03.12.E3 · modern-geometry / homotopy

Simplicial homotopy theory exercise pack (Goerss-Jardine supplement)

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Formal definition of the pack Intermediate

Goerss-Jardine develops homotopy theory inside the category $sSet$ of simplicial sets, replacing topological spaces by functors $Δ^{op} \to Set$ . The central objects are Kan complexes — simplicial sets in which every horn $Λ_{k}^{n} \to X$ extends to a simplex $Δ^{n} \to X$ — because these are exactly the objects whose combinatorial homotopy groups behave like the homotopy groups of a space. The book's spine is the Kan-Quillen model structure on $sSet$ , whose weak equivalences are the maps inducing isomorphisms on homotopy groups and whose fibrant objects are the Kan complexes.

This pack collects nine exercises drawn from Chapters I and II — two easy, four medium, three hard — each with a hint and a full solution. The exercises are grouped: horn-filling and nerve warm-ups (easy), the combinatorial homotopy groups and the long exact sequence of a Kan fibration (medium), and the model-structure axioms and the identification $π_{n} (X, x) = [(Δ^{n}, \partial Δ^{n}), (X, x)]$ as honest group structures (hard).

The conventions are Goerss-Jardine's: $Δ^{n}$ is the standard $n$ -simplex (the representable functor on $[n]$ ), $\partial Δ^{n}$ its boundary, $Λ_{k}^{n}$ the $k$ -th horn (the boundary with the $k$ -th face deleted). A map is a Kan fibration if it has the right lifting property against all horn inclusions $Λ_{k}^{n} ↪ Δ^{n}$ ; a Kan complex is one for which $X \to *$ is a Kan fibration.

Key theorem with full solution Intermediate

Before the pack proper, we work one exercise in full as the model solution. The remaining eight follow the same structure (problem, hint, full answer in <details> blocks).

Lead exercise. Show that the nerve of a category is a Kan complex if and only if the category is a groupoid.

Solution. Let $C$ be a small category and $N C$ its nerve, with $n$ -simplices the strings of $n$ composable morphisms. A horn $Λ_{k}^{n} \to N C$ is the data of all faces of an $n$ -simplex except the $k$ -th, compatibly; a filler is an $n$ -simplex restricting to them.

Inner horns ( $0 < k < n$ ) always fill. For $n = 2$ , an inner horn $Λ_{1}^{2} \to N C$ supplies morphisms $f : x \to y$ and $g : y \to z$ (the two edges meeting at the middle vertex), and the unique filler is the composite $g \circ f$ , with the missing face $x \to z$ being exactly $g \circ f$ . Higher inner horns fill by the same associativity bookkeeping. So the nerve of any category is always an inner Kan complex (a quasi-category).

Outer horns force invertibility. Consider the outer horn $Λ_{0}^{2} \to N C$ : it supplies $f : x \to y$ (edge $01$ ) and $h : x \to z$ (edge $02$ ), and a filler must produce a morphism $g : y \to z$ with $g \circ f = h$ . For arbitrary $f, h$ this requires solving $g = h \circ f^{- 1}$ , possible for all inputs precisely when $f$ is invertible. Likewise the horn $Λ_{2}^{2}$ forces right-invertibility. So all outer horns fill if and only if every morphism of $C$ is an isomorphism, i.e. $C$ is a groupoid.

Conclusion. $N C$ is a Kan complex iff every $2$ -dimensional outer horn fills iff every morphism is invertible iff $C$ is a groupoid. The higher horns then fill automatically since the nerve is $2$ -coskeletal. $□$

This is the structural fact separating quasi-categories (inner horns only, nerves of categories) from Kan complexes (all horns, nerves of groupoids). It is the entry point to the entire $\infty$ -categorical dictionary.

Exercises Intermediate

Exercise 1 (easy, proof). Geometric realization preserves products.

State the theorem that geometric realization $∣ - ∣ : sSet \to Top$ preserves finite products (landing in compactly generated spaces), and explain why this fails in the naive product topology.

Hint

$∣ X \times Y ∣ ≅ ∣ X ∣ \times ∣ Y ∣$ holds in compactly generated spaces. The issue is the product topology versus the $k$ -ification.

