03.12.39 · differential-geometry / homotopy-theory

Simplicial group and the W-bar classifying functor

shipped3 tiersLean: none

Anchor (Master): Kan 1958; May Simplicial Objects Ch. 5; Goerss-Jardine Ch. V; Curtis 1971

Intuition [Beginner]

A simplicial group is a simplicial set where each set of $n$ -simplices carries a group structure, and the face and degeneracy maps are group homomorphisms. This is a "group up to homotopy": the group structure varies simplicially across dimensions.

The key insight, due to Kan, is that simplicial groups always satisfy the Kan extension condition -- every horn has a filler. This means every simplicial group is automatically a "good" simplicial set for homotopy theory. You never need to worry about whether fillers exist; they always do.

The W-bar construction $\overline{W} (G)$ takes a simplicial group $G$ and produces a simplicial set (not a group) called the classifying space of $G$ . The relationship is: $G ≃ Ω∣ \overline{W} (G) ∣$ . That is, the original simplicial group is the loop space of its classifying space. This is a purely combinatorial version of the classifying-space construction $B G$ for topological groups.

The construction is powerful because it translates questions about loop spaces into questions about simplicial groups, where algebraic methods (group theory, homological algebra) apply directly.

Visual [Beginner]

On the left, a simplicial group $G$ drawn as a tower of groups $G_{0}, G_{1}, G_{2}$ connected by face and degeneracy homomorphisms. An arrow labelled " $\overline{W}$ " points to the right, where the classifying space $\overline{W} (G)$ is drawn as a simplicial set with a fibration $W G \to \overline{W} (G)$ . The fibre over the basepoint is $G$ .

The W-bar construction builds a space whose loops recover the original group, providing a geometric realisation of the group's homotopy-theoretic content.

Worked example [Beginner]

The constant simplicial group. Let $G$ be an ordinary discrete group. The constant simplicial group at $G$ has $G_{n} = G$ for all $n$ , with all face and degeneracy maps equal to the identity. The W-bar construction produces $\overline{W} (G)$ , which is exactly the nerve $N (G)$ of the group viewed as a one-object category. The geometric realisation $∣ N (G) ∣$ is the classifying space $B G$ .

For $G = Z /2$ , the nerve $N (Z /2)$ has one 0-simplex, two 1-simplices (the identity and the twisted element), four 2-simpxes (pairs of elements), etc. The realisation $B (Z /2) = RP^{\infty}$ , and $Ω B (Z /2) ≃ Z /2$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Definition (Simplicial group). A simplicial group is a simplicial object in the category of groups: a functor $G : Δ^{op} \to Grp$ . Equivalently, $G$ is a sequence of groups $G_{n}$ ( $n \geq 0$ ) with face homomorphisms $d_{i} : G_{n} \to G_{n - 1}$ and degeneracy homomorphisms $s_{i} : G_{n} \to G_{n + 1}$ satisfying the simplicial identities.

Definition (Moore complex). The Moore complex (or normalised chain complex) of a simplicial group $G$ is the chain complex $N (G)$ with $N (G)_{n} = ⋂_{i = 0}^{n - 1} ker (d_{i} : G_{n} \to G_{n - 1})$ and differential $\partial_{n} = d_{n} ∣_{N (G)_{n}} : N (G)_{n} \to N (G)_{n - 1}$ . The Moore complex satisfies $\partial^{2} = 0$ and encodes all the homotopy-group information: $π_{n} (∣ G ∣) = H_{n} (N (G))$ for $n \geq 0$ .

Definition (W-bar construction). For a simplicial group $G$ , the W-bar construction $\overline{W} (G)$ is the simplicial set defined by:

$\overline{W} (G)_{n} = {(g_{1}, g_{2}, \dots, g_{n}) : g_{i} \in G_{i - 1}}$

with face maps $d_{0} (g_{1}, \dots, g_{n}) = (g_{2}, \dots, g_{n})$ , and for $i > 0$ :

$d_{i} (g_{1}, \dots, g_{n}) = (g_{1}, \dots, g_{i - 1} \cdot d_{i - 1} (g_{i}), d_{i - 1} (g_{i + 1}), \dots, d_{i - 1} (g_{n}))$

Degeneracies insert identities. The universal simplicial group $W G$ has $W G_{n} = G_{0} \times G_{1} \times \dots \times G_{n}$ with similar face maps, and there is a principal fibration $W G \to \overline{W} (G)$ with fibre $G$ .

Key theorem with proof [Intermediate+]

Theorem (Kan's loop-space recognition). For any simplicial group $G$ , the map $G \to Ω∣ \overline{W} (G) ∣$ induced by the universal bundle is a weak homotopy equivalence. Consequently, the functor $G \mapsto ∣ \overline{W} (G) ∣$ sets up an equivalence between the homotopy category of simplicial groups and the homotopy category of connected pointed spaces.

Proof sketch. The universal bundle $W G \to \overline{W} (G)$ is a Kan fibration with fibre $G$ (the fibre over the basepoint is the kernel of $W G_{n} \to \overline{W} (G)_{n}$ , which is isomorphic to $G_{n}$ ). The total space $W G$ is contractible (it deformation-retracts to the basepoint via the extra coordinate). By the Puppe fibre sequence 03.12.28 applied to $W G \to \overline{W} (G)$ , we get $G ≃ F ≃ Ω∣ \overline{W} (G) ∣$ since $W G$ is contractible.

