03.12.59 · modern-geometry / homotopy

Classifying space of a crossed complex

shipped3 tiersLean: none

Anchor (Master): Brown-Higgins-Sivera §11 and §12; Brown-Higgins *The classifying space of a crossed complex*, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120; Grothendieck *Pursuing Stacks* 1983 (the homotopy hypothesis)

Intuition Beginner

A crossed complex is a tower of algebra: a groupoid of loops in degree one, commutative groups of higher fillings above, and boundary maps linking the floors. The classifying space turns that algebra back into a space. Out of the tower $C$ it builds a topological space $B C$ whose own homotopy reproduces the algebra you started from. Algebra goes in, a space comes out, and the two carry the same information.

Why bother? The whole programme started by reading a space and recording its homotopy as a crossed complex. The classifying space runs that arrow backwards. It lets you design a space by writing down its algebra first, then realising it. This is how you build spaces to order, with prescribed loops and prescribed higher fillings.

The simplest cases are old friends. If your tower is just a group $G$ sitting in degree one and nothing above, the space you get is $B G$ , the classifying space of the group, whose loops are exactly $G$ . If your tower is a single commutative group $A$ sitting up in degree $n$ and nothing else, the space you get is the Eilenberg-MacLane space $K (A, n)$ , whose only homotopy lives in dimension $n$ and equals $A$ . The classifying space of a crossed complex is the common generalisation: one machine that produces $B G$ , $K (A, n)$ , and everything built by stacking them together.

Visual Beginner

On the left, a vertical tower of algebraic boxes: a groupoid of loops in degree one at the bottom, flat commutative groups in each higher degree above, downward boundary arrows linking them. A large arrow labelled " $B$ " points to the right. On the right, a single topological space, drawn as a blob with loops and higher cells inside it. Underneath, a caption shows two special outputs: the tower that is only a group $G$ produces the space $B G$ , and the tower that is only a group $A$ in degree $n$ produces the space $K (A, n)$ .

The picture to keep is the single arrow $B$ that converts an algebraic tower into a space, with $B G$ and $K (A, n)$ as the two simplest things it can output.

Worked example Beginner

Take the two simplest towers and read off the spaces they classify.

Step 1. The tower that is only a group. Let the crossed complex be a single group $G$ in degree one, with the one-element group in every higher degree. There are no higher fillings to realise, only the loops recorded by $G$ .

Step 2. The space it builds is $B G$ . Its loops, measured by the fundamental group, give back $G$ , and it has no homotopy in any higher dimension. So $B G$ is the space $K (G, 1)$ : homotopy concentrated in degree one, equal to $G$ . For $G$ the whole numbers this is the circle; for $G$ the two-element group this is infinite real projective space.

Step 3. The tower that is only one commutative group up high. Let the crossed complex be a single commutative group $A$ sitting in degree $n$ , with the one-element group in every other degree.

Step 4. The space it builds is the Eilenberg-MacLane space $K (A, n)$ . Its only homotopy lives in dimension $n$ and equals $A$ ; every other dimension is empty. For $A$ the whole numbers and $n$ equal to two, this is infinite complex projective space.

What this tells us: the classifying space is one machine with two famous outputs. Concentrate the tower in degree one and you get $B G$ ; concentrate it in degree $n$ and you get $K (A, n)$ . Every space the machine builds is assembled from these basic pieces stacked by the boundary maps of the tower.

Check your understanding Beginner

Formal definition Intermediate+

Let $C$ be a crossed complex 03.12.55. Its classifying space is the geometric realisation of a simplicial (or cubical) set called the nerve of $C$ ^{[Brown-Higgins-Sivera §11]}.

The nerve. For each $n \geq 0$ let $Δ_{*}^{n}$ (respectively $I_{*}^{n}$ in the cubical model) denote the standard $n$ -simplex (respectively the $n$ -cube) with its skeletal filtration, and let $Π Δ_{*}^{n} = Π (Δ_{*}^{n})$ be its fundamental crossed complex. The nerve $N C$ is the simplicial set $$ (N\mathcal{C})n ;=; \mathsf{Crs}\big(\Pi\Delta^n,\ \mathcal{C}\big), $$ the set of crossed-complex morphisms from $\Pi\Delta^n_ $t o$ \mathcal{C} $, w i t h s im pl i c ia l f a ces an dd e g e n er a c i es in d u ce d b y t h eco f a ce an d co d e g e n er a cy ma p so f t h ecos im pl i c ia l cr osse d co m pl e x$ n \mapsto \Pi\Delta^n_* $. T h ec u bi c a l v er s i o n r e pl a ces$ \Delta^n_* $b y$ I^n_* $an d$ \Pi\Delta^n_* $b y$ \Pi I^n_*$; the two nerves have homeomorphic realisations on the crossed complexes that arise.

