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crossed complex in nLab

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homological algebra

(also nonabelian homological algebra)

Introduction

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diagram chasing

Schanuel's lemma

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Contents

Idea

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 ∞-groupoids: the groups appearing in the crossed complex in degree n≥2n \geq 2 are the source-fibers of the collection of nn-morphisms of the ω\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, Π 1(X 1,X 0)\Pi_1(X_1,X_0), of the ‘1-skeleton’ based at the vertices. For any vertex xx, we then have π 2(X 2,X 1,x)\pi_2(X_2,X_1,x), the relative homotopy group of the 2-dimensional stuff relative to the 1-dimensional stuff, based at xx. Varying xx 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 π 3(X 3,X 2,x)\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 Π 1(X 1,X 0)\Pi_1(X_1,X_0) on all of these. It was studying these groups , actions etc. that gave the abstract definition that follows.

Definition

As sequences of groups and groupoids

Definition

A crossed complex (“of groupoids”) CC is

  • a groupoid C 1→δ s→δ tC 0C_1 \stackrel{\overset{\delta_t}{\to}}{\underset{\delta_s} {\to}} C_0

  • together with a sequence of skeletal groupoids (C k) k=2 ∞(C_k)_{k = 2}^\infty over C 0C_0, i.e. of bundles C k=∐ x∈C 0(C k) xC_k = \coprod_{x \in C_0} (C_k)_x of groups over C 0C_0, abelian for k≥3k \geq 3, sitting in a diagram

⋯ → C 3 →δ C 2 →δ C 1 →δ s→δ t C 0 ↓ ↓ ↓ δ s ↓ = ⋯ → C 0 →= C 0 →= C 0 →= C 0 \begin{array}{c} \cdots &\to& C_3 &\stackrel{\delta}{\to}& C_2 &\stackrel{\delta}{\to}& C_1 & \stackrel{\overset{\delta_t}{\to}}{\underset{\delta_s}{\to}} & C_0 \\ && \downarrow && \downarrow && \downarrow^{\mathrlap{\delta_s}} && \downarrow^{\mathrlap{=}} \\ \cdots &\to& C_0 &\stackrel{=}{\to}& C_0 &\stackrel{=}{\to}& C_0 &\stackrel{=}{\to}& C_0 \end{array}

  • together with an action of C 1C_1 on C kC_k for k≥0k \geq 0

such that

  • the morphisms δ k\delta_k for k≥2k \geq 2 are morphisms of groupoids over C 0C_0, compatible with the action by C 1C_1

  • im(δ 2)⊂C 1im(\delta_2) \subset C_1 acts by conjugation on C 2C_2 and trivially on C kC_k for k≥3k \geq 3

  • δ k−1∘δ k=0\delta_{k-1} \circ \delta_k = 0 for k≥3k \geq 3.

There is an obvious notion of morphisms f:C→Df : C \to D of crossed complexes, being sequences of maps (f k:C k→D k)(f_k : C_k \to D_k) preserving all the above structure. The resulting category is often denoted CrsCrs or CrsCpxCrsCpx.

From strict ∞\infty-groupoids

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≥2k \geq 2 precisely only those k-morphisms whose source is a an identity k−1k-1-morphisms on an object.

Definition

(crossed complex associated to a strict ∞\infty-groupoid)

For 𝒢\mathcal{G} a globular strict ∞-groupoid

⋯→s→t𝒢 3→s→t𝒢 2→s→t𝒢 1→s→t𝒢 0 \cdots \stackrel{\overset{t}{\to}}{\underset{s}{\to}} \mathcal{G}_3 \stackrel{\overset{t}{\to}}{\underset{s}{\to}} \mathcal{G}_2 \stackrel{\overset{t}{\to}}{\underset{s}{\to}} \mathcal{G}_1 \stackrel{\overset{t}{\to}}{\underset{s}{\to}} \mathcal{G}_0

the corresponding crossed complex [𝒢][\mathcal{G}] is defined as follows:

  • the groupoid [𝒢] 1→→[𝒢] 0[\mathcal{G}]_1 \stackrel{\to}{\to} [\mathcal{G}]_0 is just the groupoid 𝒢 1→→𝒢 0\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0; underlying 𝒢\mathcal{G} by forgetting all k-morphisms for k≥2k \geq 2

  • for k≥2k \geq 2 the bundle of groups [𝒢] k[\mathcal{G}]_k is over x∈𝒢 0x \in \mathcal{G}_0 the group of k-morphisms of 𝒢\mathcal{G} whose source is the the identity on xx:

    [𝒢] k:=∐ xs k −1(Id x), [\mathcal{G}]_k := \coprod_x s_k^{-1}(Id_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≥3k \geq 3.

  • The action of [𝒢] 1[\mathcal{G}]_1 on [𝒢] k[\mathcal{G}]_k is given by whiskering/conjugation of k-morphisms by 1-morphisms in 𝒢\mathcal{G}.

  • The boundary maps δ:=t:[𝒢] k→[𝒢] k−1\delta := t : [\mathcal{G}]_{k} \to [\mathcal{G}]_{k-1} are the restrictions of the target maps t:𝒢 k→𝒢 k−1t : \mathcal{G}_k \to \mathcal{G}_{k-1}, sending a kk-morphisms with source an identity on an object to its target k−1k-1-morphism.

Write Str∞GrpdStr \infty Grpd for the 1-category of globular strict ∞-groupoids. The above construction defines an evident functor

[−]:Str∞Grpd→CrsCplx. [-] : Str \infty Grpd \to CrsCplx \,.

Proof

The idea of the proof is that a strict ∞-groupoid may completely be reconstructed from its objects, 1-morphisms and those (k≥2)(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

↗↘ f x ⇓ y ↘↗ g \array{ & \nearrow \searrow^{\mathrlap{f}} \\ x &\Downarrow& y \\ & \searrow \nearrow_{\mathllap{g}} }

is, by the exchange law, equal to the horizontal composite of the 2-morphism

↗↘ f x ⇓ y →f −1 x ↘↗ g \array{ & \nearrow \searrow^{\mathrlap{f}} \\ x &\Downarrow& y & \stackrel{f^{-1}}{\to} & x \\ & \searrow \nearrow_{\mathllap{g}} }

(whose source is the identity on xx) with the 1-morphisms ff.

A detailed proof is in

Notice that this article says “∞\infty-groupoid” for strict globular ∞\infty-groupoid and “ω\omega-groupoid” for strict cubical ∞\infty-groupoid .

This is a nonabelian and globular version of the Dold-Kan correspondence.

From chain complexes of modules

We describe a functorial construction of a crossed complex starting with a chain complex of modules over a groupoid (A n,ℋ)(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 ∞-groupoids. See also Nonabelian Algebraic Topology.

Recall the definition of the semidirect product groupoid ℋ⋉A n\mathcal{H} \ltimes A_n.

Definition

(crossed complex from a chain complex)

For AA a chain complex of modules over a groupoid ℋ\mathcal{H}, let ΘA∈Crs\Theta A \in Crs be the crossed complex

ΘA:=κ *Θ′A, \Theta A := \kappa^* \Theta' A \,,

where

Θ′A:=[A n→∂ nA n−1→⋯→A 3→∂ 3A 2→(0,∂ 2)ℋ⋉A 1] \Theta' A := \left[ A_n \stackrel{\partial_n}{\to} A_{n-1} \stackrel{}{\to} \cdots \stackrel{}{\to} A_{3} \stackrel{\partial_3}{\to} A_2 \stackrel{(0,\partial_2)}{\to} \mathcal{H}\ltimes A_1 \right]

and where

κ:P(A 0,ℋ)→ℋ⋉A 0 \kappa : P(A_0, \mathcal{H}) \to \mathcal{H} \ltimes A_0

is the canonical covering morphism from above.

