Chapter I Simplicial sets

This chapter introduces the basic elements of the homotopy theory of simplicial sets. Technically, the purpose is twofold: to prove that the category of simplicial sets has a homotopical structure in the sense that it admits the structure of a closed model category (Theorem 11.3), and to show that the resulting homotopy theory is equivalent in a strong sense to the ordinary homotopy theory of topological spaces (Theorem 11.4). Insofar as simplicial sets are algebraically defined, and the corresponding closed model structure is combinatorial in nature, we obtain an algebraic, combinatorial model for standard homotopy theory.

The substance of Theorem 11.3 is that we can find three classes of morphisms within the simplicial set category S, called cofibrations, fibrations and weak equivalences, and then demonstrate that the following list of properties is satisfied:

CM1: S is closed under all finite limits and colimits. CM2: Suppose that the following diagram commutes in S:

\begin{matrix}X w g_{N N N N P} h \\ Y ‚ ‚ ‚ ‚ “ f \end{matrix} ; Z.

If any two of f, g and h are weak equivalences, then so is the third. CM3: If f is a retract of g and g is a weak equivalence, fibration or cofibration,

then so is f. CM4: Suppose that we are given a commutative solid arrow diagram

U  w u i  Xup

V w^{i i i ij} Y

where i is a cofibration and p is a fibration. Then the dotted arrow exists, making the diagram commute, if either i or p is also a weak equivalence.

CM5: Any map f : X → Y may be factored:

(a) f = p · i where p is a fibration and i is a trivial cofibration, and (b) f = q · j where q is a trivial fibration and j is a cofibration.

The fibrations in the simplicial set category are the Kan fibrations, which are defined by a lifting property that is analogous to the notion of a Serre fibration. The cofibrations are the monomorphisms, and the weak equivalences are morphisms which induce homotopy equivalences of CW-complexes after passage to topological spaces. We shall begin to investigate the consequences of this list of axioms in subsequent chapters — they are the basis of a great deal of modern homotopy theory.

Theorem 11.3 and Theorem 11.4 are due to Quillen [75], but the development given here is different: the results are proved simultaneously, and their proofs jointly depend fundamentally on Quillen’s later result that the realization of a Kan fibration is a Serre fibration [76]. The category of simplicial sets is historically the first full algebraic model for homotopy theory to have been found, but the verification of its closed model structure is still the most difficult proof of abstract homotopy theory. These theorems and their proofs effectively summarize all of the classical homotopy theory of simplicial sets, as developed mostly by Kan in the 1950’s. Kan’s work was a natural outgrowth of the work of Eilenberg and Mac Lane on singular homology theory, and is part of a thread of ideas that used to be called “combinatorial homotopy theory” and which can be traced back to the work of Poincaré at the beginning of the twentieth century.

We give here, in the proof of the main results and the development leading to them, a comprehensive introduction to the homotopy theory of simplicial sets. Simplicial sets are defined, with examples, in Section 1, the functorial relationship with topological spaces via realization and the singular functor is described in Section 2, and we start to describe the combinatorial homotopical structure (Kan fibrations and Kan complexes) in Section 3. We introduce the Gabriel-Zisman theory of anodyne extensions in Section 4: this is the obstruction-theoretic machine that trivializes many potential difficulties related to the function complexes of Section 5, the notion of simplicial homotopy in Section 6, and the discussion of simplicial homotopy groups for Kan complexes in Section 7. The fundamental groupoid for a Kan complex is introduced in Section 8, by way of proving a major result about composition of simplicial sets maps which induce isomorphisms in homotopy groups (Theorem 8.2). This theorem, along with a lifting property result for maps which are simultaneously Kan fibrations and homotopy groups isomorphisms (Theorem 7.10 — later strengthened in Theorem 11.2), is used to demonstrate in Section 9 (Theorem 9.1) that the collection of Kan complexes and maps between them satisfies the axioms for a category of fibrant objects in the sense of Brown [15]. This is a first axiomatic approximation to the desired closed model structure, and is the platform on which the relation with standard homotopy theory is constructed with the introduction of minimal fibrations in Section 10. The basic ideas there are that every Kan fibration has a “minimal model” (Proposition 10.3 and Lemma 10.4), and the Gabriel-Zisman result that minimal fibrations induce Serre fibrations after realization (Theorem 10.9). It is then a relatively simple matter to show that the realization of a Kan fibration is a Serre fibration (Theorem 10.10).

The main theorems are proved in the final section, but Section 10 is the heart of the matter from a technical point of view once all the definitions and elementary properties have been established. We have not heard of a proof of Theorem 11.3 or Theorem 11.4 that avoids minimal fibrations. The minimality concept is very powerful wherever it appears (any rational homotopy theorist would agree), but not much has yet been made of it from a formal point of view.

1. Basic definitions.

Let Δ be the category of finite ordinal numbers, with order-preserving maps between them. More precisely, the objects for Δ consist of elements n, n ≥ 0, where n is a string of relations

0 → 1 → 2 → ⋯ → n

(in other words n is a totally ordered set with n + 1 elements). A morphism θ : m → n is an order-preserving set function, or alternatively just a functor. We usually commit the abuse of saying that Δ is the ordinal number category.

A simplicial set is a contravariant functor X : Δ^{op} → Sets, where Sets is the category of sets.

Example 1.1. There is a standard covariant functor

Δ → Top_{n↦|Δ^{n}|}.

The topological standard n-simplex |Δ^{n}| ⊂ ℝ^{n+1} is the space

|Δ^{n}| = {(t_{0}, …, t_{n}) ∈ ℝ^{n+1}| ∑_{i=0}^{n} t_{i} = 1, t_{i} ≥ 0},

with the subspace topology. The map θ_{∗} : |Δ^{n}| → |Δ^{m}| induced by θ : n → m is defined by

θ_{∗}(t_{0}, …, t_{m}) = (s_{0}, …, s_{n}),

where

s_{i} = \{ \begin{matrix}0 & θ^{−1} = ∅ \\ ∑_{j∈θ^{−1}(i)} t_{j} & θ^{−1}(i) ≠ ∅ \end{matrix}

One checks that θ ↦ θ_{∗} is indeed a functor (exercise). Let T be a topological space. The singular set S(T) is the simplicial set given by

n ↦ hom(|Δ^{n}|, T).

This is the object that gives the singular homology of the space T.

Among all of the functors m → n appearing in Δ there are special ones, namely

d^{i} : n − 1 → n  0 ≤ i ≤ n  (cofaces)
s^{j} : n + 1 → n  0 ≤ j ≤ n  (codegeneracies)

where, by definition,

d^{i}(0 → 1 → ⋯ → n − 1) = (0 → 1 → ⋯ → i − 1 → i + 1 → ⋯ → n)

(ie. compose i − 1 → i → i + 1, giving a string of arrows of length n − 1 in n), and

s^{j}(0 → 1 → ⋯ → n + 1) = (0 → 1 → ⋯ → j ⟶^{1} j → ⋯ → n)

(insert the identity 1_{j} in the j^{th} place, giving a string of length n + 1 in n). It is an exercise to show that these functors satisfy a list of identities as follows, called the cosimplicial identities:

(1.2)  \{ \begin{matrix}d^{j}d^{i} = d^{i}d^{j−1} & if i < j \\ s^{j}d^{i} = d^{i}s^{j−1} & if i < j \\ s^{j}d^{j} = 1 = s^{j}d^{j+1} \\ s^{j}d^{i} = d^{i−1}s^{j} & if i > j + 1 \\ s^{j}s^{i} = s^{i}s^{j+1} & if i ≤ j \end{matrix}

The maps d^{j}, s^{i} and these relations can be viewed as a set of generators and relations for Δ (see [66]). Thus, in order to define a simplicial set Y, it suffices to write down sets Y_{n}, n ≥ 0 (sets of n-simplices) together with maps

d_{i} : Y_{n} → Y_{n−1}, 0 ≤ i ≤ n (faces)
s_{j} : Y_{n} → Y_{n+1}, 0 ≤ j ≤ n (degeneracies)

satisfying the simplicial identities:

(1.3)  \{ \begin{matrix}d_{i}d_{j} = d_{j−1}d_{i} & if i < j \\ d_{i}s_{j} = s_{j−1}d_{i} & if i < j \\ d_{j}s_{j} = 1 = d_{j+1}s_{j} \\ d_{i}s_{j} = s_{j}d_{i−1} & if i > j + 1 \\ s_{i}s_{j} = s_{j+1}s_{i} & if i ≤ j \end{matrix}

This is the classical way to write down the data for a simplicial set Y.

From a simplicial set Y, one may construct a simplicial abelian group ℤY (ie. a contravariant functor Δ^{op} → Ab), with ℤY_{n} set equal to the free abelian group on Y_{n}. ℤY has associated to it a chain complex, called its Moore complex and also written ℤY, with

ℤY_{0} ⟵^{∂} ℤY_{1} ⟵^{∂} ℤY_{2} ⟵ … and

∂ = ∑_{i=0}^{n}(−1)^{i}d_{i}

in degree n. Recall that the integral singular homology groups H_{∗}(X; ℤ) of the space X are defined to be the homology groups of the chain complex ℤSX.

Example 1.4. Suppose that C is a (small) category. The classifying space (or nerve ) BC of C is the simplicial set with

BC_{n} = hom_{cat}(n, C),

where hom_{cat}(n, C) denotes the set of functors from n to C. In other words an n-simplex is a string

a_{0} ⟶^{α_{1}} a_{1} ⟶^{α_{2}} … ⟶^{α_{n}} a_{n}

of composeable arrows of length n in C.

We shall see later that there is a topological space |Y | functorially associated to every simplicial set Y, called the realization of Y. The term “classifying space” for the simplicial set BC is therefore something of an abuse – one really means that |BC| is the classifying space of C. Ultimately, however, it does not matter; the two constructions are indistinguishable from a homotopy theoretic point of view.

Example 1.5. If G is a group, then G can be identified with a category (or groupoid) with one object ∗ and one morphism g : ∗ → ∗ for each element g of G, and so the classifying space BG of G is defined. Moreover |BG| is an Eilenberg-Mac Lane space of the form K(G, 1), as the notation suggests; this is now the standard construction.

Example 1.6. Suppose that A is an exact category, like the category P(R) of finitely generated projective modules on a ring R (see [78]). Then A has associated to it a category QA. The objects of QA are those of A. The arrows of QA are equivalence classes of diagrams

· ↞ · ↣ ·

where both arrows are parts of exact sequences of A, and composition is represented by pullback. Then K_{i−1}(A) := π_{i}|BQA| defines the K-groups of A for i ≥ 1; in particular π_{i}|BQP(R)| = K_{i−1}(R), the i^{th} algebraic K-group of the ring R.

Example 1.7. The standard n-simplex, simplicial Δ^{n} in the simplicial set category S is defined by

Δ^{n} = hom_{Δ}( , n).

In other words, Δ^{n} is the contravariant functor on Δ which is represented by n.

A map f : X → Y of simplicial sets (or, more simply, a simplicial map) is the obvious thing, namely a natural transformation of contravariant set-valued functors defined on Δ. S will denote the resulting category of simplicial sets and simplicial maps.

The Yoneda Lemma implies that simplicial maps Δ^{n} → Y classify n-simplices of Y in the sense that there is a natural bijection

hom_{S}(Δ^{n}, Y) ≅ Y_{n}

between the set Y_{n} of n-simplices of Y and the set hom_{S}(Δ^{n}, Y) of simplicial maps from Δ^{n} to Y (see [66], or better yet, prove the assertion as an exercise). More precisely, write ι_{n} = 1_{n} ∈ hom_{Δ}(n, n). Then the bijection is given by associating the simplex φ(ι_{n}) ∈ Y_{n} to each simplicial map φ : Δ^{n} → Y. This means that each simplex x ∈ Y_{n} has associated to it a unique simplicial map ι_{x} : Δ^{n} → Y such that ι_{x}(ι_{n}) = x. One often writes x = ι_{x}, since it’s usually convenient to confuse the two.

Δ^{n} contains subcomplexes ∂Δ^{n} (boundary of Δ^{n}) and Λ_{k}^{n}, 0 ≤ k ≤ n (k^{th} horn, really the cone centred on the k^{th} vertex). ∂Δ^{n} is the smallest subcomplex of Δ^{n} containing the faces d_{j}(ι_{n}), 0 ≤ j ≤ n of the standard simplex ι_{n}. One finds that

∂Δ_{j}^{n} = \{ \begin{matrix}Δ_{j}^{n} if 0 ≤ j ≤ n − 1, \\ iterated degeneracies of elements of Δ_{k}^{n}, \\ 0 ≤ k ≤ n − 1, if j ≥ n. \end{matrix}

It is a standard convention to write ∂Δ^{0} = ∅, where ∅ is the “unique” simplicial set which consists of the empty set in each degree. ∅ is an initial object for the simplicial set category S.

The k^{th} horn Λ_{k}^{n} ⊂ Δ^{n} (n ≥ 1) is the subcomplex of Δ^{n} which is generated by all faces d_{j}(ι_{n}) except the k^{th} face d_{k}(ι_{n}). One could represent Λ_{0}^{2}, for example, by the picture

0 ‚ ‚ ‚ ‚ “ _{N N N N P} 1  2 ⊂ 0 ‚ ‚ ‚ ‚ “ _{N N N N P} {1  w \atop 2} = Δ^{2}.

2. Realization.

Let Top denote the category of topological spaces. To go further, we have to get serious about the realization functor | | : S → Top. There is a quick way to construct it which uses the simplex category Δ ↓ X of a simplicial set X. The objects of Δ ↓ X are the maps σ : Δ^{n} → X, or simplices of X. An arrow of Δ ↓ X is a commutative diagram of simplicial maps

Δ^{n} _{h h hh j} σ u θ ; X ; Δ^{m} ^{’ ’ ’’)} τ

Observe that θ is induced by a unique ordinal number map θ : m → n.

Lemma 2.1. There is an isomorphism

X ≅ \underrightarrow{lim} ; \begin{matrix}Δ^{n} → X \\ in Δ ↓ X \end{matrix} Δ^{n}.

Proof: The proof is easy; it is really the observation that any functor C → Sets, which is defined on a small category C, is a colimit of representable functors.

The realization |X| of a simplicial set X is defined by the colimit

|X| = \underrightarrow{lim} ; \begin{matrix}Δ^{n} → X \\ in Δ ↓ X \end{matrix} |Δ^{n}|.

in the category of topological spaces. |X| is seen to be functorial in simplicial sets X, by using the fact that any simplicial map f : X → Y induces a functor f_{∗} : Δ ↓ X → Δ ↓ Y in the obvious way, by composition with f.