The classifying space. The classifying space of $C$ is the geometric realisation $$ B\mathcal{C} ;=; |N\mathcal{C}|. $$ This is the value of a functor $B : Crs \to Top$ , natural in $C$ : a crossed-complex morphism $f : C \to D$ induces a simplicial map $N f : N C \to N D$ and hence a continuous map $B f : B C \to B D$ .

Homotopy groups of the nerve. For a reduced crossed complex $C$ (one object, $C_{0} = *$ ) the homotopy groups of $B C$ are the homology of $C$ : $$ \pi_1(B\mathcal{C}) \cong \pi_1\mathcal{C} = \operatorname{coker}\delta_2, \qquad \pi_n(B\mathcal{C}) \cong H_n(\mathcal{C}) = \ker\delta_n / \operatorname{im}\delta_{n+1} \quad (n \ge 2), $$ with the first acting on the higher ones as the fundamental-groupoid action of $C$ . Thus $B$ realises the algebra of $C$ as the homotopy of a space ^{[Brown-Higgins 1991]}.

Counterexamples to common slips

$B C$ is not the realisation of $C$ regarded as a chain complex. The nerve uses the full nonabelian crossed-complex hom-sets $Crs (Π Δ_{*}^{n}, C)$ , so $π_{1} (B C) = coker δ_{2}$ is genuinely nonabelian when $δ_{2}$ is; abelianising first collapses this to homology and produces the wrong fundamental group.
The classification bijection $[X, B C] ≅ [Π X_{*}, C]$ requires $X$ to carry a CW (skeletal) filtration $X_{*}$ ; it is not a statement about arbitrary spaces with arbitrary filtrations. The map sends a continuous $g : X \to B C$ to the induced $Π g : Π X_{*} \to Π B C \to C$ after a canonical $Π B C \to C$ .
$K (A, n)$ as $B$ of a one-degree crossed complex requires $A$ abelian and $n \geq 2$ ; for $n = 1$ one may use any group $G$ , abelian or not, and gets $B G = K (G, 1)$ . Placing a nonabelian group in degree $n \geq 2$ is not a crossed complex.

Key theorem with proof Intermediate+

Theorem (homotopy classification). Let $C$ be a crossed complex and let $X$ be a CW complex with its skeletal filtration $X_$. There is a natural bijection* $$ [X,\ B\mathcal{C}] ;\cong; [\Pi X_*,\ \mathcal{C}] $$ between homotopy classes of continuous maps $X \to B C$ and homotopy classes of crossed-complex morphisms $\Pi X_ \to \mathcal{C} $[r e f : T O D O_{R} E F B r o w n - H i g g in s 1991] . C o n se q u e n tl y, w h e n$ \mathcal{C} = K(A,n) $i s t h ecr osse d co m pl e x w i t h$ A $in d e g r ee$ n$ and the identity group elsewhere,* $$ [X,\ K(A,n)] ;\cong; H^n(X;\ A), $$ so $B$ recovers the Eilenberg-MacLane representability of ordinary cohomology.

The construction and the bijection follow Brown-Higgins-Sivera ^{[Brown-Higgins-Sivera §11]}.

Proof. Write $B = B C = ∣ N C ∣$ . The fundamental crossed complex of $B$ comes equipped with a natural counit. Apply $Π$ to the realisation: there is a natural crossed-complex morphism $$ \varepsilon \colon \Pi(B\mathcal{C})* \longrightarrow \mathcal{C}, $$ where $(B\mathcal{C}) $i s t h es k e l e t a l f i l t r a t i o n o f t h er e a l i s a t i o n . T h e m or p hi s m$ \varepsilon $i s b u i l t ce l l b y ce l l : an$ n $- ce l l o f$ |N\mathcal{C}| $i s an o n d e g e n er a t ee l e m e n t o f$ (N\mathcal{C})n = \mathsf{Crs}(\Pi\Delta^n, \mathcal{C}) $, t ha t i s, am or p hi s m$ \Pi\Delta^n_* \to \mathcal{C} $; e v a l u a t in g o n t h e t o p ce l l o f$ \Pi\Delta^n_* $an d a sse mb l in g o v er a l l ce l l sy i e l d s$ \varepsilon$, and the simplicial identities make it a crossed-complex morphism.