⋯→ (ΘA) 3 → (ΘA) 2 → (ΘA) 1 → P(A 0,ℋ) ↓ ↓ ↓ ↓ ⋯→ A 3 →∂ 3 A 2 →(0,∂ 2) ℋ⋉A 1 →(1,∂ 1) ℋ⋉A 0. \array{ \cdots \to & (\Theta A)_3 &\to& (\Theta A)_2 &\to& (\Theta A)_1 &\to& P(A_0, \mathcal{H}) \\ & \downarrow && \downarrow && \downarrow && \downarrow \\ \cdots \to & A_3 &\stackrel{\partial_3}{\to}& A_2 &\stackrel{(0,\partial_2)}{\to}& \mathcal{H} \ltimes A_1 &\stackrel{(1, \partial_1)}{\to}& \mathcal{H} \ltimes A_0 } \,.

Here ℋ⋉A 1\mathcal{H} \ltimes A_1 acts on A nA_n for n≥2n \geq 2 via the projection ℋ⋉A 1→ℋ\mathcal{H} \ltimes A_1 \to \mathcal{H}, i.e. A 1A_1 acts trivially. (…)

Finally set Θ(A) 0:=A 0\Theta(A)_0 := A_0.

We spell out what this boils down to explicitly.

Explicit description

Let A •A_\bullet be a chain complex of modules over the groupoid ℋ\mathcal{H}. Then the crossed complex Θ(A)\Theta(A) is the following.

  • Its set of objects is Θ(A) 0=A 0\Theta(A)_0 = A_0.

    Remember that A 0A_0 itself is a module over matcalH=(ℋ 1→→ℋ 0)\matcal{H} = (\mathcal{H}_1 \stackrel{\to}{\to} \mathcal{H}_0), so that A 0=corpdod p∈ℋ 0(A 0) pA_0 = \corpdod_{p \in \mathcal{H}_0} (A_0)_p.

  • For x∈(A 0) px \in (A_0)_p and y∈(A 0) qy \in (A_0)_q a morphism in Θ(A) 1\Theta(A)_1 from xx to yy is labeled by h∈ℋ 1h \in \mathcal{H}_1 and a∈(A 1) qa \in (A_1)_q

    x→(h,a)(y=ρ(h)(x)−∂a), x \stackrel{(h,a)}{\to} (y = \rho(h)(x) - \partial a) \,,

    where ρ\rho denotes the action of ℋ\mathcal{H} on A 0A_0.

    The composition law is given by

    y (h 1,a 1)↗ ↘ (h 2,a 2) x →(h 1∘h 2,ρ(h 2)(a 1)+a 2) z. \array{ && y \\ & {}^{\mathllap{(h_1, a_1)}}\nearrow && \searrow^{\mathrlap{(h_2,a_2)}} \\ x &&\stackrel{(h_1 \circ h_2, \rho(h_2)(a_1) + a_2)}{\to}&& z } \,.

  • For k≥2k \geq 2 the family of groups Θ(A) k\Theta(A)_k is over x∈(A 0) px \in (A_0)_p the group (A k) q(A_k)_q

    Θ(A) k≥2=∐ p∈ℋ 0∐ x∈(A 0) q(A k)q \Theta(A)_{k \geq 2} = \coprod_{p \in \mathcal{H}_0} \coprod_{x\in (A_0)_q} (A_k)q

  • The boundary maps and actions are the obvious ones…

Example

(ordinary abelian chain complex as crossed complex)

Let C •C_\bullet be an ordinary chain complex of abelian groups, i.e. a chain complex of modules over the trivial groupoid.

Then (ΘC) 1(\Theta C)_1 is the groupoid with objects C 0C_0 and morphisms {x→b(x+∂b)}\{x \stackrel{b}{\to} (x + \partial b)\}. And for n≥2n \geq 2 we have that (ΘC) n(\Theta C)_n is ∐ x∈C 0C n\coprod_{x \in C_0} C_n.

