On the classification of fibrations

Classification questions are often about understanding components of a category. It is much more desirable however to be able to understand the entire homotopy type of this category and not just the set of its components. In this paper we prove that this is possible for the category of functors indexed a small category I which assign to any morphism in I a weak equivalence in a given model category. We identify the homotopy type of this category with the mapping space out of the nerve of I into the classifying space of the space of weak equivalences of values of these functors. We use it to reprove the classical classification of fibration theorem of Stasheff and its generalizations by Dwyer-Kan.


Introduction
Classification questions are often about understanding components of a category. It is not unusual however that with a category one can associate a unique homotopy type of a simplicial set whose set of components coincide with the set of components of the category. Such a space carries more information about the category than just the set of its components. For example: 1.1. Definition. Let X and F be spaces. Define Fib(X, F ) to be the category whose objects are maps f : A → B where B is weakly equivalent to X and the homotopy fiber of f , over any base point in B, is weakly equivalent to F . The set of morphisms in Fib(X, F ) between f : A → B and f ′ : A ′ → B ′ , consists of pairs of weak equivalences φ : A → A ′ and ψ : B → B ′ for which f ′ φ = ψf . The composition of morphisms is induced by the usual composition of maps.
A classical result states that the components of Fib(X, F ) can be enumerated by the set of homotopy classes [X, Bwe(F, F )] where Bwe(F, F ) is the classifying space of the topological monoid of weak equivalences of F . This is a classical theorem proved by Stasheff in [19] and re-proved and generalized by May in [15].
Instead of looking at the set of components, it is more desirable to study the entire moduli space of fibrations. One would like to understand the homotopy type of the category Fib(X, F ) and not just the set of its components. Naively one can try to form the nerve of Fib(X, F ) and then identify its homotopy type. However, since Fib(X, F ) is not equivalent to a small category, this can not be done so directly. Instead we are going to show that Fib(X, F ) has what we call a core (see 5.2) which is a small category whose nerve approximates the homotopy type of Fib(X, F ). Our classification statement can be then formulated as follows: Theorem A. The category Fib(X, F ) has a core whose nerve admits a map to Bwe(X, X). This map has a section and its homotopy fiber is weakly equivalent to the mapping space map(X, Bwe(F, F )).
It turns out that the above theorem is a particular case of a much more general statement that holds in an arbitrary model category. The purpose of such a generalization is not only to show that analogous classification statements hold in much broader context. Statements that hold in an arbitrary model category often have more conceptual proofs in which one does not need to use the nature of objects considered but rather basic fundamental facts from homotopy theory. In this way arguments are becoming more transparent. It was Dwyer and Kan who first realized and proved that such general classification statements are true. In their sequence of papers that includes [5,6,7,8,9] they develop a strategy and techniques for dealing with classification questions. An important part of their program was a discovery of continuity in model categories. They showed that an arbitrary model category has mapping spaces whose homotopy type is unique. They also gave a particular model for them through the use of so called hammocks.
In this paper we follow, in principle, the plan of Dwyer and Kan. Our realization of their strategy is different however. For example homotopical smallness is an essential ingredient in our work. Another important difference is that we use a model for mapping spaces developed in [4]. Our general statement is about the homotopy type of the category of weak equivalences M we of a model category M. Its objects are the objects of M and morphisms are all the weak equivalences in M. To understand its homotopy type, we study the components of M we . For an object X in M, we denote by X we the full subcategory of M we that consists of all the objects in M which are weakly equivalent to X. This subcategory is also called a component of M we . Our key result states (see 17.1): Theorem B. Let I be a small category and X be an object in a model category M. The category of functors Fun(I, X we ) has a core which is weakly equivalent to the mapping space map(N (I), Bwe(X, X)).