The bijection. For a CW complex $X$ with skeletal filtration $X_{*}$ , define $$ \Theta \colon [X, B\mathcal{C}] \longrightarrow [\Pi X_*, \mathcal{C}], \qquad \Theta[g] = [\varepsilon \circ \Pi(g_*)], $$ where $g_{*} : X_{*} \to (B C)_{*}$ is the filtered map induced by $g$ after a cellular approximation (every map of CW complexes is homotopic to a filtered one), $Π (g_{*}) : Π X_{*} \to Π (B C)_{*}$ is its image under the fundamental-crossed-complex functor, and $ε$ is the counit. The map $Θ$ is well defined: homotopic maps $g ≃ g^{'}$ give filtered-homotopic cellular approximations, hence homotopic morphisms of crossed complexes.

Surjectivity. Given a morphism $ϕ : Π X_{*} \to C$ , the simplicial set $N C$ has the property that morphisms out of the fundamental crossed complex of a CW filtration are realised by simplicial maps into $N C$ : an $n$ -cell $e$ of $X$ is a characteristic map $Π Δ_{*}^{n} \to Π X_{*}$ , and the composite $ϕ \circ (cell) : Π Δ_{*}^{n} \to C$ is precisely an $n$ -simplex of $N C$ . Sending each cell of $X$ to that simplex defines a simplicial map $N (X) \to N C$ on the simplicial set $N (X)$ modelling $X$ , whose realisation $g : X \to B C$ satisfies $Θ [g] = [ϕ]$ . This uses that $Π$ takes the cellular structure of $X$ to a presentation of $Π X_{*}$ by free generators in each degree.

Injectivity. Suppose $Θ [g] = Θ [g^{'}]$ , so $ε Π (g_{*}) ≃ ε Π (g_{*}^{'})$ through a homotopy $H : Π X_{*} \otimes Π I_{*} \to C$ of crossed complexes, where $\otimes$ is the monoidal tensor product of $Crs$ and $I_{*}$ is the unit interval filtered by its endpoints. By the exponential law $Π (X_{*} \otimes I_{*}) ≅ Π X_{*} \otimes Π I_{*}$ on CW filtrations 03.12.55, $H$ corresponds to a morphism $Π (X \times I)_{*} \to C$ , which by surjectivity is realised by a continuous homotopy $X \times I \to B C$ from $g$ to $g^{'}$ . Hence $[g] = [g^{'}]$ . Thus $Θ$ is a bijection, natural in both $X$ and $C$ because $ε$ and $Π$ are natural.

The Eilenberg-MacLane corollary. Let $C = K (A, n)$ be the crossed complex with $C_{n} = A$ (abelian, $n \geq 2$ ) and the identity group in every other degree, all boundaries the zero map. A crossed-complex morphism $Π X_{*} \to K (A, n)$ is then a single equivariant homomorphism $C_{n} (X_{*}) = π_{n} (X^{(n)}, X^{(n - 1)}) \to A$ that vanishes on the image of $δ_{n + 1}$ and on the action-induced relations, i.e. a homomorphism from $C_{n} (X_{*}) / im δ_{n + 1}$ that kills $ker δ_{n}$ . By the cellular description of the crossed complex, $C_{n} (X_{*})$ is the free module on the $n$ -cells with $δ$ the cellular boundary, so a morphism to $K (A, n)$ is exactly a cellular $n$ -cocycle of $X$ with coefficients in $A$ up to coboundary. Homotopy of such morphisms is precisely cohomologous, so $$ [\Pi X_*, K(A,n)] \cong H^n(X; A). $$ Combining with the theorem gives $[X, B (K (A, n))] = [X, K (A, n)] ≅ H^{n} (X; A)$ , the classical representability. $□$

Bridge. The classification bijection is the foundational reason the classifying space deserves its name: $B C$ corepresents the functor $X \mapsto [Π X_{*}, C]$ , so a space mapping into $B C$ is the same data as an algebraic morphism out of its fundamental crossed complex. This is exactly the universal property that turns the analysis arrow $X \mapsto Π X_{*}$ of 03.12.55 into a synthesis arrow $C \mapsto B C$ , and it generalises the Eilenberg-MacLane representability $[X, K (A, n)] ≅ H^{n} (X; A)$ from a single abelian coefficient group to an entire nonabelian crossed complex of coefficients. The central insight is that the counit $ε : Π B C \to C$ together with the exponential law for $\otimes$ supplies both halves of the bijection at once. Putting these together, $B$ and $Π$ form an adjoint-like pair on homotopy categories, and this pattern builds toward the recognition of crossed complexes as a model for a class of homotopy types; the same counit appears again in the cohomology classification of 03.12.05 read through the nerve.

Exercises Intermediate+

Exercise 3 (medium, short-answer).