Answer

Theorem (Goerss-Jardine I.2). The geometric realization functor $∣ - ∣$ preserves finite limits; in particular the natural map $∣ X \times Y ∣ \to ∣ X ∣ \times ∣ Y ∣$ is a homeomorphism when the target carries the compactly generated (k-space) topology.

The realization $∣ X ∣$ is a colimit of simplices glued by the face and degeneracy maps. The natural map $∣ X \times Y ∣ \to ∣ X ∣ \times ∣ Y ∣$ is always a continuous bijection, but in the ordinary product topology it need not be a homeomorphism: a product of two CW complexes is CW only after $k$ -ification when one factor is not locally compact (Dowker's counterexample). Working in $CGHaus$ , products are computed with the compactly generated topology, and the map becomes a homeomorphism. This is why simplicial homotopy theory is set in compactly generated spaces — realization is then a strong monoidal left adjoint, matching products on the nose. Goerss-Jardine §I.2.

Exercise 2 (easy, numeric). Horns of the 2-simplex.

List the horns $Λ_{k}^{2}$ of $Δ^{2}$ , identify which are inner and which are outer, and state which horn inclusions a Kan fibration must lift against.

Hint

$Λ_{k}^{2}$ is $\partial Δ^{2}$ with the face opposite vertex $k$ removed. Inner means $0 < k < n$ .

Answer

The triangle $Δ^{2}$ has three horns, each obtained from the boundary $\partial Δ^{2}$ (three edges) by deleting one edge:

$Λ_{0}^{2}$ : keep edges $01$ and $02$ (the two edges at vertex $0$ ), delete edge $12$ . Outer.
$Λ_{1}^{2}$ : keep edges $01$ and $12$ (the two edges at vertex $1$ ), delete edge $02$ . Inner.
$Λ_{2}^{2}$ : keep edges $02$ and $12$ (the two edges at vertex $2$ ), delete edge $01$ . Outer.

A Kan fibration $p : X \to Y$ must have the right lifting property against all horn inclusions $Λ_{k}^{n} ↪ Δ^{n}$ , $0 \leq k \leq n$ , $n \geq 1$ — inner and outer alike. A left/inner fibration (quasi-categorical) lifts only against inner horns. Goerss-Jardine §I.3.

Exercise 3 (medium, proof). $π_{0}$ of a Kan complex is well-defined.

For a Kan complex $X$ , define $π_{0} (X)$ as the set of $0$ -simplices modulo the relation $x \sim y$ iff there is a $1$ -simplex with faces $x, y$ . Prove this relation is an equivalence relation, using the Kan condition.

Hint

Reflexivity uses a degenerate $1$ -simplex. Symmetry and transitivity use horn-filling on $Λ_{0}^{2}$ and $Λ_{1}^{2}$ .

Answer

Write $x \sim y$ if there exists $ω \in X_{1}$ with $d_{1} ω = x$ , $d_{0} ω = y$ .

Reflexive. The degeneracy $s_{0} x \in X_{1}$ has $d_{0} s_{0} x = d_{1} s_{0} x = x$ , so $x \sim x$ .

Symmetric. Suppose $ω : x \to y$ . Build the horn $Λ_{0}^{2} \to X$ with edge $01$ equal to $ω$ and edge $02$ equal to $s_{0} x$ (the constant path at $x$ ). The Kan condition fills it to a $2$ -simplex $σ$ whose remaining face $d_{0} σ$ is a $1$ -simplex from $y$ to $x$ . Hence $y \sim x$ .

Transitive. Suppose $ω : x \to y$ and $τ : y \to z$ . Form the horn $Λ_{1}^{2} \to X$ with edge $01 = ω$ and edge $12 = τ$ . The Kan condition fills it to a $2$ -simplex $σ$ , whose face $d_{1} σ$ is a $1$ -simplex from $x$ to $z$ . Hence $x \sim z$ .

So $\sim$ is an equivalence relation and $π_{0} (X) = X_{0} / \sim$ is well-defined. Each step used the horn-filling property; in a non-Kan simplicial set the relation need not even be symmetric. Goerss-Jardine §I.7.