Categorical constructions and notation
To describe sets we use Zermelo-Fraenkel set theory with the axiom of choice.
2.1. The term category is used as defined in [14,Section 7]. The category A op is the opposite category of A. A natural transformation between functors f, g : B → A is denoted by φ : g → f . It consists of morphisms φ b : g(b) → f (b) in A for any object b in B such that f (β)φ b1 = φ b0 g(β) for any morphism β : The category of sets is denoted by Sets. A category is small if it has a set of objects. Cat denotes the category of small categories and ∆ its full subcategory whose objects are posets [n] := {0 < . . . < n} for n ≥ 0.
The symbol B ⊂ A denotes the fact that B is a subset or a subcategory or a subspace, depending on whether A is a set or a category or a space.
The restriction of P to the the full subcategory of arrows in M of the form ∅ → X is a functorial cofibrant replacement in M.
Simplicial sets are also called spaces and their category with the standard model structure (see for example [11]) is denoted by Spaces. The full subcategory of Spaces whose objects are the standard simplices ∆[n] is isomorphic to ∆.
2.3. A system F of categories indexed by a category C consists of a category F c for any object c in C and a functor F α : F c0 → F c1 for any morphism α : c 1 → c 0 in C (note contravariancy). These functors are required to satisfy: F id c = id for any object c; and F αα ′ = F α ′ F α for any morphisms α ′ : c 2 → c 1 and α : c 1 → c 0 .
A subsystem G ⊂ F consists of a subcategory G c ⊂ F c for any object c in C such that, for any morphism α : c 1 → c 0 , F α takes G c0 to G c1 .
2.4. Let F be a system of categories indexed by C. Its Grothendieck construction, denoted by Gr C F , is the category whose objects are pairs (c, x) where c is an object in C and x in F c . The set of morphisms between (c 1 , x 1 ) and (c 0 , x 0 ) is the set of pairs (α : : c 1 → c 0 , β : x 1 → F α (x 0 )) where α is a morphism in C and β is a morphism in F c1 . The composition of (α ′ : c 2 → c 1 , β ′ : x 2 → F α ′ (x 1 )) and (α : c 1 → c 0 , β : x 1 → F α (x 0 )) is defined to be the pair: The projection π : Gr C F → C is the functor that assigns to an object (c, x) (resp. morphism (α, β)) in Gr C F the object c (resp. morphism α) in C. For any object c in C the functor F c → Gr C F which assigns to an object x the pair (c, x) and to a morphism β : x → y the pair (id c , β) is called the standard inclusion.
2.5. Let f : B → A be a functor and a be an object in A. The under category a ↑ f has pairs (b, α : a → f (b)) of an object b in B and a morphism α : a → f (b) in A as objects. The set of morphisms between (b 1 , α 1 ) and (b 0 , α 0 ) in a ↑ f is the set of morphisms β : b 1 → b 0 in B for which f (β)α 1 = α 0 . The category a ↑ id A is also denoted by a ↑A. By forgetting the second component we obtain a functor (a ↑ f ) → B called forgetful.
The over category f ↓ a has pairs (b, α : f (b) → a) of an object b in B and a morphism α in A as objects. The set of morphisms between (b 1 , α 1 ) and (b 0 , α 0 ) is the set of morphisms β : b 1 → b 0 in B for which α 0 f (β) = α 1 .
Consider the following commutative diagram of functors: Let c be an object in C. The symbol (e, h) : c ↑ g → h(c) ↑ f denotes the functor that maps an object (d, α) to (e(d), h(α)) and a morphism α : d 1 → d 0 to e(α). Let γ : a 1 → a 0 be a morphism in A. The functor γ ↑ f : a 0 ↑ f → a 1 ↑ f assigns to (b, α) the object (b, αγ) and to a morphism β the same β. The assignment a → (a ↑ f ) and λ → (γ ↑ f ) is a system of categories indexed by A denoted by − ↑ f . Its Grothendieck construction Gr A (− ↑ f ) is isomorphic to a category whose objects are pairs (b, α : a → f (b)) of an object b in B and a morphism α in A. The set of morphisms between (b 1 , α 1 : a 1 → f (b 1 )) and (b 0 , α 0 : a 0 → f (b 0 )) consists of pairs (β : b 1 → b 0 , γ : a 1 → a 0 ) where γ is a morphism in A and β is a morphism in B making the following square commutative: The functorπ : These functors fit into the following commutative diagram: Note that there is a natural transformation between the functorsfπ and id GrA(−↑f ) which for an object (b, α : a → f (b)) is given by the morphism (id b , α).
2.6. Fun(I, C) is the category of functors indexed by a small category I with values in a category C and natural transformations as morphisms. The set of natural transformations between F : I → C and G : I → C is denoted by Nat(F, G).
Let f : I → J be a functor of small categories. The composition with f functor is denoted by f * : Fun(J, C) → Fun(I, C). It assigns to a natural transformation {ψ j } j∈J the natural transformation {ψ f (i) } i∈I . If C is closed under colimits, then f * has a left adjoint f k : Fun(I, C) → Fun(J, C) called the left Kan extension. The assignment I → Fun(I, C) and f → f * is a system of categories indexed by Cat.
A natural transformation φ : F → G in Fun(I, M) is called a weak equivalence if φ i : F (i) → G(i) is a weak equivalence for any i in I. Ho(Fun(I, M)) denotes the localization of Fun(I, M) with respect to weak equivalences which exists by [3,Theorem 11.3].

