usually state results below for the group case, leaving the general case to the ‘reader’.
It is fairly clear that crossed complexes / group(oid) $T$-complexes correspond to simplicial group(oid)s in which certain equations hold. It therefore is reasonable that they are equivalent to a variety / reflective subcategory in the category, $SSet Grpd$, of simplicially enriched groupoids. (The discussion in the entry on group T-complex is relevant here.)
There is a functor $C(-)$ from simplicial groups to crossed complexes given by
in higher dimensions with at its ‘bottom end’, the crossed module,
with $\partial$ induced from the boundary in the Moore complex.
The category of crossed complexes form a variety in the category of all hypercrossed complexes. Alternatively, groupoid T-complexes (the groupoid version of group T-complex) form a variety in the category of all simplicial groups.
3-group, 2-crossed module / crossed square, differential 2-crossed module
∞-group, simplicial group, crossed complex, hypercrossed complex
Crossed complexes were defined by Blakers in 1948 (following a suggestion of Samuel Eilenberg) and developed by Whitehead in 1949 and 1950 (but these authors used different terminology). They were applied by Johannes Huebschmann to group cohomology in 1980. They were further developed in series of articles by Ronnie Brown and collaborators in the context of nonabelian algebraic topology, and partly because they were found equivalent to form of (strict) cubical $\omega$-groupoid with connections. This equivalence enabled a number of new results, including van Kampen type theorems and monoidal closed structures for crossed complexes.
Textbook treatment is in
A survey of the use of crossed complexes is in
The equivalence of strict omega-groupoids and crossed complexes is discussed in
Notice that this article says ”$\infty$-groupoid” for strict globular $\infty$-groupoid and ”$\omega$-groupoid” for strict cubical $\infty$-groupoid .
For the relation to group cohomology see