Using the classification theorem, prove that $[X, K (A, n)] ≅ H^{n} (X; A)$ for a CW complex $X$ and abelian $A$ , $n \geq 2$ , by identifying $[Π X_{*}, K (A, n)]$ with cohomology.

Hint

A morphism $Π X_{*} \to K (A, n)$ is a single equivariant homomorphism $C_{n} (X_{*}) \to A$ vanishing on $im δ_{n + 1}$ ; two such are homotopic iff they differ by a coboundary.

Answer

A morphism $ϕ : Π X_{*} \to K (A, n)$ is determined by its degree- $n$ component, a homomorphism $ϕ_{n} : C_{n} (X_{*}) \to A$ , equivariant and satisfying $ϕ_{n} \circ δ_{n + 1} = 0$ (so $ϕ_{n}$ is a cocycle on the cellular chains). Homotopy $ϕ ≃ ϕ^{'}$ of crossed-complex morphisms into $K (A, n)$ amounts to a degree- $(n - 1)$ map $h : C_{n - 1} (X_{*}) \to A$ with $ϕ_{n} - ϕ_{n}^{'} = h \circ δ_{n}$ , i.e. the cocycles differ by a coboundary. Thus $[Π X_{*}, K (A, n)] = Z^{n} / B^{n} = H^{n} (X; A)$ , and the theorem gives $[X, K (A, n)] ≅ H^{n} (X; A)$ . Rubric: full credit for the cocycle identification and the coboundary identification of homotopy.

Exercise 4 (medium, short-answer).

Show that $B : Crs \to Top$ is a functor: describe the action on a crossed-complex morphism $f : C \to D$ and verify functoriality.

Hint

$f$ induces $N f$ on nerves by post-composition $Crs (Π Δ_{*}^{n}, C) \to Crs (Π Δ_{*}^{n}, D)$ ; realisation is itself a functor.

Answer

For $f : C \to D$ , define $(N f)_{n} : (N C)_{n} \to (N D)_{n}$ by $ψ \mapsto f \circ ψ$ for $ψ \in Crs (Π Δ_{*}^{n}, C)$ . This commutes with faces and degeneracies because they act by precomposition with the cosimplicial maps of $Π Δ_{*}^{∙}$ , and pre- and post-composition commute; so $N f$ is simplicial. Geometric realisation sends $N f$ to a continuous $B f = ∣ N f ∣$ . Functoriality: $N (id) = id$ and $N (g f) = (N g) (N f)$ since post-composition is functorial, and $∣ \cdot ∣$ is a functor, so $B (id) = id$ and $B (g f) = B (g) B (f)$ . Rubric: full credit for the post-composition description, simpliciality, and the two functor identities.

Exercise 5 (medium, short-answer).

Let $C$ be a crossed complex with $δ_{2} : C_{2} \to C_{1}$ nonzero, so $π_{1} C = coker δ_{2}$ may be a proper quotient of $C_{1}$ . Explain why $π_{1} (B C)$ is $coker δ_{2}$ and not $C_{1}$ , and why this distinguishes $B C$ from the realisation of a chain complex.

Hint

The 2-cells of the nerve kill the image of $δ_{2}$ in the fundamental group; a chain complex has no nonabelian degree-one data to begin with.

Answer

In $N C$ the 1-simplices are arrows of $C_{1}$ and the 2-simplices include the fillings recording $δ_{2} c$ for $c \in C_{2}$ ; realising attaches a 2-cell along each such loop $δ_{2} c$ , so $π_{1} (B C) = C_{1} / ⟨ ⟨ im δ_{2} ⟩ ⟩ = coker δ_{2} = π_{1} C$ , the normal closure being automatic since $im δ_{2}$ is already normal by equivariance. A chain complex carries an abelian group in degree one with no group multiplication and no $δ_{2}$ -conjugation data, so its realisation can only produce abelian $π_{1}$ ; the crossed complex retains the nonabelian quotient. Rubric: full credit for the 2-cell attachment giving $coker δ_{2}$ and the contrast with the abelian chain-complex case.

Exercise 6 (hard, short-answer).

Let $C$ be a crossed module $δ_{2} : M \to P$ regarded as a crossed complex concentrated in degrees one and two. Show that $B C$ is a homotopy 2-type and identify $π_{1}$ and $π_{2}$ .

Hint

A crossed complex with $C_{n}$ the identity group for $n \geq 3$ has $H_{n} = 0$ there; $π_{1} = coker δ_{2}$ and $π_{2} = ker δ_{2}$ .