Proposition

These form a pair of adjoint functors

(∇⊣Θ):Chn→Θ←∇Crs (\nabla \dashv \Theta) : Chn \stackrel{\overset{\nabla}{\leftarrow}}{\underset{\Theta}{\to}} Crs

where…

This is proposition 7.4.29.

(…)

Examples

In low degree

Say that a crossed complex CC is nn-truncated if C kC_k is trivial for k>kk \gt k.

Then

A discussion of the kind of homotopy types generally modelled by crossed complexes, namely a linear model is in homotopy n-type.

Abelian chain complexes

The notion of crossed complex generalizes the notion of chain complex of abelian groups. Clearly in degree k≥3k \geq 3 a crossed complex with C 0=*C_0 = * is a chain complex of abelian groups. To regard the first 2 degrees A 1→δA 0A_1 \stackrel{\delta}{\to} A_0 of a chain complex of abelian groups as a crossed module, form the groupoid

A 0×A 1→p 1→p 1+δA 0 A_0 \times A_1 \stackrel{\overset{p_1 + \delta}{\to}}{\underset{p_1}{\to}} A_0

and take the action of this groupoid on all C kC_k to be trivial. This yields a functor

θ:ChainCplx(Ab)→CrsCpl \theta : ChainCplx(Ab) \to CrsCpl

that embeds chain complexes of abelian groups into crossed complexes.

Complex of modules over a groupoid

The embedding of chain complexes of abelian groups into crossed complexes generalizes to an embedding of chain complexs of modules over a groupoid

θ:ChainCplx(GrpdMod)→CrsCplx. \theta : ChainCplx(GrpdMod) \to CrsCplx \,.

For details see Nonabelian Algebraic Topology, section 7.4.v.

Fundamental crossed complex

If X *X_* is a filtered space, there is a crossed complex ΠX *\Pi X_* – the fundamental crossed complex which corresponds to a (filtered and) strict ∞-groupoid version of the fundamental ∞-groupoid of XX. In degree 1 it is the subgroupoid Π 1(X 1,X 0)\Pi_1(X_1,X_0) of the fundamental groupoid Π 1(X 1)\Pi_1(X_1) of X 1X_1 on objects in X 0X_0. In degree n>1n \gt 1 it is the family of relative homotopy groups? {π n(X n,X n−1,p):p∈X 0}\{\pi_n(X_n,X_{n-1},p) : p\in X_0\}.

This gives a functor Π\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.

Fundamental crossed complex of the nn-simplex

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 nn-cubes and nn-simplices, with their CW-filtration. We obtain Π(I n)\Pi(I^n) and Π(Δ n)\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 CC is a crossed complex, then its simplicial nerve is the simplicial set with Ner(C) n=Crs(Π(Δ n),C)Ner(C)_n = Crs(\Pi(\Delta^n),C) in dimension nn.

Example

(fundamental crossed complex of the nn-simplex)

The topological nn-simplex Δ n\Delta^n is canonically a filtered space with (Δ n) k(\Delta^n)_k being the union of its kk-faces.

Then we have that Π 1((Δ n) 1,(Δ n) 0)\Pi_1((\Delta^n)_1, (\Delta^n)_0) is the groupoid whose objects are the n+1n+1 vertices of Δ n\Delta^n and which has precisely one morphism x i→x jx_i \to x_j for each ordered pair x i,x j∈(Δ n) 0x_i,x_j \in (\Delta^n)_0 (all of them being isomorphisms)

Π 1((Δ 2) 1,(Δ 2) 0)={ x 1 ↗↙ ↘↖ x 0 →← x 2}. \Pi_1((\Delta^2)_1,(\Delta^2)_0) = \left\{ \array{ && x_1 \\ & \nearrow\swarrow && \searrow \nwarrow \\ x_0 &&\stackrel{\leftarrow}{\to}&& x_2 } \right\} \,.