Proposition 2.2. The realization functor is left adjoint to the singular functor in the sense that there is an isomorphism

hom_{Top}(|X|, Y) ≅ hom_{S}(X, SY)

which is natural in simplicial sets X and topological spaces Y.

Proof: There are isomorphisms

hom_{Top}(|X|, Y) ≅ \underleftarrow{lim}_{Δ^{n}→X} hom_{Top}(|Δ^{n}|, Y)
≅ \underleftarrow{lim}_{Δ^{n}→X} hom_{S}(Δ^{n}, S(Y))
≅ hom_{S}(X, SY)).  ■

Note that S has all colimits and the realization functor | | preserves them, since it has a right adjoint.

Proposition 2.3. |X| is a CW-complex for each simplicial set X.

Proof: Define the n^{th} skeleton sk_{n} X of X be the subcomplex of X which is generated by the simplices of X of degree ≤ n. Then

X = ⋃_{n≥0} sk_{n} X,

and there are pushout diagrams

⨆_{x∈NX_{n}} ∂Δ^{n} w z u sk_{n−1} X z u

⨆_{x∈NX_{n}} Δ^{n} wsk_{n} X

of simplicial sets, where NX_{n} ⊂ X_{n} is the set of non-degenerate simplices of degree n. In other words,

NX_{n} = {x ∈ X_{n}|x is not of the form s_{i}y for any 0 ≤ i ≤ n − 1 and y ∈ X_{n−1}}.

The realization of Δ^{n} is the space |Δ^{n}|, since Δ ↓ Δ^{n} has terminal object 1 : Δ^{n} → Δ^{n}. Furthermore, one can show that there is a coequalizer

⨆_{0≤i<j≤n} Δ^{n−2} ⇉ ⨆_{i=0}^{n} Δ^{n−1} → ∂Δ^{n}

given by the relations d^{j}d^{i} = d^{i}d^{j−1} if i < j (exercise), and so there is a coequalizer diagram of spaces

⨆_{0≤i<j≤n} |Δ^{n−2}| ⇉ ⨆_{i=0}^{n} |Δ^{n−1}| → |∂Δ^{n}|

Thus, the induced map |∂Δ^{n}| → |Δ^{n}| maps |∂Δ^{n}| onto the (n − 1)-sphere bounding |Δ^{n}|. It follows that |X| is a filtered colimit of spaces | sk_{n} X| where | sk_{n} X| is obtained from | sk_{n−1} X| by attaching n-cells according to the pushout diagram

⨆_{x∈NX_{n}} |∂Δ^{n}| w z u | sk_{n−1} X| z u

⨆_{x∈NX_{n}} |Δ^{n}| w| sk_{n} X|.

■

In particular |X| is a compactly generated Hausdorff space, and so the realization functor takes values in the category CGHaus of all such. We shall interpret | | as such a functor. Here is the reason:

Proposition 2.4. The functor | | : S → CGHaus preserves finite limits.

We won’t get into the general topology involved in proving this result; a demonstration is given in [33]. Proposition 2.4 avoids the problem that |X × Y | may not be homeomorphic to |X| × |Y | in general in the ordinary category of topological spaces, in that it implies that

|X × Y | ≅ |X| ×_{Ke} |Y |

(Kelley space product = product in CGHaus). We lose no homotopical information by working CGHaus since, for example, the definition of homotopy groups of a CW-complex does not see the difference between Top and CGHaus.

3. Kan complexes.

Recall the “presentation”

⨆_{0≤i<j≤n} Δ^{n−2} ⇉ ⨆_{i=0}^{n} Δ^{n−1} → ∂Δ^{n}

of ∂Δ^{n} that was mentioned in the last section. There is a similar presentation for Λ_{k}^{n}. Lemma 3.1. The “fork” defined by the commutative diagram

Δ^{n−2} w d^{j−1} u in_{i<j} Δ^{n−1} uin_{i}_{ȷ ȷ ȷ ȷȷ fi} d^{i}

⨆_{0≤i<j≤n} Δ^{n−2} w w ⨆_{i≠k} Δ^{n−1} w Λ_{k}^{n}

Δ^{n−2} u in_{i<j}  w d^{i}  Δ^{n−1} uin_{j}^{‚ ‚ ‚ ‚‚ ›} d^{j}

is a coequalizer in S. Proof: There is a coequalizer

⨆_{i<j} Δ^{n−1} ×_{Λ_{k}^{n}} Δ^{n−1} ⇉ ⨆ ; \begin{matrix}i ≠ k \\ 0 ≤ i ≤ n \end{matrix} Δ^{n−1} → Λ_{k}^{n}.

But the fibre product Δ^{n−1} ×_{Λ_{k}^{n}} Δ^{n−1} is isomorphic to

Δ^{n−1} ×_{Δ^{n}} Δ^{n−1} ≅ Δ^{n−2}

since the diagram

n − 2  w d^{j−1} u d^{i}  n − 1 ud^{i}
n − 1 d^{j} wn

is a pullback in Δ. In effect, the totally ordered set {0 … î… ĵ … n} is the intersection of the subsets {0 … î… n} and {0 … ĵ … n} of {0 … n}, and this poset is isomorphic to n − 2. ■ The notation {0 … î… n} means that i isn’t there.

Corollary 3.2. The set hom_{S}(Λ_{k}^{n}, X) of simplicial set maps from Λ_{k}^{n} to X is in bijective correspondence with the set of n-tuples (y_{0}, …, ŷ_{k}, …, y_{n}) of (n−1)- simplices y_{i} of X such that d_{i}y_{j} = d_{j−1}y_{i} if i < j, and i, j ≠ k.

We can now start to describe the internal homotopy theory carried by S. The central definition is that of a fibration of simplicial sets. A map p : X → Y of simplicial sets is said to be a fibration if for every commutative diagram of simplicial set homomorphisms

\begin{matrix}Λ_{k}^{n}  w z u i \\ Xup \end{matrix} ; {Δ^{n}  w^{´´´´} \atop ˆ Y}

there is a map θ : Δ^{n} → X (dotted arrow) making the diagram commute. i is the obvious inclusion of Λ_{k}^{n} in Δ^{n}.

This requirement was called the extension condition at one time (see [58], [67], for example), and fibrations were (and still are) called Kan fibrations. The condition amounts to saying that if (x_{0} … x̂_{k} … x_{n}) is an n-tuple of simplices of X such that d_{i}x_{j} = d_{j−1}x_{i} if i < j, i, j ≠ k, and there is an n-simplex y of Y such that d_{i}y = p(x_{i}), then there is an n-simplex x of X such that d_{i}x = x_{i}, i ≠ k, and such that p(x) = y. It is usually better to formulate it in terms of diagrams.

The same language may be used to describe Serre fibrations: a continuous map of spaces f : T → U is said to be a Serre fibration if the dotted arrow exists in each commutative diagram of continuous maps

\begin{matrix}|Λ_{k}^{n}|  w z u \\ Tuf \end{matrix} ; {|Δ^{n}|  w^{´´´´} \atop ˆ U}

making it commute. By adjointness 2.2, all such diagrams may be identified with diagrams

\begin{matrix}Λ_{k}^{n}  w z u \\ S(T) uS(f) \end{matrix} ; \begin{matrix}Δ^{n}  w^{\ \ \ \ ]} \\ S(U), \end{matrix}

so that f : T → U is a Serre fibration if and only if S(f) : S(T) → S(U) is a (Kan) fibration. This is partial motivation for the definition of fibration of simplicial sets. The simplicial set |Λ_{k}^{n}| is a strong deformation retract of |Δ^{n}|, so that we’ve proved

Lemma 3.3. For each space X, the map S(X) → ∗ is a fibration.

∗ is different notation for the simplicial set Δ^{0}. It consists of a singleton set in each degree, and is therefore a terminal object in the category of simplicial sets.

A fibrant simplicial set (or Kan complex) is a simplicial set Y such that the canonical map Y → ∗ is a fibration. Alternatively, Y is a Kan complex if and only if one of the following equivalent conditions is met:

K1: Every map α : Λ_{k}^{n} → Y may be extended to a map defined on Δ^{n} in the sense that there is a commutative diagram

{Λ_{k}^{n} w α z u \atop Y} ; Δ^{n}^{````}ˆ

K2: For each n-tuple of (n−1)-simplices (y_{0} … ŷ_{k} … y_{n}) of Y such that d_{i}y_{j} = d_{j−1}y_{i} if i < j, i, j ≠ k, there is an n-simplex y such that d_{i}y = y_{i}.

The standard examples of fibrant simplicial sets are singular complexes, as we’ve seen, as well as classifying spaces BG of groups G, and simplicial groups. A simplicial group H is a simplicial object in the category of groups; this means that H is a contravariant functor from Δ to the category Grp of groups. I generally reserve the symbol e for the identities of the groups H_{n}, for all n ≥ 0.

Lemma 3.4 (Moore). The underlying simplicial set of any simplicial group H is fibrant.

Proof: Suppose that (x_{0}, …, x_{k−1}, x_{ℓ−1}, x_{ℓ}, …, x_{n}), ℓ ≥ k + 2, is a family of (n − 1)-simplices of H which is compatible in the sense that d_{i}x_{j} = d_{j−1}x_{i} for i < j whenever the two sides of the equation are defined. Suppose that there is an n-simplex y of H such that d_{i}y = x_{i} for i ≤ k − 1 and i ≥ ℓ. Then the family

(e, …, e^{k−1}, x_{ℓ−1}d_{ℓ−1}(y^{−1})^{ℓ−1}, e, …, e)

is compatible, and d_{i}(s_{ℓ−2}(x_{ℓ−1}d_{ℓ−1}y^{−1})y) = x_{i} for i ≤ k − 1 and i ≥ ℓ − 1. This is the inductive step in the proof of the Proposition. ■

Recall that a groupoid is a category in which every morphism is invertible. Categories associated to groups as above are obvious examples, so that the following result specializes to the assertion that classifying spaces of groups are Kan complexes.

Lemma 3.5. Suppose that G is a groupoid. Then BG is fibrant.

Proof: If C is a small category, then its nerve BC is a 2-coskeleton in the sense that the set of simplicial maps f : X → BC is in bijective correspondence with commutative (truncated) diagrams

X_{2}  w f_{2} uu  BC_{2} uu

X_{1}  w f_{1} uu  BC_{1} uu

X_{0} w f_{0} BC_{0}

in which the vertical maps are the relevant simplicial structure maps. It suffices to prove this for X = Δ^{n} since X is a colimit of simplices. But any simplicial map f : Δ^{n} → BC can be identified with a functor f : n → C, and this functor is completely specified by its action on vertices (f_{0}), and morphisms (f_{1}), and the requirement that f respects composition (f_{2}, and d_{i}f_{2} = f_{1}d_{i}). Another way of saying this is that a simplicial map X → BC is completely determined by its restriction to sk_{2} X.

The inclusion Λ_{k}^{n} ⊂ Δ^{n} induces an isomorphism

sk_{n−2} Λ_{k}^{n} ≅ sk_{n−2} Δ^{n}.

To see this, observe that every simplex of the form d_{i}d_{j}ι_{n}, i < j, is a face of some d_{r}ι_{n} with r ≠ k: if k ≠ i, j use d_{i}(d_{j}ι_{n}), if k = i use d_{k}(d_{j}ι_{n}), and if k = j use d_{i}d_{k}ι_{n} = d_{k−1}(d_{i}ι_{n}. It immediately follows that the extension problem

\begin{matrix}Λ_{k}^{n} w α y u \\ BG \end{matrix} ; Δ^{n}^{´´´´}ˆ

is solved if n ≥ 4, for in that case sk_{2} Λ_{k}^{n} = sk_{2} Δ^{n}.

Suppose that n = 3, and consider the extension problem

\begin{matrix}Λ_{0}^{3} w α y u \\ BG \end{matrix} ; Δ^{3}^{´´´´}ˆ

Then sk_{1} Λ_{0}^{3} = sk_{1} Δ^{3} and so we are entitled to write α_{1} : a_{0} → a_{1}, α_{2} : a_{1} → a_{2} and α_{3} : a_{2} → a_{3} for the images under the simplicial map α of the 1-simplices 0 → 1, 1 → 2 and 2 → 3, respectively. Write x : a_{1} → a_{3} for the image of 1 → 3 under α. Then the boundary of d_{0}ι_{3} maps to the graph

\begin{matrix}a_{1} w α_{2}_{```}ˆ x \\ a_{2} 4 4 4 7 α_{3} \end{matrix} ; a_{3}

in the groupoid G under α, and this graph bounds a 2-simplex of BG if and only if x = α_{3}α_{2} in G. But the images of the 2-simplices d_{2}ι_{3} and d_{1}ι_{3} under α together determine a commutative diagram

a_{0} h h h h k α_{1}  u_{α_{3}(α_{2}α_{1})}_{4 4 4 4 6} α_{2}α_{1}

a_{1}_{4 4 4 4 6} x  a_{2} h h h h k α_{3}

a_{3}

in G, so that

xα_{1} = α_{3}(α_{2}α_{1}),

and x = α_{3}α_{2}, by right cancellation. It follows that the simplicial map α : Λ_{0}^{3} → BG extends to ∂Δ^{3} = sk_{2} Δ^{3}, and the extension problem is solved.

The other cases corresponding to the inclusions Λ_{i}^{3} ⊂ Δ^{3} are similar.

If n = 2, then, for example, a simplicial map α : Λ_{0}^{2} → BG can be identified with a diagram

\begin{aligned}& a_{0} ȷ ȷ ȷ fl α_{1} _{[ [ []}x \\ a_{1} & & a_{2} \end{aligned}

and α can be extended to a 2-simplex of BG if and only if there is an arrow α_{2} : a_{1} → a_{2} of G such that α_{2}α_{1} = x. But obviously α_{2} = xα_{1}^{−1} does the trick. The other cases in dimension 2 are similar. ■ The standard n-simplex Δ^{n} = Bn fails to be fibrant for n ≥ 2, precisely because the last step in the proof of Lemma 3.5 fails in that case.

4. Anodyne extensions.

The homotopy theory of simplicial sets is based on the definition of fibration given above. Originally, all of the work done in this area was expressed in terms of the extension condition, and some rather grisly-looking combinatorics was involved. The Gabriel-Zisman theory of anodyne extensions [33] gives a way to short-circuit most of the pain.