Answer

With $C_{1} = P$ , $C_{2} = M$ , and the identity group in degrees $\geq 3$ , the homology vanishes above degree two, so $π_{n} (B C) = 0$ for $n \geq 3$ and $B C$ is a homotopy 2-type. The classification gives $π_{1} (B C) = coker δ_{2} = P / δ_{2} M$ and $π_{2} (B C) = H_{2} (C) = ker δ_{2}$ , with the $π_{1}$ -action on $π_{2}$ induced by the $P$ -action on $M$ . This is the standard correspondence between crossed modules and (pointed, connected) homotopy 2-types. Rubric: full credit for the vanishing above degree two, $π_{1} = coker δ_{2}$ , $π_{2} = ker δ_{2}$ , and the action.

Exercise 7 (hard, symbolic).

Let $A$ be abelian and consider the crossed complex $K (A, n)$ for $n \geq 2$ . Using $π_{k} (B C) ≅ H_{k} (C)$ , compute all homotopy groups of $B (K (A, n))$ symbolically and confirm the loop-space relation $Ω K (A, n) ≃ K (A, n - 1)$ at the level of homotopy groups.

Hint

$K (A, n)$ as a crossed complex has $A$ in degree $n$ and the identity group elsewhere, so $H_{k} = A$ if $k = n$ and $0$ otherwise; looping shifts homotopy down by one.

Answer

$H_{k} (K (A, n)) = A$ for $k = n$ and $0$ otherwise, so $π_{k} (B (K (A, n))) = A$ if $k = n$ and $0$ if $k \neq = n$ : thus $B (K (A, n))$ is the topological $K (A, n)$ . For the loop space, $π_{k} (Ω K (A, n)) = π_{k + 1} (K (A, n)) = A$ iff $k + 1 = n$ , i.e. $k = n - 1$ , and $0$ otherwise; these are exactly the homotopy groups of $K (A, n - 1)$ , so $Ω K (A, n) ≃ K (A, n - 1)$ . Rubric: full credit for the concentrated homotopy of $B (K (A, n))$ and the index-shift verification of the loop relation.

Advanced results Master

The classifying space completes the circuit of the Brown-Higgins programme. The fundamental crossed complex $Π : FTop \to Crs$ of 03.12.55 reads a filtered space as algebra; the classifying space $B : Crs \to Top$ writes a crossed complex back as a space, and the homotopy classification $[X, B C] ≅ [Π X_{*}, C]$ binds the two into a representability statement. Three structural facts organise the consequences.

Crossed-complex cohomology and the representing object. For a crossed complex $C$ and a CW complex $X$ , the set $[X, B C]$ is the $C$ -cohomology of $X$ . When $C = K (A, n)$ this is $H^{n} (X; A)$ ; for a general $C$ it is a nonabelian, multi-degree invariant that simultaneously records cohomology in several degrees together with the action and boundary data, with a long-exact-sequence machinery coming from short exact sequences of crossed complexes ^{[Brown-Higgins 1991]}. The classifying space is the corepresenting object: it converts the algebraic hom-functor $[Π (-)_{*}, C]$ into an ordinary homotopy functor on $Top$ . This is the precise sense in which $B C$ generalises the Eilenberg-MacLane space from a single abelian coefficient to a whole crossed complex of coefficients, and it specialises to the Eilenberg-MacLane representability of 03.12.05 when the coefficients are concentrated in one degree.

Homotopy 2-types and the strict-model boundary. Crossed modules — crossed complexes concentrated in degrees one and two — model exactly the pointed connected homotopy 2-types via $B$ : the homotopy category of crossed modules is equivalent to that of pointed connected spaces $X$ with $π_{n} X = 0$ for $n \geq 3$ , with $π_{1} = coker δ_{2}$ and $π_{2} = ker δ_{2}$ . This is the Mac Lane-Whitehead theorem in modern form, and it is sharp: crossed complexes are a strict $ω$ -groupoid model, and strict $ω$ -groupoids do not model all homotopy types. The first failure is in dimension three. The Whitehead product and the higher $k$ -invariants of a $3$ -type already require the weak structure that crossed complexes flatten; $B$ on crossed complexes lands precisely in the homotopy types with vanishing Whitehead products and vanishing Postnikov interaction beyond the linear boundary data. Crossed complexes are therefore the linear or abelianised-above-degree-two slice of homotopy theory, faithful through 2-types and increasingly lossy above.