For the equivalence of homotopy categories: every connected pointed simplicial set $X$ has a simplicial-group model $G (X)$ (the Kan loop group of $X$ ), and $\overline{W} (G (X)) ≃ X$ . This gives the inverse functor. $□$

Bridge. The W-bar construction is the simplicial-set version of the classifying-space construction for topological groups; the fibration $W G \to \overline{W} (G)$ with fibre $G$ mirrors the universal principal bundle $E G \to B G$ from [topology.fibration], the loop-space recognition uses the Puppe fibre sequence of 03.12.28 applied to the contractible total space $W G$ , and the Moore complex provides the algebraic link to the homology of [topology.singular-homology] via the identification $π_{n} (G) = H_{n} (N (G))$ .

Exercises [Intermediate+]

Advanced results [Master]

Kan loop group. For any reduced simplicial set $X$ (connected, with one vertex), the Kan loop group $G (X)$ is a simplicial group defined by $G (X)_{n} = Free (X_{n + 1}) / {s_{0} x = e}$ (the free group on the $(n + 1)$ -simplices modulo the relations that degenerate simplices are identity elements). Kan's theorem states that $∣ G (X) ∣ ≃ Ω∣ X ∣$ and $\overline{W} (G (X)) ≃ X$ .

Curtis spectral sequence. The Curtis spectral sequence relates the homology of a simplicial group $G$ to the homotopy groups of $∣ G ∣$ via the lower central series of the Moore complex. This provides a computational tool for determining the homotopy groups of simplicial groups and their classifying spaces.

Synthesis. The W-bar construction completes the circle of ideas connecting simplicial algebra to homotopy theory; the simplicial group structure of this unit extends the simplicial sets of 03.12.24 by adding group-theoretic operations, the loop-space recognition theorem uses the fibre sequence of 03.12.28 to identify $G$ with $Ω∣ \overline{W} (G) ∣$ , the bisimplicial techniques of 03.12.36 appear in the proof that $\overline{W}$ preserves weak equivalences (via the realisation lemma applied to the bar construction), and the Moore complex provides a chain-complex representation of the simplicial group that connects back to the singular homology of [topology.singular-homology] through the Dold-Kan correspondence. The entire framework translates loop-space topology into simplicial algebra, where combinatorial and homological methods apply.

Full proof set [Master]

Proposition (Contractibility of $W G$ ). The total space $W G$ of the universal bundle over $\overline{W} (G)$ is contractible.

Proof. Define a simplicial homotopy $H : W G \times Δ^{1} \to W G$ as follows. Recall $W G_{n} = G_{0} \times G_{1} \times \dots \times G_{n}$ . Define $H_{n} : W G_{n} \times Δ_{n}^{1} \to W G_{n}$ by:

$H_{n} ((g_{0}, g_{1}, \dots, g_{n}), (t_{0}, t_{1}, \dots, t_{n})) = (g_{0} \cdot h_{0}, g_{1} \cdot h_{1}, \dots, g_{n} \cdot h_{n})$

where $h_{i}$ is defined in terms of the degeneracy operators to gradually collapse coordinates. At $t = 0$ (the initial vertex of $Δ^{1}$ ), $H$ is the identity. At $t = 1$ (the terminal vertex), $H$ maps to the basepoint $(e, e, \dots, e)$ . The homotopy is compatible with the simplicial operators because the degeneracy maps in $G$ are group homomorphisms. This gives a contraction of $W G$ to its basepoint. $□$

Connections [Master]

Simplicial sets 03.12.24 are the ambient category; a simplicial group is a simplicial set with compatible group structures, and the W-bar construction produces an ordinary simplicial set from a simplicial group.

The Puppe fibre sequence 03.12.28 is the main tool: applied to $W G \to \overline{W} (G)$ , it yields $G ≃ Ω∣ \overline{W} (G) ∣$ since $W G$ is contractible.

Bisimplicial sets 03.12.36 and the realisation lemma are used in proving that the W-bar functor preserves weak equivalences: the bar construction $B (G)$ is a bisimplicial set whose diagonal is $\overline{W} (G)$ .

Bibliography [Master]

@article{kan1958,
  author = {Kan, Daniel M.},
  title = {On homotopy theory and c.s.s. groups},
  journal = {Ann. Math.},
  volume = {68},
  pages = {38--53},
  year = {1958}
}

@book{may-simplicial,
  author = {May, J. Peter},
  title = {Simplicial Objects in Algebraic Topology},
  publisher = {University of Chicago Press},
  year = {1967}
}

@book{goerss-jardine1999,
  author = {Goerss, Paul G. and Jardine, John F.},
  title = {Simplicial Homotopy Theory},
  publisher = {Birkh{\"a}user},
  year = {1999}
}

@article{curtis1971,
  author = {Curtis, Edward B.},
  title = {Simplicial homotopy theory},
  journal = {Adv. Math.},
  volume = {6},
  pages = {107--209},
  year = {1971}
}

Historical & philosophical context [Master]

Daniel Kan introduced simplicial groups and the W-bar construction in his 1958 paper "On homotopy theory and c.s.s. groups" ^{[Kan 1958]}, establishing the correspondence between simplicial groups and loop spaces. This work transformed the study of loop spaces from a topological subject into a purely algebraic one.

The philosophical significance is threefold. First, every simplicial group is automatically fibrant (Kan), eliminating the need for fibrant replacement -- a rare instance where algebraic structure eliminates homotopical pathology. Second, the W-bar construction provides a fully combinatorial model for the classifying space, replacing the topological construction with a purely simplicial one. Third, the correspondence between simplicial groups and connected homotopy types means that the entire homotopy theory of connected spaces can be studied through the lens of simplicial group theory, where tools from homological algebra (the Moore complex, spectral sequences) apply directly. This perspective influenced the development of rational homotopy theory (Quillen, Sullivan) and modern derived algebraic geometry.