At any x ix_i the relative homotopy group π 2((Δ n) 2,(Δ n) 1,x i)\pi_2((\Delta^n)_2,(\Delta^n)_1, x_i) is a group on the set of 2-faces that have x ix_i as a 0-face: there is a unique homotopy class of disks in Δ n\Delta^n that sits in the 2-faces (Δ n) 2(\Delta^n)_2, whose base point is at x jx_j and whose boundary runs along the boundary of a given 2-face of Δ n\Delta^n.

So (using the equivalence of crossed complexes with strict ω\omega-groupoids) for instance ΠΔ 2\Pi \Delta^2 is generated from Π 1((Δ 2) 1,(Δ 2) 0)\Pi_1((\Delta^2)_1,(\Delta^2)_0) as above and a 2-cell

x 1 ↙ ⇓ ↖ x 0 → x 2 \array{ && x_1 \\ & \swarrow &\Downarrow& \nwarrow \\ x_0 &&\to&& x_2 }

under whiskering and composition. For instance whiskering this with x 1→x 2x_1 \to x_2 yields the 2-morphism

x 1 ↙ ⇙ ↘ x 0 → x 2. \array{ && x_1 \\ & \swarrow &\swArrow& \searrow \\ x_0 &&\to&& x_2 } \,.

One sees that ΠΔ 2\Pi \Delta^2 is the strict groupoidification of the second oriental.

Generally, ΠΔ n\Pi \Delta^n is the nn-groupoid freely generated from kk-morphisms for each kk-face of Δ n\Delta^n.

Crossed complexes as Moore complexes

Crossed complexes (of groups) correspond to group T-complexes. Any group TT-complex is a simplicial group and in the entry for them it is mentioned that a simplicial group has a group TT-complex structure if and only if 𝒩G∩D\mathcal{N}G\cap D is the trivial graded subgroup, where D=(D n) n≥1D = (D_n)_{n\geq 1} is the graded subgroup of GG generated by the degenerate elements. If GG is such a group TT-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, 𝒩G∩D\mathcal{N}G\cap D. (The Whitehead products for GG 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, CC, there is a simplicial group, K(C)K(C), constructed using an analogue of the inverse in the Dold-Kan correspondence, which is a group TT-complex and whose Moore complex is isomorphic to CC.

The generalisation to general crossed complexes (of groupoids) and simplicially enriched groupoids is quite easy to do. We will usually state results below for the group case, leaving the general case to the ‘reader’.

From simplicial group(oid)s to crossed complexes

It is fairly clear that crossed complexes / group(oid) TT-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, SSetGrpdSSet Grpd, of simplicially enriched groupoids. (The discussion in the entry on group T-complex is relevant here.)

There is a functor C(−)C(-) from simplicial groups to crossed complexes given by

C(G) n+1=𝒩G n(𝒩G n∩D n)d 0(𝒩G n+1∩D n+1),{C}(G)_{n+1} = \frac{\mathcal{N}G_n}{(\mathcal{N}G_n\cap D_n)d_0(\mathcal{N}G_{n+1}\cap D_{n+1})},

in higher dimensions with at its ‘bottom end’, the crossed module,

𝒩G 1d 0(𝒩G 2∩D 2)→𝒩G 0\frac{\mathcal{N}G_1}{d_0(\mathcal{N}G_2\cap D_2)} \to \mathcal{N}G_0

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.

References

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 a form of (strict) cubical ω\omega-groupoid with connections. This equivalence enabled a number of new results, including van Kampen type theorems and monoidal closed structures for crossed complexes.

Monographs:

Survey:

  • Ronnie Brown, Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems, to appear in Michiel Hazewinkel (ed.), Handbook of Algebra, volume 6, Elsevier, 2008/2009. (arxiv:math.AT/0212274 v7).

  • Andrew Tonks: Theory and Applications of Crossed Complexes, PhD thesis, Bangor (1993) [pdf, pdf]

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 “ω\omega-groupoid” for strict cubical ∞\infty-groupoid .

For the relation to group cohomology see

Last revised on June 13, 2024 at 11:32:52. See the history of this page for a list of all contributions to it.