The homotopy hypothesis. Grothendieck's homotopy hypothesis ^{[Grothendieck 1983]} asserts that weak $\infty$ -groupoids model all homotopy types: the realisation/nerve adjunction between an appropriate category of weak $\infty$ -groupoids and topological spaces is an equivalence of homotopy theories. The crossed-complex classifying space is the strict, computable shadow of this conjecture. Where strict crossed complexes capture 2-types exactly and higher types partially, the weak $\infty$ -groupoids of Grothendieck, Batanin, Leinster, and the quasi-category model of Joyal and Lurie are designed to capture every homotopy type with no loss. Homotopy type theory makes the same hypothesis foundational: in the univalent setting a type is an $\infty$ -groupoid, identity types are path spaces, and the homotopy hypothesis is built into the semantics rather than proved about it. The classifying-space functor $B$ on $Crs$ is the first nonabelian rung of this ladder — the place where the passage from algebra to space stops being a matter of chain complexes and abelian groups and becomes genuinely homotopical.

Synthesis. The classifying space is the foundational reason the entire crossed-complex programme is more than an invariant: it makes $Crs$ a category of coefficients whose objects build spaces, so that the analysis functor $Π$ and the synthesis functor $B$ together present crossed complexes as a model for a slice of homotopy theory. The central insight is that the natural bijection $[X, B C] ≅ [Π X_{*}, C]$ is exactly an adjunction between $Π$ and $B$ on homotopy categories, and this is dual to the way Eilenberg-MacLane spaces represent cohomology — putting these together, $K (A, n)$ is recovered as $B$ of a one-degree crossed complex and $H^{n} (X; A)$ as the corepresented functor, so ordinary cohomology becomes the abelian, single-degree corner of crossed-complex cohomology. This pattern recurs one level up: the strict model captures 2-types faithfully, generalises the Mac Lane-Whitehead correspondence, and then meets its limit at 3-types, which is where Grothendieck's homotopy hypothesis takes over and the bridge is the replacement of strict crossed complexes by weak $\infty$ -groupoids. The crossed-complex classifying space builds toward that hypothesis and appears again in homotopy type theory, where the algebra-to-space passage is made foundational rather than constructed; the crossed complex is thus the precise nonabelian generalisation of a chain complex, and its classifying space the precise generalisation of an Eilenberg-MacLane space, faithful exactly as far as strictness allows.

Full proof set Master

Proposition (homotopy groups of the nerve). Let $C$ be a reduced crossed complex (one object). Then $π_{1} (B C) ≅ coker δ_{2} = π_{1} C$ and $π_{n} (B C) ≅ H_{n} (C) = ker δ_{n} / im δ_{n + 1}$ for $n \geq 2$ , with the $π_{1}$ -action on $π_{n}$ induced by the fundamental-groupoid action of $C$ .

Proof. Use the classification theorem with $X = S^{k}$ , the $k$ -sphere with its minimal CW filtration. The fundamental crossed complex $Π (S^{k})_{*}$ is the free crossed complex on a single generator in degree $k$ (for $k \geq 2$ , with identity groups elsewhere; for $k = 1$ it is the free group on one generator in degree one). Hence $$ \pi_k(B\mathcal{C}) = [S^k, B\mathcal{C}]* \cong [\Pi(S^k), \mathcal{C}]_, $$ the based homotopy classes. For $k = 1$ , a based morphism $Π (S^{1})_{*} \to C$ is an element of $C_{1}$ at the basepoint, and two such are homotopic exactly when they differ by an element of $im δ_{2}$ (a homotopy is a $1$ -cell of $C$ realising a $δ_{2}$ -relation), so $π_{1} (B C) = C_{1} / im δ_{2} = coker δ_{2}$ . For $k \geq 2$ , a based morphism $Π (S^{k})_{*} \to C$ is an element $c \in C_{k}$ with $δ_{k} c = 0$ (the generator has zero boundary in $Π (S^{k})_{*}$ , and morphisms preserve $δ$ ), so it is a $k$ -cycle; a homotopy of two such morphisms is an element of $C_{k + 1}$ whose boundary is their difference, so two cycles are homotopic exactly when they differ by an element of $im δ_{k + 1}$ . Therefore $π_{k} (B C) = ker δ_{k} / im δ_{k + 1} = H_{k} (C)$ . The action of $π_{1}$ is transported from the $C_{1}$ -action on $C_{k}$ through the same identifications, descending to $π_{1} C$ by axiom CC4 of 03.12.55. $□$

Proposition ( $B$ of a one-degree crossed complex is an Eilenberg-MacLane space). For abelian $A$ and $n \geq 2$ , let $K (A, n)$ be the crossed complex with $A$ in degree $n$ , the identity group elsewhere, and all boundaries zero. Then $B (K (A, n))$ is an Eilenberg-MacLane space of type $(A, n)$ , and for $n = 1$ and any group $G$ , $B (G) = K (G, 1) = B G$ .