A class M of (pointwise) monomorphisms of S is said to be saturated if the following conditions are satisfied:

A: All isomorphisms are in M. B: M is closed under pushout in the sense that, in a pushout square

\begin{matrix}A  w u i \\ Cui_{∗} \end{matrix} ; \begin{matrix}B  w \\ B ∪_{A} C, \end{matrix}

if i ∈ M then so is i_{∗} (Exercise: Show that i_{∗} is monic). C: Each retract of an element of M is in M. This means that, given a diagram

1  u
A′  w ui′  A  w ui  A′ ui′

B′  wB  wB′,
1  u

if i is in M then i′ is in M. D: M is closed under countable compositions and arbitrary direct sums,

meaning respectively that: D1: Given

A_{1} ⟶^{i_{1}} A_{2} ⟶^{i_{2}} A_{3} ⟶^{i_{3}} …

with i_{j} ∈ M, the canonical map A_{1} → \underrightarrow{lim} A_{i} is in M. D2: Given i_{j} : A_{j} → B_{j} in M, j ∈ I, the map

⊔ i_{j} : ⨆_{j∈I} A_{j} → ⨆_{j∈I} B_{j}

is in M.

A map p : X → Y is said to have the right lifting property (RLP is the standard acronym) with respect to a class of monomorphisms M if in every solid arrow diagram

A  w u i  Xup

B w^{i i i ij} Y

with i ∈ M the dotted arrow exists making the diagram commute.

Lemma 4.1. The class M_{p} of all monomorphisms which have the left lifting property (LLP) with respect to a fixed simplicial map p : X → Y is saturated.

Proof: (trivial) For example, we prove the axiom B. Suppose given a commutative diagram

\begin{matrix}A  w u i \\ C  w u j \\ Xup \end{matrix} ; \begin{matrix}B  w \\ B ∪_{A} C  w \\ Y, \end{matrix}

where the square on the left is a pushout. Then there is a map θ : B → X such that the “composite” diagram

A  w u i  Xup

B  w^{h h hh j}^{θ}  Y

commutes. But then θ induces the required lifting θ_{∗} : B ∪_{A} C → X by the universal property of the pushout. ■

The saturated class M_{B} generated by a class of monomorphisms B is the intersection of all saturated classes containing B. One also says that M_{B} is the saturation of B.

Consider the following three classes of monomorphisms:

B_{1} := the set of all inclusions Λ_{k}^{n} ⊂ Δ^{n}, 0 ≤ k ≤ n, n > 0 B_{2} := the set of all inclusions

(Δ^{1} × ∂Δ^{n}) ∪ ({e} × Δ^{n}) ⊂ (Δ^{1} × Δ^{n}), e = 0, 1

B_{3} := the set of all inclusions

(Δ^{1} × Y) ∪ ({e} × X) ⊂ (Δ^{1} × X),

where Y ⊂ X is an inclusion of simplicial sets, and e = 0, 1.

Proposition 4.2. The classes B_{1}, B_{2} and B_{3} have the same saturation.

M_{B_{1}} is called the class of anodyne extensions.

Corollary 4.3. A fibration is a map which has the right lifting property with respect to all anodyne extensions.

Proof of Proposition 4.2: We shall show only that M_{B_{2}} = M_{B_{1}}; it is relatively easy to see that M_{B_{2}} = M_{B_{3}}, since a simplicial set X can be built up from a subcomplex Y by attaching cells. To show that M_{B_{2}} ⊂ M_{B_{1}}, observe that any saturated set is closed under finite composition. The simplicial set Δ^{n} × Δ^{1} has non-degenerate simplices

h_{j} : Δ^{n+1} → Δ^{n} × Δ^{1}, 0 ≤ j ≤ n,

where the h_{j} may be identified with the strings of morphisms

\begin{aligned}(0, 0)  w \\ (0, 1)  w \\ … & w \\ & (0, j) u \end{aligned} ; \begin{aligned}(1, j)  w \\ … & w \\ & (1, n) \end{aligned}

of length n + 1 in n × 1 (anything longer must have a repeat). One can show that there are commutative diagrams

(4.4)  \begin{matrix}Δ^{n}  w d^{0} \\ Δ^{n+1} uh_{0} \end{matrix} \begin{matrix}Δ^{n} × \{1\} u ≅  y  w \\ Δ^{n} × Δ^{1} \end{matrix}  \begin{matrix}Δ^{n}  w d^{n+1} \\ Δ^{n+1} uh_{n} \end{matrix} \begin{matrix}Δ^{n} × \{0\} u ≅  y  w \\ Δ^{n} × Δ^{1} \end{matrix}

(4.5)  \{ \begin{matrix}Δ^{n}  w d^{i} u h_{j−1} \\ Δ^{n} uh_{j} \end{matrix} \begin{matrix}Δ^{n−1} × Δ^{1}  w d^{i} × 1 \\ Δ^{n} × Δ^{1} \end{matrix}  if i < j \begin{matrix}Δ^{n}  w d^{j+1} u d^{j+1} \\ Δ^{n+1} uh_{j+1} \end{matrix} \begin{matrix}Δ^{n+1}  w h_{j} \\ Δ^{n} × Δ^{1} \end{matrix} \begin{matrix}Δ^{n}  w d^{i} u h_{j} \\ Δ^{n+1} uh_{j} \end{matrix} \begin{matrix}Δ^{n−1} × Δ^{1}  w d^{i−1} × 1 \\ Δ^{n} × Δ^{1} \end{matrix}  if i > j + 1.

Moreover d_{j+1}h_{j} ∉ ∂Δ^{n}×Δ^{1} for j ≥ 0 since it projects to ι_{n} under the projection map Δ^{n} × Δ^{1} → Δ^{n}. Finally, d_{j+1}h_{j} is not a face of h_{i} for j ≥ i + 1 since it has vertex (0, j).

Let (Δ^{n} × Δ^{1})^{(i)}, i ≥ 1 be the smallest subcomplex of Δ^{n} × Δ^{1} containing ∂Δ^{n} × Δ^{1} and the simplices h_{0}, …, h_{i}. Then (Δ^{n} × Δ^{1})^{(n)} = Δ^{n} × Δ^{1} and there is a sequence of pushouts, each having the form

Λ_{i+2}^{n+1}  w (d_{0}h_{i+1}, …, { \atop d_{i+2}h_{i+1}}, …, d_{n+1}h_{i+1}) z u (Δ^{n} × Δ^{1})^{(i)} z u \begin{matrix}Δ^{n+1}  w h_{i+1} \\ (Δ^{n} × Δ^{1})^{(i+1)} \end{matrix}  n−1 ≥ i ≥ −1

by the observation above.

To see that M_{B_{1}} ⊂ M_{B_{2}}, suppose that k < n, and construct the functors

n ⟶^{i} n × 1 ⟶^{r_{k}} n,

where i(j) = (j, 1) and r_{k} is defined by the diagram

0 w u 1 w u … wk − 1 w u  k  w u  k  w u  … wku
0 w1 w… wk − 1 wk wk + 1 w… wn

in n. Then r · i = 1_{n}, and r and i induce a retraction diagram of simplicial set maps

Λ_{k}^{n} w z u

(Λ_{k}^{n} × Δ^{1}) ∪ (Δ^{n} × {0}) w z u

Λ_{k}^{n} z u

Δ^{n} w Δ^{n} × Δ^{1} w Δ^{n}

(apply the classifying space functor B). It follows that the inclusion Λ_{k}^{n} ⊂ Δ^{n} is in M_{B_{2}} if k < n.

Similarly, if k > 0, then the functor v_{k} : n × 1 → n defined by the diagram

0 w u 1 w u … wk w u k + 1 w u … wnu
k wk w… wk wk + 1 w… wn

may be used to show that the inclusion Λ_{k}^{n} ⊂ Δ^{n} is a retract of

(Λ_{k}^{n} × Δ^{1}) ∪ (Δ^{n} × {1}).

Thus, Λ_{k}^{n} ⊂ Δ^{n} is in the class M_{B_{2}} for all n and k. ■

Corollary 4.6. Suppose that i : K ↪ L is an anodyne extension and that Y ↪ X is an arbitrary inclusion. Then the induced map

(K × X) ∪ (L × Y) → (L × X)

is an anodyne extension.

Proof: The set of morphisms K′ → L′ such that

(K′ × X) ∪ (L′ × Y) → (L′ × X)

is anodyne is a saturated set. Consider the inclusion

(Δ^{1} × Y′) ∪ ({e} × X′) ⊂ (Δ^{1} × X′) (Y′ ⊂ X′)

and the induced inclusion

(((Δ^{1} × Y′) ∪ ({e} × X′)) × X) ∪ ((Δ^{1} × X′) × Y) z  w u ≅  ((Δ^{1} × X′) × X) u≅

(((Δ^{1} × ((Y′ × X) ∪ (X′ × Y))) ∪ ({e} × (X′ × X))  Δ^{1} × (X′ × X)

This inclusion is anodyne, and so the saturated set in question contains all anodyne morphisms. ■

5. Function complexes.

Let X and Y be simplicial sets. The function complex Hom(X, Y) is the simplicial set defined by

Hom(X, Y)_{n} = hom_{S}(X × Δ^{n}, Y).

If θ : m → n is an ordinal number map, then the induced function

θ^{∗} : Hom(X, Y)_{n} → Hom(X, Y)_{m}

is defined by

(X × Δ^{n} ⟶^{f} Y) ↦ (X × Δ^{m} ⟶^{1×θ} X × Δ^{n} ⟶^{f} Y).

In other words, one thinks of X × Δ^{n} as a cosimplicial “space” in the obvious way.

There is an evaluation map

ev : X × Hom(X, Y) → Y

defined by (x, f) ↦ f(x, ι_{n}). To show, for example, that ev commutes with face maps d_{j}, one has to check that

f · (1 × d^{j})(d_{j}x, ι_{n−1}) = d_{j}f(x, ι_{n}).

But

f · (1 × d^{j})(d_{j}x, ι_{n−1}) = f(d_{j}x, d_{j}ι_{n}) = d_{j}f(x, ι_{n}).

More generally, ev commutes with all simplicial structure maps and is thus a simplicial set map which is natural in X and Y.

Proposition 5.1 (Exponential Law). The function

ev_{∗} : hom_{S}(K, Hom(X, Y)) → hom_{S}(X × K, Y),

which is defined by sending the simplicial map g : K → Hom(X, Y) to the composite

X × K ⟶^{1×g} X × Hom(X, Y) ⟶^{ev} Y,

is a bijection which is natural in K, X and Y.

Proof: The inverse of ev_{∗} is the map

hom_{S}(X × K, Y) → hom_{S}(K, Hom(X, Y))

defined by sending g : X × K → Y to the map g_{∗} : K → Hom(X.Y), where, for x ∈ K_{n}, g_{∗}(x) is the composite

X × Δ^{n} ⟶^{1×ι_{x}} X × K ⟶^{g} Y. ■

The relation between function complexes and the homotopy theory of simplicial sets is given by

Proposition 5.2. Suppose that i : K ↪ L is an inclusion of simplicial sets and p : X → Y is a fibration. Then the map

Hom(L, X) ⟶^{(i^{∗},p_{∗})} Hom(K, X) ×_{Hom(K,Y)} Hom(L, Y),

which is induced by the diagram

Hom(L, X) w p_{∗} u i^{∗}

Hom(L, Y) ui^{∗}

Hom(K, X) w p_{∗} Hom(K, Y),

is a fibration.

Proof: Diagrams of the form

Λ_{k}^{n} w z u

Hom(L, X) u(i^{∗}, p_{∗})

Δ^{n} w

Hom(K, X) ×_{Hom(K,Y)} Hom(L, Y)

may be identified with diagrams

(Λ_{k}^{n} × L) ∪_{(Λ_{k}^{n}×K)} (Δ^{n} × K)  w z u j  Xup

Δ^{n} × L wY

by the Exponential Law (Proposition 5.1). But j is an anodyne extension by 4.6, so the required lifting exists. ■

Corollary 5.3. (1) If p : X → Y is a fibration, then so is p_{∗} : Hom(K, X) → Hom(K, Y)

(2) If X is fibrant then the induced map i^{∗} : Hom(L, X) → Hom(K, X) is

a fibration. Proof:

(1) The diagram

Hom(K, Y) w 1 u

Hom(K, Y) u

∗ w∗

is a pullback, and the following commutes:

Hom(K, X)_{ȷ ȷ ȷ ȷ fi} N N N N N N N N N N N N N P p_{∗}_{’ ’ ’ ’ ’ ’’)}
Hom(K, Y) w 1 u

Hom(K, Y) u

Hom(∅, X) w p_{∗} Hom(∅, Y)
∗  ∗

for a uniquely determined choice of map Hom(K, Y) → Hom(∅, X).

(2) There is a commutative diagram

Hom(L, X)_{ȷ ȷ ȷ ȷ fi} N N N N N N N N N N N N N P_{’ ’ ’ ’ ’ ’’)} i^{∗}

Hom(K, X)  w u 1  Hom(L, ∗) u  ∗

Hom(K, X)  wHom(K, ∗)  ∗

where the inner square is a pullback. ■

6. Simplicial homotopy.

Let f, g : K → X be simplicial maps. We say that there is a simplicial homotopy f ⟶^{≃} g from f to g if there is a commutative diagram

K × Δ^{0} u 1 × d^{1}  K_{4 4 44 6} f

K × Δ^{1}  w h  X

K × Δ^{0} u 1 × d^{0}  K^{h h hh j} g

The map h is called a homotopy.

It’s rather important to note that the commutativity of the diagram defining the homotopy h implies that h(x, 0) = f(x) and h(x, 1) = g(x) for all simplices x ∈ K. We have given a definition of homotopy which is intuitively correct elementwise – it is essentially the reverse of the definition that one is usually tempted to write down in terms of face (or coface) maps.

Suppose i : L ⊂ K denotes an inclusion and that the restrictions of f and g to L coincide. We say that there is a simplicial homotopy from f to g, (rel L) and

write f ⟶^{≃} g (rel L), if the diagram exists above, and the following commutes as well:

{K × Δ^{1}  w h \atop X} ; L × Δ^{1}  w pr_{L} u z i × 1  Luα

where α = f|_{L} = g|_{L}, and pr_{L} is projection onto the left factor (pr_{R} will denote projection on the right). A homotopy of the form

L × Δ^{1} ⟶^{pr_{L}} L ⟶^{α} X

is called a constant homotopy (at α).

The homotopy relation may fail to be an equivalence relation in general. Consider the maps ι_{0}, ι_{1} : Δ^{0} ⇉ Δ^{n}, (n ≥ 1), which classify the vertices 0 and 1, respectively, of Δ^{n}. There is a simplex [0, 1] : Δ^{1} → Δ^{n} determined by these

vertices, and so ι_{0} ⟶^{≃} ι_{1} (alternatively, 0 ⟶^{≃} 1). But there is no 1-simplex which

could give a homotopy ι_{1} ⟶^{≃} ι_{0}, since 0 ≤ 1. This observation provides a second means (see Lemma 3.5) of seeing that Δ^{n} not fibrant, since we can prove

Lemma 6.1. Suppose that X is a fibrant simplicial set. Then simplicial homotopy of vertices x : Δ^{0} → X of X is an equivalence relation.