Exercise 4 (medium, proof). The simplicial homotopy groups carry a group structure.

For a pointed Kan complex $(X, x)$ and $n \geq 1$ , define $π_{n} (X, x)$ as homotopy classes of maps $(Δ^{n}, \partial Δ^{n}) \to (X, x)$ , equivalently $n$ -simplices $α$ with all faces degenerate at $x$ , modulo homotopy. Sketch how the Kan condition produces the group multiplication.

Hint

Given representatives $α, β$ , build a horn $Λ_{k}^{n + 1}$ whose relevant faces are $α$ , $β$ , and basepoint-degenerate simplices; the missing face is the product.

Answer

Represent classes by $n$ -simplices $α, β \in X_{n}$ all of whose faces are the constant $s_{0}^{n - 1} x$ (the basepoint). To multiply, construct a horn $Λ_{n}^{n + 1} \to X$ : assign to the faces $d_{n - 1}, d_{n + 1}$ the simplices $α$ and $β$ , and to every other face the basepoint-degenerate $n$ -simplex. These faces agree on overlaps because all their own faces are basepoint, so they assemble into a well-defined horn. The Kan condition fills it to an $(n + 1)$ -simplex $σ$ ; the missing face $d_{n} σ$ is again an $n$ -simplex with all faces at the basepoint, and we define $[α] \cdot [β] := [d_{n} σ]$ .

The Kan condition also supplies the homotopies showing this is independent of the chosen fillers, that the basepoint-degenerate simplex is a two-sided unit, that inverses exist (fill a horn missing a different face), and — for $n \geq 2$ — that the operation is abelian (the Eckmann-Hilton argument runs because two independent "directions" of multiplication exist). So $π_{n} (X, x)$ is a group, abelian for $n \geq 2$ , isomorphic to $π_{n} (∣ X ∣, x)$ . Goerss-Jardine §I.7.

Exercise 5 (medium, proof). Long exact sequence of a Kan fibration.

Let $p : E \to B$ be a Kan fibration of pointed Kan complexes with fibre $F = p^{- 1} (b)$ . State the long exact sequence relating $π_{*} (F)$ , $π_{*} (E)$ , $π_{*} (B)$ , and indicate where the Kan condition enters in defining the connecting map.

Hint

The sequence has the same shape as for a Serre fibration of spaces. The connecting map $\partial : π_{n} (B) \to π_{n - 1} (F)$ lifts a representing simplex through $p$ .

Answer

Theorem. A Kan fibration $p : E \to B$ with fibre $F$ over the basepoint yields a long exact sequence of homotopy groups

\dots \to π_{n} (F) \to π_{n} (E) p_{*} π_{n} (B) \partial π_{n - 1} (F) \to \dots \to π_{0} (E) \to π_{0} (B) .

The maps $π_{n} (F) \to π_{n} (E)$ and $p_{*}$ are induced by the inclusion $F ↪ E$ and by $p$ .

The connecting homomorphism $\partial$ uses the Kan condition directly. Given a class $[β] \in π_{n} (B)$ represented by $β : Δ^{n} \to B$ with $\partial Δ^{n}$ at the basepoint, the basepoint of $E$ supplies a lift of $Λ_{0}^{n} \to E$ (all faces map to the basepoint of $E$ , which lies over the basepoint of $B$ ). Filling against $p$ — possible because $p$ is a Kan fibration, so the lifting problem $Λ_{0}^{n} \to E$ , $Δ^{n} \to B$ has a solution $\tilde{β} : Δ^{n} \to E$ — produces a simplex whose last face $d_{n} \tilde{β}$ lands in $F$ and represents $\partial [β] \in π_{n - 1} (F)$ .

Exactness at each spot is verified by further horn-fillings; the argument is the combinatorial mirror of the topological fibration sequence. Goerss-Jardine §I.7. This is the workhorse computational tool of simplicial homotopy theory.

Exercise 6 (medium, numeric). Homotopy groups of a minimal $K (Z, 1)$ .

The simplicial abelian group $K (Z, 1) = N Z$ , the nerve of $Z$ regarded as a one-object groupoid, is a Kan complex. Compute its simplicial homotopy groups $π_{n} (K (Z, 1))$ .