Proof. Apply the previous proposition to $C = K (A, n)$ . The homology is $H_{k} (C) = A$ for $k = n$ (the whole group is a cycle, no boundaries enter or leave) and $H_{k} (C) = 0$ for $k \neq = n$ (the complex is the identity group in those degrees). For $n \geq 2$ this gives $π_{n} (B (K (A, n))) = A$ and $π_{k} = 0$ for $k \neq = n$ , the defining property of an Eilenberg-MacLane space $K (A, n)$ in the sense of 03.12.05. For $n = 1$ , $C = G$ concentrated in degree one with $δ_{2}$ identity-valued gives $π_{1} (B G) = coker δ_{2} = G$ and $π_{k} = H_{k} = 0$ for $k \geq 2$ , so $B G = K (G, 1)$ , the classifying space of the group. $□$

Proposition (crossed modules model homotopy 2-types). Let $δ_{2} : M \to P$ be a crossed module, regarded as a crossed complex $C$ concentrated in degrees one and two. Then $B C$ is a connected homotopy 2-type with $π_{1} (B C) = coker δ_{2}$ , $π_{2} (B C) = ker δ_{2}$ , and the $π_{1}$ -action on $π_{2}$ induced by the $P$ -action on $M$ ; conversely every pointed connected homotopy 2-type arises this way up to homotopy.

Proof. For the direct statement apply the homotopy-group proposition: $H_{k} (C) = 0$ for $k \geq 3$ since $C_{k}$ is the identity group there, so $π_{k} (B C) = 0$ for $k \geq 3$ and $B C$ is a 2-type; $π_{1} = coker δ_{2}$ and $π_{2} = H_{2} (C) = ker δ_{2}$ (no $C_{3}$ , so no boundaries into degree two), with the action as stated. For the converse, given a pointed connected 2-type $Y$ take its skeletal filtration; the fundamental crossed complex $Π Y_{*}$ is concentrated in degrees one and two up to homotopy because $π_{k} Y = 0$ for $k \geq 3$ forces $H_{k} (Π Y_{*}) = 0$ there, and the truncation $τ_{\leq 2} Π Y_{*}$ is the Whitehead crossed module of the pair $(Y^{(2)}, Y^{(1)})$ of 03.12.55; its classifying space recovers $Y$ up to homotopy by the classification theorem applied to $id_{Y}$ . Hence the homotopy categories of crossed modules and of pointed connected 2-types are equivalent. $□$

Connections Master

The crossed complex of a filtered space 03.12.55 is the input to $B$ : the classifying space inverts the analysis functor $Π : FTop \to Crs$ , and the homotopy classification $[X, B C] ≅ [Π X_{*}, C]$ is an adjunction between $Π$ and $B$ . The exponential law $Π (X_{*} \otimes Y_{*}) ≅ Π X_{*} \otimes Π Y_{*}$ established there is exactly what makes the homotopies in the injectivity half of the classification theorem realisable, so the monoidal closed structure of $Crs$ is load-bearing for the representability statement.
The Eilenberg-MacLane space 03.12.05 is the abelian, single-degree special case: $B$ of the crossed complex $K (A, n)$ is the topological $K (A, n)$ , and the classification specialises to $[X, K (A, n)] ≅ H^{n} (X; A)$ , the representability of ordinary cohomology. The crossed-complex classifying space is the nonabelian, multi-degree generalisation of this representing object, so cohomology with a single abelian coefficient group sits inside crossed-complex cohomology as its linear corner.
The 2-group / internal-category picture 03.12.56 is the truncated source of the homotopy 2-types produced by $B$ : a crossed module is a strict 2-group by the Brown-Spencer equivalence, and its classifying space is the homotopy 2-type with $π_{1} = coker δ_{2}$ and $π_{2} = ker δ_{2}$ . The Mac Lane-Whitehead correspondence between crossed modules and 2-types is realised concretely by the functor $B$ studied here.
The cubical $ω$ -groupoid of a filtered space 03.12.57 supplies the cubical nerve used to build $B C$ : the cubical model $(N C)_{n} = Crs (Π I_{*}^{n}, C)$ has a homeomorphic realisation to the simplicial one, and the equivalence of cubical $ω$ -groupoids with crossed complexes is what lets the easy cubical subdivision arguments transfer to the classifying-space construction. The exponential law that powers the classification proof is proved on the cubical side.