Proof: There is a homotopy x ⟶^{≃} y if and only if there is a 1-simplex v of X such that d_{1}v = x and d_{0}v = y (alternativley ∂v = (y, x); in general the boundary ∂σ of an n-simplex σ is denoted by ∂σ = (d_{0}σ, …, d_{n}σ)). But then the equation ∂(s_{0}x) = (x, x) gives the reflexivity of the homotopy relation.

Suppose that ∂v_{2} = (y, x) and ∂v_{0} = (z, y). Then d_{0}v_{2} = d_{1}v_{0}, and so v_{0} and v_{2} determine a map (v_{0}, , v_{2}) : Λ_{1}^{2} → X in the obvious way. Choose a lifting

Λ_{1}^{2}  w (v_{0}, , v_{2}) z u  X.

Δ^{2}^{N N N NN P} θ

Then

∂(d_{1}θ) = (d_{0}d_{1}θ, d_{1}d_{1}θ)
= (d_{0}d_{0}θ, d_{1}d_{2}θ)
= (z, x),

and so the relation is transitive. Finally, given ∂v_{2} = (y, x), set v_{1} = s_{0}x. Then d_{1}v_{1} = d_{1}v_{2} and so v_{1} and v_{2} define a map ( , v_{1}, v_{2}) : Λ_{0}^{2} → X. Choose an extension

Λ_{0}^{2}  w ( , v_{1}, v_{2}) z u  X.

Δ^{2}^{N N N NN P} θ′

Then

∂(d_{0}θ′) = (d_{0}d_{0}θ′, d_{1}d_{0}θ′)
= (d_{0}d_{1}θ′, d_{0}d_{2}θ′)
= (x, y),

and the relation is symmetric. ■

Corollary 6.2. Suppose X is fibrant and that L ⊂ K is an inclusion of simplicial sets. Then

(a) homotopy of maps K → X is an equivalence relation, and

(b) homotopy of maps K → X (rel L) is an equivalence relation.

Proof: (a) is a special case of (b), with L = ∅. But homotopy of maps K → X (rel L) corresponds to homotopy of vertices in the fibres of the Kan fibration

i^{∗} : Hom(K, X) → Hom(L, X)

via the Exponential Law 5.1. ■

7. Simplicial homotopy groups.

Let X be a fibrant simplicial set and let v ∈ X_{0} be a vertex of X. Define π_{n}(X, v), n ≥ 1, to be the set of homotopy classes of maps α : Δ^{n} → X (rel ∂Δ^{n}) for maps α which fit into diagrams

Δ^{n} w α X

∂Δ^{n}  w u z  Δ^{0}. uv

One often writes v : ∂Δ^{n} → X for the composition

∂Δ^{n} → Δ^{0} ⟶^{v} X.

Define π_{0}(X) to be the set of homotopy classes of vertices of X. π_{0}(X) is the set of path components of X. The simplicial set X is said to be connected if π_{0}(X) is trivial (ie. a one-element set). We shall write [α] for the homotopy class of α, in all contexts.

Suppose that α, β : Δ^{n} → X represent elements of π_{n}(X, v). Then the simplices

\{ \begin{matrix}v_{i} = v, & 0 ≤ i ≤ n − 2, \\ v_{n−1} = α, & and \\ v_{n+1} = β \end{matrix}

satisfy d_{i}v_{j} = d_{j−1}v_{i} if i < j and i, j ≠ n, since all faces of all simplices v_{i} map through the vertex v. Thus, the v_{i} determine a simplicial set map (v_{0}, …, v_{n−1}, , v_{n+1}) : Λ_{n}^{n+1} → X, and there is an extension

\begin{matrix}Λ_{n}^{n+1} w (v_{0}, …, v_{n−1}, , v_{n+1}) z u \\ X. \end{matrix} ; Δ^{n+1} ^{’ ’ ’ ’ ’ ’ ’ ’ ’)} ω

Observe that

∂(d_{n}ω) = (d_{0}d_{n}ω, …, d_{n−1}d_{n}ω, d_{n}d_{n}ω)
= (d_{n−1}d_{0}ω, …, d_{n−1}d_{n−1}ω, d_{n}d_{n+1}ω)
= (v, …, v),

and so d_{n}ω represents an element of π_{n}(X, v).

Lemma 7.1. The homotopy class of d_{n}ω (rel ∂Δ^{n}) is independent of the choices of representatives of [α] and [β] and of the choice of ω.

Proof: Suppose that h_{n−1} is a homotopy α ⟶^{≃} α′ (rel ∂Δ^{n}) and h_{n+1} is a

homotopy β ⟶^{≃} β′ (rel ∂Δ^{n}). Suppose further that

∂ω = (v, …, v, α, d_{n}ω, β)

and

∂ω′ = (v, …, v, α′, d_{n}ω′, β′).

Then there is a map

(Δ^{n+1} × ∂Δ^{1}) ∪ (Λ_{n}^{n+1} × Δ^{1}) ⟶^{(ω′,ω,(v,...,h_{n−1}, ,h_{n+1}))} X

which is determined by the data. Choose an extension

(Δ^{n+1} × ∂Δ^{1}) ∪ (Λ_{n}^{n+1} × Δ^{1})  w (ω′, ω, (v, …, h_{n−1}, , h_{n+1})) z u  X.

Δ^{n+1} × Δ^{1} ^{```````````````````}ˆ w

Then the composite

Δ^{n} × Δ^{1} ⟶^{d^{n}×1} Δ^{n+1} × Δ^{1} ⟶^{w} X

is a homotopy d_{n}ω ⟶^{≃} d_{n}ω′ (rel ∂Δ^{n}). ■ It follows from Lemma 7.1 that the assignment

([α], [β]) ↦ [d_{n}ω], where ∂ω = (v, …, v, α, d_{n}ω, β),

gives a well-defined pairing

m : π_{n}(X, v) × π_{n}(X, v) → π_{n}(X, v).

Let e ∈ π_{n}(X, v) be the homotopy class which is represented by the constant map

Δ^{n} → Δ^{0} ⟶^{v} X.

Theorem 7.2. With these definitions, π_{n}(X, v) is a group for n ≥ 1, which is abelian if n ≥ 2.

Proof: We shall demonstrate here that the π_{n}(X, v) are groups; the abelian property for the higher homotopy groups will be proved later.

It is easily seen (exercise) that α · e = e · α = α for any α ∈ π_{n}(X, v), and that the map π_{n}(X, v) → π_{n}(X, v) induced by left multiplication by α is bijective. The result follows, then, if we can show that the multiplication in π_{n}(X, v) is associative in general and abelian if n ≥ 2.

To see that the multiplication is associative, let x, y, z : Δ^{n} → X represent elements of π_{n}(X, v). Choose (n + 1)-simplices ω_{n−1}, ω_{n+1}, ω_{n+2} such that

∂ω_{n−1} = (v, …, v, x, d_{n}ω_{n−1}, y),
∂ω_{n+1} = (v, …, v, d_{n}ω_{n−1}, d_{n}ω_{n+1}, z), and
∂ω_{n+2} = (v, …, v, y, d_{n}ω_{n+2}, z).

Then there is a map

Λ_{n}^{n+2} ⟶^{(v,...,v,ω_{n−1}, ,ω_{n+1},ω_{n+2})} X

which extends to a map u : Δ^{n+2} → X. But then

∂(d_{n}ω) = (v, …, v, x, d_{n}ω_{n+1}, d_{n}ω_{n+2}),

and so

([x][y])[z] = [d_{n}ω_{n−1}][z]
= [d_{n}ω_{n+1}]
= [d_{n}d_{n}u]
= [x][d_{n}ω_{n+1}]
= [x]([y][z]).  ■

In order to prove that π_{n}(X, v) abelian for n ≥ 2, it is most instructive to show that there is a loop-space ΩX such that π_{n}(X, v) ≅ π_{n−1}(ΩX, v) and then to show that π_{i}(ΩX, v) is abelian for i ≥ 1. This is accomplished with a series of definitions and lemmas, all of which we’ll need in any case. The first step is to construct the long exact sequence of a fibration.

Suppose that p : X → Y is a Kan fibration and that F is the fibre over a vertex ∗ ∈ Y in the sense that the square

\begin{matrix}F  w i u \\ Xup \end{matrix} ; {Δ^{0}  w ∗ \atop Y}

is cartesian. Suppose that v is a vertex of F and that α : Δ^{n} → Y represents an element of π_{n}(Y, ∗). Then in the diagram

Λ_{0}^{n}  w ( , v, …, v) z u  Xup

Δ^{n} w α^{A A A A AA C} θ

Y,

the element [d_{0}θ] ∈ π_{n−1}(F, v) is independent of the choice of θ and representative of [α]. The resulting function

∂ : π_{n}(Y, ∗) → π_{n−1}(F, v)

is called the boundary map.

Lemma 7.3.

(a) The boundary map ∂ : π_{n}(Y, ∗) → π_{n−1}(F, v) is a group homomorphism if n > 1.

(b) The sequence

⋯ → π_{n}(F, v) ⟶^{i_{∗}} π_{n}(X, v) ⟶^{p_{∗}} π_{n}(Y, ∗) ⟶^{∂} π_{n−1}(F, v) → …

… ⟶^{p_{∗}} π_{1}(Y, ∗) ⟶^{∂} π_{0}(F) ⟶^{i_{∗}} π_{0}(X) ⟶^{p_{∗}} π_{0}(Y)

is exact in the sense that kernel equals image everywhere. Moreover, there is an action of π_{1}(Y, ∗) on π_{0}(F) such that two elements of π_{0}(F) have the same image under i_{∗} in π_{0}(X) if and only if they are in the same orbit for the π_{1}(Y, ∗)-action.

Most of the proof of Lemma 7.3 is easy, once you know

Lemma 7.4. Let α : Δ^{n} → X represent an element of π_{n}(X, v). Then [α] = e if and only if there is an (n + 1)-simplex ω of X such that ∂ω = (v, …, v, α).

The proof of Lemma 7.4 is an exercise. Proof of Lemma 7.3: (a) To see that ∂ : π_{n}(Y, ∗) → π_{n−1}(F, v) is a homomorphism if n ≥ 2, suppose that we are given diagrams

\begin{matrix}Λ_{0}^{n}  w v z u \\ Xup \end{matrix} ; {Δ^{n}  w α_{i}^{````}ˆ θ_{i} \atop Y} i = n − 1, n, n + 1,

where the α_{i} represent elements of π_{n}(Y, ∗). Suppose that there is an (n + 1)- simplex ω such that

∂ω = (∗, …, ∗, α_{n−1}, α_{n}, α_{n+1}).

Then there is a commutative diagram

Λ_{0}^{n+1}  w ( , v, …, v, θ_{n−1}, θ_{n}, θ_{n+1}) z u  Xup

Δ^{n+1} w ω^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} γ

Y,

and

∂(d_{0}γ) = (d_{0}d_{0}γ, d_{1}d_{0}γ, …, d_{n}d_{0}γ)
= (d_{0}d_{1}γ, d_{0}d_{2}γ, …, d_{0}d_{n−1}γ, d_{0}d_{n}γ, d_{0}d_{n+1}γ)
= (v, …, v, d_{0}θ_{n−1}, d_{0}θ_{n}, d_{0}θ_{n+1})

Thus [d_{0}θ_{n}] = [d_{0}θ_{n−1}][d_{0}θ_{n+1}], and so ∂([α_{n−1}][α_{n+1}]) = ∂[α_{n−1}]∂[α_{n+1}] in π_{n−1}(F, v).

(b) We shall show that the sequence

π_{n}(X, v) ⟶^{p∗} π_{n}(Y, ∗) ⟶^{∂} π_{n−1}(F, v)

is exact; the rest of the proof is an exercise. The composite is trivial, since in the diagram

\begin{matrix}Λ_{0}^{n}  w v z u \\ Xup \end{matrix} ; {Δ^{n}  w pα^{````}ˆ α \atop Y}

with [α] ∈ π_{n}(X, v), we find that d_{0}α = v. On the other hand, suppose that γ : Δ^{n} → Y represents a class [γ] such that ∂[γ] = e. Choose a diagram

\begin{matrix}Λ_{0}^{n}  w v z u \\ Xup \end{matrix} ; {Δ^{n}  w γ^{````}ˆ θ \atop Y}

so that [d_{0}θ] = ∂[γ]. But then there is a simplex homotopy

Δ^{n−1} × Δ^{1} ⟶^{h_{0}} F

giving d_{0}θ ≃ v (rel ∂Δ^{n}). Thus, there is a diagram

(Δ^{n} × 1) ∪ (∂Δ^{n} × Δ^{1})  w (θ, (h_{0}, v, …, v)) z u  X.

Δ^{n} × Δ^{1} ^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} h

Moreover p · h is a homotopy γ ≃ p · (h · d^{1}) (rel ∂Δ^{n}). ■

Now for some definitions. For a Kan complex X and a vertex ∗ of X, the path space P X is defined by requiring that the following square is a pullback.

P X  w pr u  Hom(Δ^{1}, X) u(d^{0})^{∗}

Δ^{0} w ∗ Hom(Δ^{0}, X) ≅ X.

Furthermore, the map π : P X → X is defined to be the composite

P X ⟶^{pr} Hom(Δ^{1}, X) ⟶^{(d^{1})^{∗}} Hom(Δ^{0}, X) ≅ X.

The maps (d^{ϵ})^{∗} are fibrations for ϵ = 0, 1, by 5.3. In particular, P X is fibrant.

Lemma 7.5. π_{i}(P X, v) is trivial for i ≥ 0 and all vertices v, and π is a fibration.

Proof: d^{ϵ} : Δ^{0} → Δ^{1} is an anodyne extension, and so (d^{0})^{∗} has the right lifting property with respect to all maps ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0, (see the argument in 5.2). Thus, the map P X → Δ^{0} = ∗ has the right lifting property with respect to all such maps. Any two vertices of P X are homotopic, by finding extensions

\begin{matrix}∂Δ^{1} w z u \\ P X. \end{matrix} ; Δ^{1}^{\ \ \ \ ]}

If α : Δ^{n} → P X represents an element of π_{n}(P X, v), then there is a commutative diagram

\begin{matrix}∂Δ^{n+1} w (v, …, v, α) z u \\ P X, \end{matrix} ; Δ^{n+1}^{5 5 5 5 5 5 5 6}

and so [α] = e in π_{n}(P X, v). Finally, the map π sits inside the pullback diagram

P X  w u π  Hom(Δ^{1}, X) ui^{∗}

Hom(∂Δ^{1}, X) u≅

X w (∗, 1_{X}) X × X

and so π is a fibration since i^{∗} is, by 5.3. ■

Define the loop space ΩX to be the fibre of π : P X → X over the base point ∗. A simplex of ΩX is a simplicial map f : Δ^{n} × Δ^{1} → X such that the restriction of f to Δ^{n} × ∂Δ^{1} maps into ∗. Now we can prove

Lemma 7.6. π_{i}(ΩX, ∗) is abelian for i ≥ 1.