Hint

The nerve of a group $G$ realizes to $B G$ , the classifying space, which is a $K (G, 1)$ .

Answer

The simplicial set $N Z$ (nerve of $Z$ as a one-object groupoid) has $n$ -simplices the $n$ -tuples $(a_{1}, \dots, a_{n}) \in Z^{n}$ (composable strings), face maps multiplying adjacent entries, degeneracies inserting identities. It is a Kan complex because $Z$ is a group (a groupoid), as established in the lead exercise.

Its realization $∣ N Z ∣ = B Z ≃ S^{1}$ , the circle. So

π_{n} (K (Z, 1)) = π_{n} (S^{1}) = {Z 0 n = 1 n \neq = 1.

This is the defining property of an Eilenberg-MacLane space $K (Z, 1)$ : a single homotopy group $Z$ concentrated in degree $1$ . The simplicial model $N Z$ is the standard combinatorial $K (Z, 1)$ , and the Dold-Kan construction generalizes it to $K (A, n)$ for any abelian group $A$ and degree $n$ . Goerss-Jardine §I.

Exercise 7 (hard, proof). $Ex^{\infty}$ is a fibrant replacement.

Kan's functor $Ex$ has a natural transformation $X \to Ex X$ , and the colimit $Ex^{\infty} X = colim (X \to Ex X \to Ex^{2} X \to \dots)$ is always a Kan complex. State the two key properties of $Ex^{\infty}$ and explain why it serves as a fibrant replacement in the Kan-Quillen model structure.

Hint

$Ex$ is right adjoint to subdivision $sd$ . The map $X \to Ex^{\infty} X$ is a weak equivalence and the target is Kan.

Answer

Kan's extension functor $Ex$ is defined by $(Ex X)_{n} = Hom_{sSet} (sd Δ^{n}, X)$ , where $sd$ is barycentric subdivision; it is right adjoint to $sd$ . The unit gives a natural map $η : X \to Ex X$ , and iterating yields $Ex^{\infty} X = colim_{k} Ex^{k} X$ .

Two key properties (Goerss-Jardine III.4).

$Ex^{\infty} X$ is a Kan complex for every simplicial set $X$ . (Any horn $Λ_{k}^{n} \to Ex^{\infty} X$ factors through some finite stage $Ex^{k} X$ since horns are finite; subdivision-then-adjunction provides a filler one stage up, which lands in the colimit.)
The natural map $X \to Ex^{\infty} X$ is a weak equivalence (it induces an isomorphism on all homotopy groups; subdivision does not change the homotopy type of the realization).

Together these say: $X \to Ex^{\infty} X$ is a weak equivalence into a Kan (fibrant) object. Since one checks that the map is also a cofibration (a monomorphism), it is an acyclic cofibration into a fibrant object — precisely a fibrant replacement of $X$ in the Kan-Quillen model structure. Unlike the abstract small-object-argument replacement, $Ex^{\infty}$ is explicit and functorial, which is why it appears throughout the theory. Goerss-Jardine §III.4.

Exercise 8 (hard, proof). The Kan-Quillen model structure: verify the lifting axiom in one case.

In the Kan-Quillen model structure on $sSet$ , cofibrations are monomorphisms, fibrations are Kan fibrations, and weak equivalences are maps inducing $π_{*}$ -isomorphisms on fibrant replacements. Prove that every acyclic cofibration has the left lifting property against every fibration in the case of horn inclusions.

Hint

Horn inclusions $Λ_{k}^{n} ↪ Δ^{n}$ are the generating acyclic cofibrations. The lifting property against Kan fibrations is the definition of a Kan fibration.

Answer

In the Kan-Quillen model structure, the generating acyclic cofibrations are exactly the horn inclusions ${Λ_{k}^{n} ↪ Δ^{n} : n \geq 1, 0 \leq k \leq n}$ , and the generating cofibrations are the boundary inclusions ${\partial Δ^{n} ↪ Δ^{n}}$ .

Claim. A map $p : X \to Y$ has the right lifting property against all horn inclusions iff $p$ is a Kan fibration.