Historical & philosophical context Master

The classifying space of a crossed complex was constructed and its homotopy-classification property proved by Ronald Brown and Philip J. Higgins in The classifying space of a crossed complex (Math. Proc. Camb. Phil. Soc. 110, 1991, 95-120) ^{[Brown-Higgins 1991]}, which defined the nerve $N C$ , the realisation $B C = ∣ N C ∣$ , and the natural bijection $[X, B C] ≅ [Π X_{*}, C]$ for CW complexes $X$ . The construction extends J.H.C. Whitehead's homotopy-system description of 2- and 3-dimensional complexes and the Eilenberg-MacLane theory of $K (A, n)$ and ordinary cohomology to a single functor on crossed complexes; the special cases $B G = K (G, 1)$ and $K (A, n)$ are recovered as one-degree instances. The full development, including the cohomology interpretation and the tensor product, was consolidated in §11-§12 of Brown, Higgins, and Sivera, Nonabelian Algebraic Topology (EMS Tracts in Mathematics 15, 2011) ^{[Brown-Higgins-Sivera §11]}.

The wider frame is Alexander Grothendieck's homotopy hypothesis, set out in the 1983 manuscript Pursuing Stacks ^{[Grothendieck 1983]}, the proposal that weak $\infty$ -groupoids model all homotopy types via a nerve-realisation equivalence. Crossed complexes are a strict and computable model that is faithful through homotopy 2-types and lossy above; the classifying space is the explicit algebra-to-space functor in that strict setting, and it locates the nonabelian algebraic topology of Brown and Higgins as the first genuinely nonabelian rung between the Eilenberg-MacLane theory of abelian coefficients and the weak $\infty$ -groupoid models pursued by Batanin, Leinster, Joyal, and Lurie. Homotopy type theory, in the univalent foundations of Voevodsky, takes the same identification of types with $\infty$ -groupoids as a primitive, making the passage from algebra to space a feature of the semantics.

Bibliography Master

@article{BrownHiggins1991ClassifyingSpace,
  author  = {Brown, Ronald and Higgins, Philip J.},
  title   = {The classifying space of a crossed complex},
  journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
  volume  = {110},
  number  = {1},
  year    = {1991},
  pages   = {95--120}
}

@book{BrownHigginsSivera2011NAT,
  author    = {Brown, Ronald and Higgins, Philip J. and Sivera, Rafael},
  title     = {Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids},
  series    = {EMS Tracts in Mathematics},
  volume    = {15},
  publisher = {European Mathematical Society},
  year      = {2011}
}

@unpublished{Grothendieck1983PursuingStacks,
  author = {Grothendieck, Alexander},
  title  = {Pursuing Stacks},
  note   = {Manuscript, 1983; typeset edition ed. G. Maltsiniotis, Documents Math\'ematiques, SMF},
  year   = {1983}
}

@article{MacLaneWhitehead1950,
  author  = {Mac Lane, Saunders and Whitehead, J. H. C.},
  title   = {On the $3$-type of a complex},
  journal = {Proceedings of the National Academy of Sciences USA},
  volume  = {36},
  year    = {1950},
  pages   = {41--48}
}

Prerequisites

03.12.55
03.12.05

Tier anchors

beginner: Brown-Higgins-Sivera *Nonabelian Algebraic Topology* §11 informal (a crossed complex builds a space whose homotopy it controls; $BG$ and $K(A,n)$ as the simplest cases)
intermediate: Brown-Higgins-Sivera §11 (the cubical/simplicial nerve $N\mathcal{C}$, $B\mathcal{C} = |N\mathcal{C}|$, and the homotopy classification $[X, B\mathcal{C}] \cong [\Pi X_*, \mathcal{C}]$); Brown-Higgins *The classifying space of a crossed complex* 1991
master: Brown-Higgins-Sivera §11 and §12; Brown-Higgins *The classifying space of a crossed complex*, Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120; Grothendieck *Pursuing Stacks* 1983 (the homotopy hypothesis)

References

Brown, Higgins, Sivera — Nonabelian Algebraic Topology · §11 the classifying space $B\mathcal{C} = |N\mathcal{C}|$ of a crossed complex and the homotopy classification theorem $[X, B\mathcal{C}] \cong [\Pi X_*, \mathcal{C}]$; §12 the cohomology interpretation; free at https://groupoids.org.uk/pdffiles/NAT-book.pdf
Brown, Higgins — The classifying space of a crossed complex · Math. Proc. Camb. Phil. Soc. 110 (1991) 95-120; the nerve $N\mathcal{C}$, the realisation $B\mathcal{C}$, and the natural bijection $[X, B\mathcal{C}] \cong [\Pi X_*, \mathcal{C}]$
Grothendieck — Pursuing Stacks · Manuscript 1983 (letter to Quillen); the homotopy hypothesis that $\infty$-groupoids model homotopy types

Estimated time

beginner: 16m
intermediate: 42m
master: 82m