Proof: π_{n}(ΩX, ∗), as a set, consists of homotopy classes of maps of the form

Δ^{n} × Δ^{1}  w α  X,

(∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × ∂Δ^{1}) u z ^{h h hh j} ∗

rel the boundary (∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × ∂Δ^{1}). Show that π_{n}(ΩX, ∗) has a second multiplication [α] ⋆ [β] (in the 1-simplex direction) such that [∗] is an identity for this multiplication and that ⋆ and the original multiplication satisfy the interchange law

([α_{1}] ⋆ [β_{1}])([α_{2}] ⋆ [β_{2}]) = ([α_{1}][α_{2}]) ⋆ ([β_{1}][β_{2}]).

It follows that [α][β] = [α] ⋆ [β], and that both multiplications are abelian. ■

Corollary 7.7. Suppose that X is fibrant. Then π_{i}(X, ∗) is abelian if i ≥ 2.

The proof of Theorem 7.2 is now complete.

Let G be a group, and recall that the classifying space BG is fibrant, by 3.5. BG has exactly one vertex ∗. We can now show easily that BG is an Eilenberg-Mac Lane space.

Proposition 7.8. There are natural isomorphisms

π_{i}(BG, ∗) ≅ \{ \begin{matrix}G & if i = 1, \\ 0 & if i ≠ 1. \end{matrix}

Proof: BG is a 2-coskeleton (see 3.5), and so π_{i}(BG, ∗) = 0 for i ≥ 2, by 7.4. It is an elementary exercise to check that the identification BG_{1} = G induces an

isomorphism of groups π_{1}(BG, ∗) ⟶^{≅} G. π_{0}(BG) is trivial, since BG has only one vertex. ■

Suppose that f : Y → X is a map between fibrant simplicial sets. f is said to be a weak equivalence if

(7.9) \{ \begin{matrix}for each vertex y of Y the induced map f_{∗} : π_{i}(Y, y) → π_{i}(X, fy) \\ is an isomorphism for i ≥ 1, and \\ the map f_{∗} : π_{0}(Y) → π_{0}(X) is a bijection. \end{matrix}

Theorem 7.10. A map f : Y → X between fibrant simplicial sets is a fibration and a weak equivalence if and only if f has the right lifting property with respect to all maps ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0.

Proof: (⇒) The simplicial homotopy Δ^{n} × Δ^{1} → Δ^{n}, given by the diagram

0  w u  0  w u  …  w0u
0  w1  w…  wn

in n, contracts Δ^{n} onto the vertex 0. This homotopy restricts to a homotopy Λ_{0}^{n} × Δ^{1} → Λ_{0}^{n} which contracts Λ_{0}^{n} onto 0.

Now suppose that we’re given a diagram

D  \begin{matrix}∂Δ^{n}  w α z u i \\ Xup \end{matrix} ; \begin{matrix}Δ^{n}  w β \\ Y. \end{matrix}

7. Simplicial homotopy groups 35

If there is a diagram

∂Δ^{n} z u i [ [ [ [ [ [ [ [ [ [ [[] α_{4 4 44 6} d^{0}
∂Δ^{n} × Δ^{1}  w h z u i × 1  Xup
Δ^{n} × Δ^{1} w g Y

Δ^{n}^{N N N NN P} d^{0}^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} β

such that the lifting exists in the diagram

\begin{matrix}∂Δ^{n}  w h · d^{1} z u \\ Xup \end{matrix} ; \begin{matrix}Δ^{n}  w g · d^{1}^{[ [ [ []} θ \\ Y, \end{matrix}

then the lifting exists in the original diagram D. This is a consequence of the fact that there is a commutative diagram

α u

∂Δ^{n} w d^{0} z u i (∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {1}) w (h, θ) z u Xup

Δ^{n}  w d^{0}  Δ^{n} × Δ^{1}  w g^{4 4 4 4 4 4 44 6} γ  Y
β  u

36 I. Simplicial sets Now, the contracting homotopy H : Λ_{0}^{n} × Δ^{1} → Λ_{0}^{n} determines a diagram

Λ_{0}^{n} y  w j u d^{0}  ∂Δ^{n} uα
Λ_{0}^{n} × Δ^{1} w h_{1} X
Λ_{0}^{n}  w u d^{1}  Δ^{0}, uα(0)

where h_{1} = α · j · H. There is a diagram

(∂Δ^{n} × {1}) ∪ (Λ_{0}^{n} × Δ^{1})  w (α, h_{1}) z u  X
∂Δ^{n} × Δ^{1}^{4 4 4 4 4 4 44 6} h

since X is fibrant. Moreover, there is a diagram

(Δ^{n} × {1}) ∪ (∂Δ^{n} × Δ^{1})  w (β, ph) z u  Y
Δ^{n} × Δ^{1}^{4 4 4 4 4 4 44 6} g

since Y is fibrant. It therefore suffices to solve the problem for diagrams

D_{1}  \begin{matrix}∂Δ^{n}  w (x_{0}, ∗, …, ∗) z u \\ Xup \end{matrix} ; {Δ^{n}  w ω \atop Y}

for some vertex ∗ (= α(0)) of X, since the composite diagram

∂Δ^{n} w d^{1} z u i ∂Δ^{n} × Δ^{1} w h z ui × 1 Xup

Δ^{n} w d^{1} Δ^{n} × Δ^{1} w g Y

has this form. Then x_{0} represents an element [x_{0}] of π_{n−1}(X, ∗) such that p_{∗}[x_{0}] = e in π_{n−1}(Y, p∗). Thus, [x_{0}] = e in π_{n−1}(X, ∗), and so the trivializing homotopy h_{0} : Δ^{n−1} × Δ^{1} → X for X_{0} determines a homotopy

h′ = (h_{0}, ∗, …, ∗) : ∂Δ^{n} × Δ^{1} → X.

But again there is a diagram

(Δ^{n} × {1}) ∪ (∂Δ^{n} × Δ^{1})  w (ω, ph) z u  Y,

Δ^{n} × Δ^{1} ^{’ ’ ’ ’ ’ ’ ’ ’’)} g′

so it suffices to solve the lifting problem for diagrams

D_{2}  \begin{matrix}∂Δ^{n}  w ∗ z u \\ Xup \end{matrix} ; \begin{matrix}Δ^{n}  w β \\ Y. \end{matrix}

p_{∗} is onto, so β ≃ pα (rel ∂Δ^{n}) via some homotopy h′′ : Δ^{n} × Δ^{1} → Y, and so there is a commutative diagram

∂Δ^{n} w d^{0} z u (∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {0}) w (∗, α) z u Xup

Δ^{n}  w d^{0}  Δ^{n} × Δ^{1}  w h′′^{5 5 5 5 5 5 55 6} Y.

D_{2} is the composite of these two squares, and the lifting problem is solved. (⇐) Suppose that p : X → Y has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0. Then p has the right lifting property with respect to all inclusions L ⊂ K, and is a Kan fibration in particular. It is then easy to see that p_{∗} : π_{0}X → π_{0}Y is a bijection. Also, if x ∈ X is any vertex of X and F_{x} is the fibre over p(x), then F_{x} → ∗ has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0. Then F_{x} is fibrant, and π_{0}(F_{x}) = ∗ and π_{i}(F_{x}, x) = 0, i ≥ 1, by the argument of Lemma 7.5. Thus, p_{∗} : π_{i}(X, x) → π_{i}(Y, px) is an isomorphism for all i ≥ 1. ■

8. Fundamental groupoid.

Let X be a fibrant simplicial set. Provisionally, the fundamental groupoid π_{f}X of X is a category having as objects all vertices of X. An arrow x → y in π_{f}X is a homotopy class of 1-simplices ω : Δ^{1} → X (rel ∂Δ^{1}) where the diagram

Δ^{0} u d^{1} _{4 4 44 6} x

Δ^{1} w ω X

Δ^{0} u d^{0} ^{h h hh j} y

commutes. If v_{2} represents an arrow x → y of π_{f}X and v_{0} represents an arrow y → z; then the composite [v_{0}][v_{2}] is represented by d_{1}ω, where ω is a 2-simplex such that the following diagram commutes

Λ_{1}^{2}  w (v_{0}, , v_{2}) z u  X.

Δ^{2}^{N N N NN P} ω

The fact that this is well-defined should be clear. The identity at x is represented by s_{0}x. This makes sense because, if v_{2} : x → y and v_{0} : y → z then ∂s_{0}v_{0} = (v_{0}, v_{0}, s_{0}y), and ∂(s_{1}v_{2}) = (s_{0}y, v_{2}, v_{2}). The associativity is proved as it was for π_{1}. In fact, π_{f}X(x, x) = π_{1}(X, x) specifies the group of homomorphisms π_{f}X(x, x) from x to itself in π_{f}X, by definition. By solving the lifting problem

Λ_{0}^{2}  w ( , s_{0}x, v_{2}) z u  X

Δ^{2}^{B B B B BB C} ω

for v_{2} : x → y, one finds a v_{0} : y → x (namely d_{0}ω) such that [v_{0}][v_{2}] = 1_{x}. But then [v_{2}] is also epi since it has a right inverse by a similar argument. Thus [v_{2}][v_{0}][v_{2}] = [v_{2}] implies [v_{2}][v_{0}] = 1_{y}, and so π_{f}X really is a groupoid.

Now, let α : Δ^{n} → X represent an element of π_{n}(X, x) and let ω : Δ^{1} → X represent an element of π_{f}X(x, y). Then there is a commutative diagram

(∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {0}) z u _{‚ ‚ ‚ ‚ ‚ ‚ ›} (ω · pr_{R}, α)

Δ^{n} × Δ^{1}  w h_{(ω,α)}  X,

Δ^{n} u d^{0} ^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} ω_{∗}α

and ω_{∗}α represents an element of π_{n}(X, y).

Proposition 8.1. The class [ω_{∗}α] is independent of the relevant choices of representatives. Moreover, [α] ↦ [ω_{∗}α] is a group homomorphism which is functional in [ω], and so the assignment x ↦ π_{n}(X, x) determines a functor on π_{f}X.

Proof: We shall begin by establishing independence from the choice of representative for the class [ω]. Suppose that G : ω ⟶^{≃} η (rel ∂Δ^{1}) is a homotopy of paths from x to y. Then there is a 2-simplex σ of X such that

∂σ = (s_{0}y, η, ω).

Find simplices of the form h_{(η,α)} and h_{(ω,α)} according to the recipe given above, and let h_{σ} be the composite

∂Δ^{n} × Δ^{2} ⟶^{pr} Δ^{2} ⟶^{σ} X.

There is a commutative diagram

(∂Δ^{n} × Δ^{2}) ∪ (Δ^{n} × Λ_{0}^{2}) z u i w (h_{σ}, ( , h(η, α), h(ω, α))) X,

Δ^{n} × Δ^{2} ^{‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ›} θ_{1}

since the inclusion i is anodyne. Then the composite

Δ^{n} × Δ^{1} ⟶^{1×d^{0}} Δ^{n} × Δ^{2} ⟶^{θ_{1}} X

is a homotopy from ω_{∗}α to η_{∗}α (rel ∂Δ^{n}), and so [ω_{∗}α] = [η_{∗}α] in π_{n}(X, y).

Suppose that H : Δ^{n} × Δ^{1} → X gives α ⟶^{≃} β (rel ∂Δ^{n}), and choose a homotopy h(ω, β) : Δ^{n} × Δ^{1} → X as above. Let h_{s_{0}ω} denote the composite

∂Δ^{n} × Δ^{2} ⟶^{pr} Δ^{2} ⟶^{s_{0}ω} X.

Then there is a commutative diagram

(∂Δ^{n} × Δ^{2}) ∪ (Δ^{n} × Λ_{1}^{2})  w (h_{s_{0}ω}, (h(ω, β), , H)) z u j  X,

Δ^{n} × Δ^{2} ^{‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ›} γ

for some map γ, since the inclusion j is anodyne. But then the simplex given by the composite

Δ^{n} ⟶^{d^{0}} Δ^{n} × Δ^{1} ⟶^{1×d^{1}} Δ^{n} × Δ^{2} ⟶^{γ} X

is a construction for both ω_{∗}α and ω_{∗}β, so that [ω_{∗}α] = [ω_{∗}β] in π_{n}(X, y).

For the functoriality, suppose that ω : Δ^{1} → X and η : Δ^{1} → X represent elements of π_{f}X(x, y) and π_{f}X(y, z) respectively, and choose a 2-simplex γ such that ∂γ = (η, d_{1}γ, ω). Then [d_{1}γ] = [η] · [ω] in π_{f}X. Choose h_{(ω,α)} and h_{(η,ω_{∗}α)} according to the recipe above. Then there is a diagram

(∂Δ^{n} × Δ^{2}) ∪ (Δ^{n} × Λ_{1}^{2}) z u [ [ [ [ [ [ [ [ [ [ [ [[] (γ ∘ pr, (h(η, ω_{∗}α), , h(ω, α))

Δ^{n} × Δ^{2}  X,

Δ^{n} × Δ^{1} u 1 × d^{1} ^{`````````````````}ˆ ξ

and hence a diagram

(∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {0}) z u _{A A AA C} (h_{d_{1}γ}, α)

Δ^{n} × Δ^{1}  w ξ  X,

Δ^{n} u d^{1} ^{’ ’ ’ ’ ’ ’ ’ ’’)} η_{∗}ω_{∗}α = (d_{1}γ)_{∗}α

where h_{d_{1}γ} is the composite

∂Δ^{n} × Δ^{1} ⟶^{pr} Δ^{1} ⟶^{d_{1}γ} X.

The statement that ω_{∗} is a group homomorphism is easily checked. ■

Theorem 8.2. Suppose that the following is a commutative triangle of simplicial set maps:

{X w g_{N N N N P} h \atop Y} ; Z^{‚ ‚ ‚‚ ›} f

with X, Y, and Z fibrant. If any two of f, g, or h are weak equivalences, then so is the third.