This is immediate from the definitions: the right lifting property of $p$ against $Λ_{k}^{n} ↪ Δ^{n}$ says exactly that every commuting square

Λ_{k}^{n} ↓ Δ^{n} \to \to X ↓ p Y

admits a diagonal lift $Δ^{n} \to X$ — which is the definition of $p$ being a Kan fibration. So a Kan fibration lifts against every generating acyclic cofibration.

The general acyclic cofibrations are retracts of transfinite compositions of pushouts of these horn inclusions (the small-object argument). Lifting properties are closed under retracts, pushouts, and transfinite composition, so a Kan fibration lifts against every acyclic cofibration, not just the generators. This verifies one half of the lifting axiom CM4; the other half (cofibrations lift against acyclic fibrations) is dual, using the boundary inclusions. Goerss-Jardine §I.11, §II.3.

Exercise 9 (hard, proof). Quasi-categories versus Kan complexes.

A quasi-category is a simplicial set with fillers for all inner horns. Prove that a quasi-category in which every $1$ -simplex is invertible (in the homotopy category) is a Kan complex.

Hint

The only horns a quasi-category might fail to fill are the outer ones $Λ_{0}^{n}$ and $Λ_{n}^{n}$ . Reduce outer-horn filling for $n \geq 2$ to invertibility of the relevant edge.

Answer

Let $X$ be a quasi-category — inner horns $Λ_{k}^{n}$ ( $0 < k < n$ ) all fill. The homotopy category $ho (X)$ has the $0$ -simplices as objects and homotopy classes of $1$ -simplices as morphisms, with composition supplied by inner-horn fillers (the lead exercise's $Λ_{1}^{2}$ argument). Suppose every morphism of $ho (X)$ is invertible, i.e. every $1$ -simplex of $X$ is an equivalence.

We must fill the outer horns. Consider $Λ_{0}^{n} \to X$ for $n \geq 2$ . Joyal's theorem (the "special outer horn" criterion) states: in a quasi-category, an outer horn $Λ_{0}^{n} \to X$ fills whenever the initial edge (the $1$ -simplex spanning vertices $0, 1$ ) is an equivalence. By hypothesis every edge is an equivalence, so this initial edge is, and the horn fills. Symmetrically $Λ_{n}^{n}$ fills when its terminal edge is an equivalence, which again holds. The case $n = 1$ is vacuous.

Therefore every horn — inner and outer — fills, so $X$ is a Kan complex.

This is the simplicial form of the slogan "a Kan complex is an $\infty$ -groupoid": Kan complexes are exactly the quasi-categories whose every morphism is invertible. The general filling of special outer horns is Joyal's lifting theorem, proved by induction on dimension using inner-horn fillers. Goerss-Jardine, and Joyal/Lurie for the outer-horn criterion.

Exercise pack EP3. Goerss-Jardine supplement: simplicial sets, Kan complexes, the combinatorial homotopy groups, the long exact sequence of a Kan fibration, and the Kan-Quillen model structure across Ch. I-II.

Prerequisites

03.12.25 pending
03.12.33 pending
03.12.42 pending
03.12.40 pending
03.12.31 pending
03.12.32 pending
03.12.41 pending
03.12.43 pending

Tier anchors

beginner
intermediate: Goerss-Jardine, Simplicial Homotopy Theory, Ch. I (simplicial sets), Ch. I.7-11 (Kan complexes, simplicial homotopy groups), Ch. II (model structure), Ch. VI (Reedy/diagrams) exercises
master

References

Goerss-Jardine — Simplicial Homotopy Theory · Ch. I (simplicial sets, nerve, $\mathrm{Ex}^\infty$), Ch. I §7-11 (Kan complexes, the combinatorial homotopy groups $\pi_n$, the long exact sequence of a fibration), Ch. II (the Kan-Quillen model structure on $\mathbf{sSet}$), Ch. VI (function complexes and simplicial model categories)
May — Simplicial Objects in Algebraic Topology · §1-§8 — Kan complexes and combinatorial homotopy groups
Riehl — Categorical Homotopy Theory · Ch. 14-17 — simplicial sets and the homotopy theory of model categories

Estimated time

intermediate: 6h