Proof: There is one non-trivial case, namely to show that f is a weak equivalence if g and h are. This is no problem at all for π_{0}. Suppose y ∈ Y is a vertex. We must show that f_{∗} : π_{n}(Y, y) → π_{n}(Z, fy) is an isomorphism. y may not be in the image of g, but there is an x ∈ X and a path ω : y → gx since π_{0}(g) is epi. But then there is a diagram

π_{n}(Y, y)  w [ω]_{∗} u f_{∗}  π_{n}(Y, gx) uf_{∗}  π_{n}(X, x). u g_{∗} ` ` ` `˜ h_{∗} ; \begin{matrix}π_{n}(Z, fy)  w [fω]_{∗} \\ π_{n}(Z, fgx) \end{matrix}

The maps g_{∗}, h_{∗}, [ω]_{∗}, and [fω]_{∗} are isomorphisms, and so both of the maps labelled f_{∗} are isomorphisms. ■

There are three competing definitions for the fundamental groupoid of an arbitrary simplicial set X. The most obvious choice is the classical fundamental groupoid π|X| of the realization of X; in the notation above, this is π_{f}S|X|. Its objects are the elements of |X|, and its morphisms are homotopy classes of paths in |X|. The second choice is the model GP_{∗}X of Gabriel and Zisman. GP_{∗}X is the free groupoid associated to the path category P_{∗}X of X. The path category has, as objects, all the vertices (elements of X_{0}) of X. It is generated, as a category, by the 1-simplices of X, subject to the relation that, for each 2-simplex σ of X, the diagram

\begin{matrix}v_{0} w d_{2}σ_{[ [ []} d_{1}σ \\ v_{1} ȷ ȷ ȷ fl d_{0}σ \end{matrix} ; v_{2}

commutes. The free groupoid G(Δ ↓ X) associated to the simplex category Δ ↓ X is also a good model. We shall see later on, in Section 12 that π|X|, GP_{∗}X and G(Δ ↓ X) are all naturally equivalent, once we have developed the techniques for doing so.

9. Categories of fibrant objects.

Let S_{f} be the full subcategory of the simplicial set category whose objects are the Kan complexes. S_{f} has all finite products. We have two distinguished classes of maps in S_{f}, namely the fibrations (defined by the lifting property) and the weak equivalences (defined via simplicial homotopy groups). A trivial fibration p : X → Y in S_{f} is defined to be a map which is both a fibration and a weak equivalence. A path object for X ∈ S_{f} is a commutative diagram

X^{I} u(d_{0}, d_{1})

X^{N N N NN P} s w Δ X × X

where s is a weak equivalence and (d_{0}, d_{1}) is a fibration. The maps d_{0} and d_{1} are necessarily trivial fibrations. Any Kan complex X has a natural choice of path object, namely the diagram

Hom(Δ^{1}, X) u  (d_{0}, d_{1})

Hom(∂Δ^{1}, X) u≅

X  w^{’ ’ ’ ’’)} s  X × X u

where s is the map

X ≅ Hom(Δ^{0}, X) ⟶^{(s^{0})^{∗}} Hom(Δ^{1}, X).

(s^{0})^{∗} = s is a weak equivalence; in effect, it is a right inverse for the map

Hom(Δ^{1}, X) ⟶^{(d^{0})^{∗}} Hom(Δ^{0}, X),

and (d^{0})^{∗} is a trivial fibration, by 7.10, since it has the right lifting property with respect to all inclusions ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0. (d^{0})^{∗} is isomorphic to one of the components of the map Hom(Δ^{1}, X) → X × X.

The following list of properties of S_{f} is essentially a recapitulation of things that we’ve seen:

(A) Suppose given a commutative diagram

\begin{matrix}X w g_{N N N N P} h \\ Y ‚ ‚ ‚ ‚ “ f \end{matrix} ; Z.

If any two of f, g and h are weak equivalences, then so is the third. (B) The composite of two fibrations is a fibration. Any isomorphism is a

fibration. (C) Pullback diagrams of the form

\begin{matrix}Z ×_{Y} X  w u pr \\ Xup \end{matrix} ; {Z  w \atop Y}

exist in the case where p is a fibration. Furthermore, pr is a fibration which is trivial if p is trivial.

(D) For any object X there is at least one path space X^{I}.

(E) For any object X, the map X → ∗ is a fibration.

Recall that we proved (A) outright in 8.2. (B) is an easy exercise. (C) holds because fibrations and trivial fibrations are defined by lifting properties, by 7.10. (D) was discussed above, and (E) isn’t really worth mentioning.

Following K. Brown’s thesis [15] (where the notion was introduced), a category C which has all finite products and has distinguished classes of maps called fibrations and weak equivalences which together satisfy axioms (A) – (E) is called a category of fibrant objects (for a homotopy theory). We’ve proved:

Theorem 9.1. S_{f} is a category of fibrant objects for a homotopy theory.

Other basic examples for us are the category CGHaus of compactly generated Hausdorff spaces, and the category Top of topological spaces. In fact, more is true. The fibrations of CGHaus are the Serre fibrations, and the weak equivalences are the weak homotopy equivalences. A map i : U → V in CGHaus is said to be a cofibration if it has the left lifting property with respect to all trivial fibrations.

Proposition 9.2. The category CGHaus and these three classes of maps satisfy the following list of axioms:

CM1: CGHaus is closed under all finite limits and colimits.

CM2: Suppose that the following diagram commutes in CGHaus:

\begin{matrix}X w g_{N N N N P} h \\ Y ‚ ‚ ‚ ‚ “ f \end{matrix} ; Z.

If any two of f, g and h are weak equivalences, then so is the third.

CM3: If f is a retract of g and g is a weak equivalence, fibration or cofibration, then so is f.

CM4: Suppose that we are given a commutative solid arrow diagram

U  w u i  Xup

V w^{i i i ij} Y

where i is a cofibration and p is a fibration. Then the dotted arrow exists, making the diagram commute, if either i or p is also a weak equivalence.

CM5: Any map f : X → Y may be factored:

(a) f = p · i where p is a fibration and i is a trivial cofibration, and

(b) f = q · j where q is a trivial fibration and j is a cofibration.

Proof: The category CGHaus has all small limits and colimits, giving CM1 (see [66, p.182]). This fact is also used to prove the factorization axioms CM5; this is the next step.

The map p : X → Y is a Serre fibration if and only if it has the right lifting property with respect to all inclusions j : |Λ_{k}^{n}| → |Δ^{n}|. Each such j is necessarily a cofibration. Now consider all diagrams

D  \begin{matrix}|Λ_{k_{D}}^{n_{D}}|  w α_{D} z u \\ Xuf \end{matrix} ; {|Δ^{n_{D}}  w β_{D} | \atop Y}

and form the pushout

⨆_{D} |Λ_{k_{D}}^{n_{D}}|  w (α_{D}) z u  Xu i_{1}  X_{0}_{’ ’ ’ ’)} f = f_{0}

⨆_{D} |Δ^{n_{D}}| w X_{1} w f_{1} Y.

Then we obtain a factorization

f = f_{0} = f_{1} · i_{1},

where i_{1} is a cofibration since it’s a pushout of such, and also a weak equivalence since it is a pushout of a map which has a strong deformation retraction. We repeat the process by considering all diagrams.

\begin{matrix}|Λ_{k_{D}}^{n_{D}}|  w α_{D} z u \\ X_{1} uf_{1} \end{matrix} ; {|Δ^{n_{D}}  w β_{D} | \atop Y}

and so on. Thus, we obtain a commutative diagram

X  X_{0}  w i_{1}_{4 4 4 4 6} f_{0}  X_{1}  w i_{2} u f_{1}  X_{2}  w h h h h k f_{2}  …,

Y

which induces a diagram

X_{0} w τ_{0}_{[ [ [[]} f = f_{0} \underrightarrow{lim} X_{i} ȷ ȷ ȷ ȷ fl f_{∞}

Y

But now τ_{0} has the left lifting property with respect to all trivial fibrations, so it’s a cofibration. Moreover, τ_{0} is a weak equivalence since any compact subset of \underrightarrow{lim} X_{i} lies in some finite stage X_{i}, and all the X_{i} → X_{i+1} are weak equivalences. Finally, f_{∞} is a fibration; in effect, for each diagram

|Λ_{k}^{n}| w α z u \underrightarrow{lim} X_{i} uf_{∞}

|Δ^{n}|  w β  Y,

there is an index i and a map α_{i} making the following diagram commute:

|Λ_{k}^{n}| w α z u _{N N NN P} α_{i}

\underrightarrow{lim} X_{i} uf_{∞}

X_{i}^{h h hh j}_{ȷ ȷ ȷ ȷ ȷ fi} f_{i}

|Δ^{n}|  w β  Y.

But then

\begin{matrix}|Λ_{k}^{n}|  w α_{i} z u \\ X_{i} uf_{i} \end{matrix} ; {|Δ^{n}|  w β \atop Y}

is one of the diagrams defining f_{i+1} and there is a diagram

|Λ_{k}^{n}| w α_{i} z u X_{i} uf_{i} A A A A D

X_{i+1}_{`````}ˆ

|Δ^{n}|  w β^{4 4 44 6} θ_{i}  Y

which defines the lifting.

The other lifting property is similar, using

Lemma 9.4. The map p : X → Y is a trivial fibration if and only if p has the right lifting property with respect to all inclusions |∂Δ^{n}| ⊂ |Δ^{n}|.

The proof is an exercise.

Quillen calls this proof a small object argument [75]. CM4 is really a consequence of this argument as well. What we showed, in effect, was that any map f : X → Y has a factorization f = p · i such that p is a fibration and i is a weak equivalence which has the left lifting property with respect to all fibrations.

Suppose now that we have a diagram

U  w α u i  Xup

V w β^{i i i ij} Y,

where p is a fibration and i is a trivial cofibration. We want to construct the dotted arrow (giving the non-trivial part of CM4). Then there is a diagram

U  w j u i  Wuπ

V  w 1_{V}^{i i i ij} r  V,

where j is a weak equivalence which has the left lifting property with respect to all fibrations, and π is a (necessarily trivial) fibration. Thus, the dotted arrow r exists. But then i is a retract of j, and so i has the same lifting property. All of the other axioms are trivial, and so the proof of Proposition 9.2 is complete.

A closed model category is a category C, together with three classes of maps called cofibrations, fibrations and weak equivalences, such that the axioms CM1 – CM5 are satisfied. Proposition 9.2 is the statement that CGHaus has the structure of a closed model category.

Proposition 9.5. CGHaus is a category of fibrant objects for a homotopy theory. In fact the subcategory of fibrant objects in any closed model category C is a category of fibrant objects for a homotopy theory.

Proof: (E) is part of the definition. For (D), the map Δ : X → X × X may be factored

X^{I} u(d_{0}, d_{1})

X w Δ^{N N N NN P} s X × X,

where s is a trivial cofibration and (d_{0}, d_{1}) is a fibration. For (B) and (C), we prove:

Lemma 9.6.

(a) A map f : X → Y in C has the right lifting property with respect to all cofibrations (respectively trivial cofibrations) if and only if f is a trivial fibration (respectively fibration).

(b) U → V in C has the left lifting property with respect to all fibrations (respectively trivial fibratons) if and only if i is a trivial cofibration (respectively cofibration).

Proof: We’ll show that f : X → Y has the right lifting property with respect to all cofibrations if and only if f is a trivial fibration. The rest of the proof is an exercise.

Suppose that f has the advertised lifting property, and form the diagram

X  w 1_{X} u i  Xuf

V  w p^{h h hh j} r  Y,

where i is a cofibration, p is a trivial fibration, and r exists by the lifting property. Then f is a retract of p and is therefore a trivial fibration. The reverse implication is CM4. ■

Finally, since fibrations (respectively trivial fibrations) are those maps having the right lifting property with respect to all trivial cofibrations (respectively all fibrations), they are stable under composition and pullback and include all isomorphisms, yielding (B) and (C). (A) is just CM2. This completes the proof of Proposition 9.5.

We shall see that the category of fibrant objects structure that we have displayed for S_{f} is the restriction of a closed model structure on the entire simplicial set category, as in the corollary above. This will be proved in the next section.

10. Minimal fibrations.

Minimal Kan complexes play roughly the same role in the homotopy theory of simplicial sets as minimal models play in rational homotopy theory (there ought to be an abstract theory of such things). Minimal Kan complexes appear as fibres of minimal fibrations; it turns out that minimal fibrations are exactly the right vehicle for relating the homotopy theories of S and CGHaus.

A simplicial set map q : X → Y is said to be a minimal fibration if q is a fibration, and for every diagram

(10.1)

∂Δ^{n} × Δ^{1}  w pr z u  ∂Δ^{n} u

Δ^{n} × Δ^{1}  w h u pr  Xuq

Δ^{n} wY

the composites

Δ^{n} w d^{0} w d^{1} Δ^{n} × Δ^{1} w h X

are equal. This means that, if two simplices x and y in X_{n} are fibrewise homotopic (rel ∂Δ^{n}), then x = y.

Note that minimal fibrations are stable under base change.

More generally, write x ≃_{p} y if there is a diagram of the form (10.1) such that h(Δ^{n} × 0) = x, and h(Δ^{n} × 1) = y. The relation ≃_{p} is an equivalence relation (exercise).

Lemma 10.2. Suppose that x and y are degenerate r-simplices of a simplicial set X such that ∂x = ∂y. Then x = y.

Proof: (See also [67], p. 36.) Suppose that x = s_{m}z and y = s_{n}w. If m = n, then

z = d_{m}x = d_{m}y = w,

and so x = y. Suppose that m < n. Then

z = d_{m}x = d_{m}s_{n}w = s_{n−1}d_{m}w,

and so

x = s_{m}s_{n−1}d_{m}w = s_{n}s_{m}d_{m}w.

Thus

s_{m}d_{m}w = d_{n}x = d_{n}y = w.

Therefore x = s_{n}w = y. ■ Now we can prove:

Proposition 10.3. Let p : X → Y be a Kan fibration. Then p has a strong fibre-wise deformation retract q : Z → Y which is a minimal fibration.

Proof: Let Z^{(0)} be the subcomplex of X which is generated by a choice of vertex in each p-class, and let i^{(0)} : Z^{(0)} ⊂ X be the canonical inclusion. There is a map r^{(0)} : sk_{0} X → Z^{(0)} which is determined by choices of representatives. Moreover pi^{(0)}r^{(0)} = p|_{sk_{0} X}, and j_{0} ≃ i^{(0)}r^{(0)}, where j_{0} : sk_{0} X ⊂ X is the obvious inclusion, via a homotopy h_{0} : sk_{0} X × Δ^{1} → X such that h_{0}(x, 0) = x and h_{0}(x, 1) = r^{(0)}(x), and h_{0} is constant on simplices of Z^{(0)}. h_{0} can be constructed fibrewise in the sense that p·h_{0} is constant, by using the homotopies implicit in the definition of ≃_{p} . The subcomplex Z^{(0)} has a unique simplex in each p-equivalence class that it intersects, by Lemma 10.2.

Let Z^{(1)} be the subcomplex of X which is obtained by adjoining to Z^{(0)} a representative for each homotopy class of 1-simplices x such that ∂x ⊂ Z^{(0)} and x is not p-related to a 1-simplex of Z^{(0)}. Again, Z^{(1)} has a unique simplex in each p-equivalence class that it intersects, by construction in degrees ≤ 1 and Lemma 10.2 in degrees > 1.

Let x be a non-degenerate 1-simplex of X. Then there is a commutative diagram

(Δ^{1} × {0}) ∪ (∂Δ^{1} × Δ^{1})  w (x, h_{0}|_{∂x}) z u  Xup

Δ^{1} × Δ^{1} w pr_{L}^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} h_{x}

Δ^{1} w px

Y,

by the homotopy lifting property, where the constant homotopy is chosen for h_{x} if x ∈ Z^{(1)}. But then ∂(h_{x}(Δ^{1} × {1})) ⊂ Z^{(0)} and so h_{x}(Δ^{1} × {1},) is p-related to a unique 1-simplex r^{(1)}(x) of Z^{(1)} via some diagram

∂Δ^{1} × Δ^{1} w pr_{L} z u ∂Δ^{1} u∂h_{x}(Δ^{1} × {1})

Δ^{1} × Δ^{1}  w g_{x} u pr  Xup

Δ^{1}  w px  Y,

where g_{x} is constant if x ∈ Z^{(1)}, r^{(1)}(x) = g_{x}(Δ^{1} × 1), and

g_{x}(Δ^{1} × {0}) = h_{x}(Δ^{1} × {1}).

This defines r^{(1)} : sk_{1} X → Z^{(1)}.

We require a homotopy h_{1} : j_{1} ≃_{p} i^{(1)}r^{(1)}, such that i^{(1)} : Z^{(1)} ⊂ X and j_{1} : sk_{1} X ⊂ X are the obvious inclusions, and such that h_{1} is consistent with h_{0}. We also require that the restriction of h_{1} to Z^{(1)} be constant. This is done for the simplex x by constructing a commutative diagram

(∂Δ^{1} × Δ^{2}) ∪ (Δ^{1} × Λ_{1}^{2}) w (s_{1}h_{0}, (g_{x}, , h_{x})) z u Xup

Δ^{1} × Δ^{2} w pr^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} θ_{x}

Δ^{1} w px

Y,

where the lifting θ_{x} is chosen to be the composite

Δ^{1} × Δ^{2} ⟶^{pr_{L}} Δ^{1} ⟶^{x} X

if x ∈ Z^{(1)}. Then h_{0} can be extended to the required homotopy h_{1} : j_{1} ≃_{p} i^{(1)}r^{(1)} by requiring that h_{1}|_{x} = θ_{x} · (1 × d^{1}).

Proceeding inductively gives i : Z = \underrightarrow{lim} Z^{(n)} ⊂ X and r : X → Z such that 1_{X} ≃ ir fibrewise, and such that q : Z → Y has the minimality property. Finally, q is a Kan fibration, since it is a retract of a Kan fibration. ■

Lemma 10.4. Suppose that

\begin{matrix}Z w f_{N N N N P} q \\ Z′ ‚ ‚ ‚ ‚ “ q′ \end{matrix} ; Y

is a fibrewise homotopy equivalence of minimal fibrations q and q′. Then f is an isomorphism of simplicial sets.

To prove Lemma 10.4, one uses:

Sublemma 10.5. Suppose that two maps

\begin{matrix}X  w f_{N N N N P} p \\ Z ‚ ‚ ‚ ‚ “ q \end{matrix} ; Y  \begin{matrix}X  w g_{N N N N P} p \\ Z ‚ ‚ ‚ ‚ “ q \end{matrix} ; Y

are fibrewise homotopic, where g is an isomorphism and q is minimal. Then f is an isomorphism.

Proof of sublemma: Let the diagram

Xu d^{0} _{ȷ ȷ ȷ ȷȷ fi} f

X × Δ^{1} w h Z

Xu d^{1} ^{N N N NN P} g

represent the homotopy. Suppose that f(x) = f(y) for n-simplices x and y of X. Then inductively d_{i}x = d_{i}y, 0 ≤ i ≤ n, and so the composites

∂Δ^{n} × Δ^{1} ⟶^{i×1} Δ^{n} × Δ^{1} ⟶^{x×1} X × Δ^{1} ⟶^{h} Z,

and

∂Δ^{n} × Δ^{1} ⟶^{i×1} Δ^{n} × Δ^{1} ⟶^{y×1} X × Δ^{1} ⟶^{h} Z,

are equal (to a map h_{∗} : ∂Δ^{n} × Δ^{1} → Y). Write h_{x} for the composite homotopy

Δ^{n} × Δ^{1} ⟶^{x×1} X × Δ^{1} ⟶^{h} Y.

Then there is a commutative diagram

(Δ^{n} × Λ_{2}^{2}) ∪ (∂Δ^{n} × Δ^{2}) w ((h_{x}, h_{y}, ), s_{0}h_{∗}) z u Zuq

Δ^{n} × Δ^{2} w pr_{L}^{‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ›} G Δ^{n} w px = py Y,

and the homotopy G · (1 × d^{2}) shows that x = y. Thus, f is monic.

To see that f is epi, suppose inductively that f : X_{i} → Z_{i} is an isomorphism for 0 ≤ i ≤ n − 1, and let x : Δ^{n} → Z be an n-simplex of Z. Then there is a commutative diagram

\begin{matrix}∂Δ^{n}  w (x_{0}, …, x_{n}) z u \\ Xuf \end{matrix} ; {Δ^{n}  w x \atop Z}

10. Minimal fibrations 55

by the inductive assumption, and so one can find a diagram

(∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {1})  w (h|_{(x_{0},...,x_{n})}, x) z u  Zuq
Δ^{n} × Δ^{1} w pr_{L}^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} h_{1}

Δ^{n} w qx

Y.

Then there is a diagram

Xug
Δ^{n} w d^{1}^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} z

Δ^{n} × Δ^{1} w h_{1}

Z,

since g is epi. The restriction of z to ∂Δ^{n} is the composite g · (x_{0}, …, x_{n}), so that ∂z = (x_{0}, …, x_{n}) since g is monic. Thus, there is a diagram

(Δ^{n} × Λ_{0}^{2}) ∪ (∂Δ^{n} × Δ^{2}) w (( , h_{1}, h_{z}), s_{1}h|_{(x_{0},...,x_{n})}) z u Zuq

Δ^{n} × Δ^{2} w pr_{L}^{‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚‚ ›} G′ Δ^{n} w qx Y.

Finally, the composite

Δ^{n} × Δ^{1} ⟶^{1×d^{0}} Δ^{n} × Δ^{2} ⟶^{G′} Z

is a fibrewise homotopy from f(z) to x, and so x = f(z). ■ Lemma 10.6. Suppose given a Kan fibration p and pullback diagrams

\begin{matrix}f_{i}^{−1}p  w u p_{i} \\ Xup \end{matrix} ; {A  w \atop Y} i = 0, 1.

Suppose further that there is a homotopy h : f_{0} ⟶^{≃} f_{1}. Then there is a fibrewise homotopy equivalence

\begin{matrix}f_{1}^{−1}p w ≃_{[ [ []} p_{1} \\ f_{2}^{−1}p. ȷ ȷ ȷ fl p_{2} \end{matrix} ; A

Proof: Consider the diagrams of pullbacks

\begin{matrix}f_{ϵ}^{−1}p  w x_{ϵ} u p_{ϵ} \\ h^{−1}p  w up^{h} \\ Xup \end{matrix} A  w d^{ϵ} A × Δ^{1}  w h Y ϵ = 0, 1.

Then there is a commutative diagram

\begin{matrix}f_{0}^{−1}p  w x_{0} u d^{0} \\ h^{−1}p up^{h} \end{matrix} ; \begin{matrix}f_{0}^{−1}p × Δ^{1}  w p_{0} × 1^{[ [ [ [[]} θ \\ A × Δ^{1}, \end{matrix}

by the homotopy lifting property. It follows that there is a diagram

f_{0}^{−1}p w d^{1}_{ȷ ȷ ȷ ȷ ȷ fi} θ_{∗}_{````````}ˆ p_{0} f_{0}^{−1}p × Δ^{1}_{’ ’ ’ ’’)} θ

f_{1}^{−1}p  w x_{1} up_{1}  h^{−1}p up^{h}
A  w d^{1}  A × Δ^{1}

and hence an induced map θ_{∗} as indicated. Similarly there are diagrams

\begin{matrix}f_{1}^{−1}p  w x_{1} u d^{1} \\ h^{−1}p up^{h} \end{matrix} ; \begin{matrix}f_{1}^{−1}p × Δ^{1}  w p_{1} × 1^{[ [ [ [[]} ω \\ A × Δ^{1}, \end{matrix}

and

f_{1}^{−1}p w d^{0}_{ȷ ȷ ȷ ȷ ȷ fi} ω_{∗}_{````````}ˆ p_{1} f_{1}^{−1}p × Δ^{1}_{’ ’ ’ ’’)} ω

f_{0}^{−1}p  w x_{0} up_{0}  h^{−1}p up^{h}
A  w d^{0}  A × Δ^{1}.

Form the diagram

f_{0}^{−1}p × Λ_{0}^{2}  w ( , θ, ω(θ_{∗} × 1)) z u  h^{−1}p up^{h}
f_{0}^{−1}p × Δ^{2}  w p_{0} × s^{1}^{’ ’ ’ ’ ’ ’ ’ ’’)} γ  A × Δ^{1},

by using the homotopy lifting property and the relations d_{1}θ = x_{1}θ_{∗} = d_{1}(ω(θ_{∗} × 1)). Then there is a commutative diagram

f_{0}^{−1}p × Δ^{1} w 1 × d^{0}_{ȷ ȷ ȷ ȷ ȷ fi} γ_{∗} u pr_{L} f_{0}^{−1}p × Δ^{2}_{’ ’ ’ ’’)} γ

f_{0}^{−1}p  w x_{0} u p_{0}  h^{−1}p up^{h}
f_{0}^{−1}p  w p_{0}  A  w d^{0}  A × Δ^{1},

by the simplicial identities. Then γ_{∗} : ω_{∗}θ_{∗} ≃ 1 is a fibrewise homotopy. There is a similar fibrewise homotopy θ_{∗}ω_{∗} ≃ 1. ■

Corollary 10.7. Suppose that q : Z → Y is a minimal fibration, and that f_{i} : X → Y, i = 0, 1 are homotopic simplicial maps. Then there is a commutative diagram

\begin{matrix}f_{0}^{−1}q w ≅_{[ [ []} q_{0} \\ f_{1}^{−1}q. ȷ ȷ ȷ fl q_{1} \end{matrix} ; X

In particular, the pullbacks f_{0}^{−1}q and f_{1}^{−1}q are isomorphic.

Corollary 10.8. Suppose that q : Z → Y is a minimal fibration with Y connected. Suppose that F is the fibre of q over a base point ∗ of Y. Then, for any simplex σ : Δ^{n} → Y there is a commutative diagram

\begin{matrix}F × Δ^{n} w ≅_{[ [ []} pr_{R} \\ σ^{−1}q. ȷ ȷ ȷ fl \end{matrix} ; Δ^{n}

Proof: Suppose that v and w are vertices of Y such that there is a 1-simplex z of Y with ∂z = (v, w). Then the classifying maps v : Δ^{0} → Y and w : Δ^{0} → Y are homotopic, and so there is an isomorphism F_{v} ≅ F_{w} of fibres induced by the homotopy. In particular, there is an isomorphism F_{v} ≅ F for any vertex v of Y. Now let i_{0} : Δ^{0} → Δ^{n} be the map that picks out the vertex 0 of Δ^{n}. Finally, recall (see the proof of 7.10) that the composite

Δ^{n} → Δ^{0} ⟶^{i_{0}} Δ^{n}

is homotopic to the identity on Δ^{n}. ■

Theorem 10.9 (Gabriel-Zisman). Suppose that q : X → Y is a minimal fibration. Then its realization |q| : |X| → |Y | is a Serre fibration.

Proof: It is enough to suppose that Y has only finitely many non-degenerate simplices, since the image of any continuous map |Δ^{n}| → |Y | is contained in some finite subcomplex of |Y |. We may also suppose that Y is connected. The idea of the proof is to show that |q| : |X| → |Y | is locally trivial with fibre |F |, where F is the fibre over some base point ∗ of Y.

Now suppose that there is a pushout diagram

∂Δ^{n}  w α z u  Zz u

Δ^{n} w β Y,

where Z is subcomplex of Y with fewer non-degenerate simplices, and suppose that U is an open subset of |Z| such that there is a fibrewise homeomorphism

\begin{matrix}U ×_{|Y |} |X| w ω ≅_{[ [ []} \\ U × |F |. ȷ ȷ ȷ fl pr_{L} \end{matrix} ; U

Let U^{1} = |α|^{−1}(U) ⊂ |∂Δ^{n}|. Then there is an induced fibrewise homeomorphism

\begin{matrix}U^{1} ×_{|Y |} |X| w ω ≅_{[ [ []} \\ U^{1} × |F |. ȷ ȷ ȷ fl pr_{L} \end{matrix} ; U^{1}

On the other hand the simplicial fibrewise homeomorphism

Δ^{n} × F  w δ ≅_{[ [ []} pr_{L}  Δ^{n} ×_{Y} X ȷ ȷ ȷ fl

Δ^{n}

induces a homeomorphism

\begin{matrix}V^{1} × F w δ ≅_{[ [ []} pr_{L} \\ V^{1} ×_{|Y |} |X| ȷ ȷ ȷ fl \end{matrix} ; V^{1}

over some open subset V^{1} of |Δ^{n}| such that V^{1} ∩|∂Δ^{n}| = U^{1} and U^{1} is a retract of V^{1}. The map δ restricts to a homeomorphism

\begin{matrix}U^{1} × F w δ ≅_{[ [ []} pr_{L} \\ U^{1} ×_{|Y |} |X| ȷ ȷ ȷ fl \end{matrix} ; U^{1}

over U^{1}. Now consider the fibrewise homeomorphism

\begin{matrix}U^{1} × |F | w δ^{−1}ω ≅_{[ [ []} pr_{L} \\ U^{1} × |F |. ȷ ȷ ȷ fl pr_{L} \end{matrix} ; U^{1}

There is a homeomorphism

\begin{matrix}V^{1} × |F | w r^{∗}(δ^{−1}ω) ≅_{[ [ []} pr_{L} \\ V^{1} × |F |. ȷ ȷ ȷ fl pr_{L} \end{matrix} ; V^{1}

which restricts to δ^{−1}ω over U^{1}. In effect r^{∗}(δ^{−1}ω)(v′, f) = (v′, φ(rv′, f)), where δ^{−1}ω(w, f) = (w, φ(w, f)), and this definition is “functorial”. Thus, the fibrewise isomorphism

\begin{matrix}V^{1} × F w δr^{∗}(δ^{−1}ω) ≅_{[ [ []} pr_{L} \\ V^{1} ×_{|Y |} |X| ȷ ȷ ȷ fl \end{matrix} ; V^{1}

restricts to ω over U^{1}. It follows that there is a fibrewise homeomorphism

\begin{matrix}(V^{1} ∪_{U^{1}} U) × F w ≅_{[ [ []} pr_{L} \\ (V^{1} ∪_{U^{1}} U) ×_{|Y |} |X| ȷ ȷ ȷ fl \end{matrix} ; (V^{1} ∪_{U^{1}} U)

over the open set. V^{1} ∪_{U^{1}} U of |Y |. ■

The following result of Quillen [76] is the key to both the closed model structure of the simplicial set category, and the relation between simplicial homotopy theory and ordinary homotopy theory. These results will appear in the next section.

Theorem 10.10 (Quillen). The realization of a Kan fibration is a Serre fibration.

Proof: Let p : X → Y be a Kan fibration. According to Proposition 10.3, one can choose a commutative diagram

Z  w j_{4 4 44 6} q  X  w g u p  Z h h h h k q

Y

where q is a minimal fibrations, gj = 1_{Z} and jg is fibrewise homotopic to 1_{X}. In view of Theorem 10.9, it clearly suffices to prove the following two results:

Lemma 10.11. g : X → Z has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0.

62 I. Simplicial sets Lemma 10.12. Suppose that g : X → Z has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0. Then |g| : |X| → |Z| is a Serre fibration. Proof of Lemma 10.11: Suppose that the diagrams

\begin{matrix}X × Δ^{1}  w h u pr \\ Xup \end{matrix} ; {X  w p \atop Y}

Xu d^{0} _{ȷ ȷ ȷ ȷȷ fi} 1_{X}
X × Δ^{1} w h X
Xu d^{1}  w g  Zuj

represent the fibrewise homotopy, and suppose that the diagram

\begin{matrix}∂Δ^{n}  w u z u i \\ Xug \end{matrix} ; {Δ^{n}  w v \atop Z}

commutes. Then there are commutative diagrams

∂Δ^{n} × Δ^{1} w u × 1 z u i × 1 X × Δ^{1} w h Xup

Δ^{n} × Δ^{1} w pr_{L} Δ^{n} w qv Y

∂Δ^{n}  w u z u i  X  w g Z  w j Xup
Δ^{n}  w v  Z^{‚ ‚ ‚‚ ›} j w q Y,

10. Minimal fibrations 63

and hence a diagram

(∂Δ^{n} × Δ^{1}) ∪ (Δ^{n} × {0})  w (h(u × 1), jv) z u  Xup
Δ^{n} × Δ^{1} w pr_{L}^{ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷ ȷȷ fi} h_{1}

Δ^{n} w qv

Y.

Let v_{1} be the simplex classified by the composite

Δ^{n} ⟶^{d^{0}} Δ^{n} × Δ^{1} ⟶^{h_{1}} X.

The idea of the proof is now to show that gv_{1} = v. The diagram

{∂Δ^{n} w u z u i \atop X} ; Δ^{n}^{[ [ [ []} v_{1}

commutes, and consider the composite

Δ^{n} × Δ^{1} ⟶^{v_{1}×1} X × Δ^{1} ⟶^{h} X ⟶^{g} Z.

Then gh(v_{1} × 1) is a homotopy gjgv_{1} = gv_{1} ⟶^{≃} gv_{1}. Moreover, the homotopy on the boundary is gh(u × 1). It follows that there is a commutative diagram

(∂Δ^{n} × Δ^{2}) ∪ (Δ^{n} × Λ_{2}^{2}) w (s_{0}(gh(u × 1)), (gh_{1}, gh(v_{1} × 1), )) z u Zuq

Δ^{n} × Δ^{2} w pr_{L}^{```````````````````}ˆ ξ Δ^{n} w qv Y.

Then the diagram

∂Δ^{n} × Δ^{1}  w pr_{L} z u i × 1  ∂Δ^{n} ugu
Δ^{n} × Δ^{1}  w ξ(1 × d^{2}) u pr_{L}  Zuq
Δ^{n}  w qv  Y

commutes, and so gv_{1} = gjv = v by the minimality of q. ■

Proof of Lemma 10.12: Suppose that f : X → Y has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0, and hence with respect to all inclusions of simplicial sets. Then there is a commutative diagram

X  w 1_{X} u (1_{X}, f)  Xuf

X × Y  w pr_{R}^{h h hh j} r  Y,

and so f is a retract of the projection pr : X × Y → Y. But then |f| is a Serre fibration. ■

This also completes the proof of Theorem 10.10.

11. The closed model structure.

The results stated and proved in this section are the culmination of all of the hard work that we have done up to this point. We shall prove here that the entire simplicial set category S (not just the subcategory of fibrant objects) has a closed model structure, and that the resulting homotopy theory is equivalent to the ordinary homotopy theory of topological spaces. These are the central organizational theorems of simplicial homotopy theory.

Proposition 11.1. Suppose that X is a Kan complex. Then the canonical map η_{X} : X → S|X| is a weak equivalence in the sense that it induces an isomorphism in all possible simplicial homotopy groups.

Proof: Recall that S|X| is also a Kan complex.

η_{X} induces an isomorphism in π_{0}: every map v : |Δ^{0}| → |X| factors through the realization of a simplex |σ| : |Δ^{n}| → |X| and so S|X| is connected if π_{0}X = ∗. On the other hand X is a disjoint union of its path components and S| | preserves disjoint unions, so that π_{0}X → π_{0}S|X| is monic.

Suppose that we have shown that η_{X} induces an isomorphism

(η_{X})_{∗} : π_{i}(X, x) ⟶^{≅} π_{i}(S|X|, ηx)

for all choice of base points x ∈ X and i ≤ n. Then, using 10.10 for the pathloop fibration ΩX → P X → X determined by x (see the discussion following the proof of 7.3), one finds a commutative diagram

\begin{matrix}π_{n+1}(X, x)  w η_{X} u ∂ ≅ \\ π_{n+1}(S|X|, ηx) u∂ \end{matrix} ; \begin{matrix}π_{n}(ΩX, x)  w ≅ η_{ΩX} \\ π_{n}(S|ΩX|, ηx), \end{matrix}

and so we’re done if we can show that P X and hence S|P X| contracts onto its base point. But there is a diagram

(Δ^{0} × Δ^{1}) ∪ (P X × ∂Δ^{1})  w (x, (1_{P X}, x)) z u  P X ut

P X × Δ^{1}  w^{’ ’ ’ ’ ’ ’ ’ ’ ’’)} h  Δ^{0},

and h exists because t has the right lifting property with respect to all ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0. ■

If X is a Kan complex and x is any vertex of X, then it follows from Proposition 11.1 and adjointness that η_{X} induces a canonical isomorphism

π_{n}(X, x) ≅ π_{n}(|X|, x), n ≥ 1,

where the group on the right is the ordinary homotopy group of the space |X|. It follows that a map f : X → Y of Kan complexes is a (simplicial) weak equivalence if and only if the induced map |f| : |X| → |Y | is a topological weak equivalence. Thus, we are entitled to define a map f : X → Y of arbitrary simplicial sets to be a weak equivalence if the induced map |f| : |X| → |Y | is a weak equivalence of spaces. Our last major technical result leading to the closed model structure of S is

Theorem 11.2. Suppose that g : X → Y is a map between arbitrary simplicial sets. Then g is a Kan fibration and a weak equivalence if and only if g has the right lifting property with respect to all inclusions ∂Δ^{n} ⊂ Δ^{n}, n ≥ 0.

Proof: Suppose that g : X → Y is a Kan fibration with the advertised lifting property. We have to show that S|g| : S|X| → S|Y | is a weak equivalence. Y is an arbitrary simplicial set, so we must define π_{0}Y to be the set of equivalence classes of vertices of Y for the relation generated by the vertex homotopy relation. In other words, y ≃ z if and only if there is a string of vertices

y = y_{0}, y_{1}, …, y_{n} = z

and a string of 1-simplices

v_{1}, …, v_{n}

of X such that ∂v_{i} = (y_{i−1}, y_{i}) or ∂v_{i} = (y_{i}, y_{i−1}) for i = 1, …, n. If Y is a Kan complex, then this definition of π_{0}Y coincides with the old definition. Moreover,

the canonical map η_{Y} : Y → S|Y | induces an isomorphism π_{0}Y ⟶^{≅} π_{0}S|Y | for all simplicial sets Y. The lifting property implies that g_{∗} : π_{0}X → π_{0}Y is an isomorphism, so that the induced map π_{0}S|X| → π_{0}S|Y | is an isomorphism as well. Finally, it suffices to show that the induced maps π_{i}(S|X|, x) → π_{i}(S|Y |, gx) of simplicial homotopy groups are isomorphisms for all vertices x of X and all i ≥ 1. But Theorem 10.9 implies that the fibre of the fibration S|g| : S|X| → S|Y | over g(x) is S|F_{x}|, where

\begin{matrix}F_{x}  w u \\ Xug \end{matrix} ; {Δ^{0}  w gx \atop Y}

is a pullback in the simplicial set category. F_{x} is a contractible Kan complex (see the corresponding argument for P X in 11.1), and so S|F_{x}| is contractible as well. The result then follows from a long exact sequence argument.

For the reverse implication, it suffices (see the proof of Theorem 10.10) to assume that g : X → Y is a minimal fibration and a weak equivalence and then prove that it has the lifting property. We may also assume that Y is connected. Consider a diagram

\begin{matrix}∂Δ^{n}  w α z u \\ Xug \end{matrix} ; {Δ^{n}  w β \atop Y}

and the induced diagram

∂Δ^{n}  w α_{∗} z u  Δ^{n} ×_{Y} X u

Δ^{n}  w 1  Δ^{n}.

It suffices to find a lifting for this last case. But there is a fibrewise isomorphism

Δ^{n} ×_{Y} X  w ≅_{[ [ []}  Δ^{n} × F_{y} ȷ ȷ ȷ fl pr_{L}

Δ^{n}

by 10.8, where F_{y} is the fibre over some vertex y of Y. Thus, it suffices to find a lifting of the following sort:

\begin{matrix}∂Δ^{n} w z u \\ F_{y}. \end{matrix} ; Δ^{n}^{\ \ \ \ ]}

But this can be done, since F_{y} is a Kan complex such that π_{0}(S|F_{y}|) is trivial, and π_{i}(S|F_{y}|, ∗) = 0, i ≥ 1 for any base point ∗, and η : F_{y} → S|F_{y}| is a weak equivalence by Proposition 11.1. ■

A cofibration of simplicial sets is an inclusion map.

Theorem 11.3. The simplicial set category S, together with the specified classes of Kan fibrations, cofibrations and weak equivalences, is a closed model category.

Proof: CM1 is satisfied, since S is complete and cocomplete. CM2 follows from CM2 for CGHaus. CM3 (the retract axiom) is trivial.

To prove the factorization axiom CM5, observe that a small object argument and the previous theorem together imply that any simplicial set map f : X → Y may be factored as:

{X  w f_{N N N N P} i \atop Y} ; Z^{‚ ‚ ‚‚ ›} p and \begin{matrix}X  w f_{[ [ []} j \\ Y, \end{matrix} ; W^{‚ ‚ ‚‚ ›} q

where i is anodyne, p is a fibration, j is an inclusion, and q is a trivial fibration. The class of inclusions i : U → V of simplicial sets such that |i| : |U| → |V | is a trivial cofibration is saturated, by adjointness, and includes all Λ_{k}^{n} ⊂ Δ^{n}. Thus all anodyne extensions are trivial cofibrations of S. To prove CM4 we must show that the lifting (dotted arrow) exists in any commutative diagram

U  w u i  Xup

V w^{i i i ij} Y,

where p is a fibration and i is a cofibration, and either i or p is trivial. The case where p is trivial is the previous theorem. On the other hand, if i is a weak equivalence, then there is a diagram

U  w j u i  Zup

V  w 1_{V}^{h h hh j} s  V

where j is anodyne and p is a (necessarily) trivial fibration, so that s exists. But then i is a retract of an anodyne extension, so i has the left lifting property with respect to all fibrations (compare the proof of 9.4). ■

The homotopy category Ho(S) is obtained from S by formally inverting the weak equivalences. There are several ways to do this [75], [33], [15]. One may also form the category Ho(T op) by formally inverting the weak homotopy equivalences; this category is equivalent to the category of CW-complexes and ordinary homotopy classes of maps. For the same reason (see [75]), Ho(S) is equivalent to the category of Kan complexes and simplicial homotopy classes of maps. The realization functor preserve weak equivalences, by definition. One may use Theorem 10.10 (see the argument in 11.1) to show that the canonical map ϵ : |S(Y)| → Y is a weak equivalence, for any topological space Y, and so the singular functor preserves weak equivalences as well. It follows that the realization and singular functors induce functors

Ho(S) w | |_{∗} u S_{∗}

Ho(Top)

of the associated homotopy categories.

Theorem 11.4. The realization and singular functors induce an equivalence of categories of Ho(S) with Ho(Top).

Proof: We have just seen that ϵ : |S(Y)| → Y is a weak equivalence for all topological spaces Y. It remains to show that η : X → S|X| is a weak equivalence for all simplicial sets X. But η is a weak equivalence if X is a Kan complex, by 11.1, and every simplicial set is weakly equivalent to a Kan complex by CM5. The composite functor S| | preserves weak equivalences. ■

The original proof of Theorem 11.3 appears in [75], modulo some fiddling with axioms (see [77]). Theorem 11.4 has been known in some form or other since the late 1950’s (see [67], [33], [59]).

Although it may now seem like a moot point, the function complex trick of Proposition 5.2 was a key step in the proof of Theorem 11.3. We can now amplify the statement of Proposition 5.2 as follows:

Proposition 11.5. The category S of simplicial sets satisfies the simplicial model axiom

SM7: Suppose that i : U → V is a cofibration and p : X → Y is a fibration. Then the induced map

Hom(V, X) ⟶^{(i^{∗},p_{∗})} Hom(U, X) ×_{Hom(U,Y)} Hom(V, X)

is a fibration, which is trivial if either i or p is trivial.

Proof: Use Proposition 5.2 and Theorem 11.2. ■