2.7.
A connected component of a category C containing an object c is the class of all objects d for which there is a finite sequence of morphisms d = x 0 → x 1 ← · · · ← x n = c in C. The symbol π 0 (C) denotes the discrete category (identities are the only morphisms) whose objects are connected components of C and π 0 : C → π 0 (C) denotes the unique functor mapping an object to its component.
Part I. Categories and homotopy. In this part we review two ways of doing homotopy theory on categories. In one the homotopy relation is induced by natural transformations. This notion however is too strong for us. We need weak equivalences. A standard way of introducing them is to transport weak notions from simplicial sets using the nerve. This works for small categories. Our aim is to extend the weak notions to categories such as Fib(X, F ) that can be approximated by small categories. Dwyer and Kan called them homotopically small ( [6]). We call them essentially small and their study is the main aim of this part.

Categorical homotopy
Here is a standard dictionary of homotopy notions on arbitrary categories: • Functors f, g : B → A are homotopic if there are functors {h k : B → A} 0≤k≤n and natural transformations f = h 0 → h 1 ← · · · ← h n = g. • f : B → A is a homotopy equivalence if it has a homotopy inverse, i.e., a functor g : A → B for which f g and gf are homotopic to id A and id B .
is a homotopy equivalence. • A strong homotopy pull-back is a commutative square of functors: is a homotopy equivalence for any object c in C.

Small categories
It is the nerve construction used to translate weak homotopy notions from Spaces into Cat. It is a functor N : Cat → Spaces that assigns to a small category I a simplicial set N (I) whose set of n-dimensional simplices is, for n > 0, the set of n-composable morphisms in I and, if n = 0, it is the set of objects in I.
Here is a basic dictionary (compare with notions recalled in Section 3): • A functor f : J → I of small categories is called a weak equivalence, if N (f ) : N (J) → N (I) is a weak equivalence of spaces.
• A functor f : J → I of small categories is called a quasi-fibration if α ↑ f : i 0 ↑ f → i 1 ↑ f is a weak equivalence for any α : i 1 → i 0 in I. • A commutative square of small categories is called a homotopy pull-back if after applying the nerve we obtain a homotopy pull-back of spaces. It is well known that if functors of small categories are homotopic as functors, then their nerves are homotopic as maps. Consequently a homotopy equivalence, resp. strong fibration, of small categories is a weak equivalence, resp. quasi fibration. To prove that for small categories strong homotopy pull-backs are homotopy pull-backs we need the so called Thomason and Puppe's theorems: 4.1. Proposition. Let I be a small category and F, G : I op → Cat be functors.
(1) N (Gr I F ) is weakly equivalent to hocolim I op N (F ).
(2) Let f : F → G be a natural transformation. Assume f i is a weak equivalence for any object i in I. Then Gr I f is a weak equivalence.
Then, for any object i in I, the following is a homotopy pull-back square: where π is the projection, F (i) → Gr I F is the standard inclusion (see 2.4), and i : [0] → I is the functor that sends the object 0 to i.

Essentially small categories
The aim of this section is to explain how certain "big" categories can be approximated by small categories. It is based on the following well known fact: . is a sequence of small categories where each inclusion is a weak equivalence, then I 0 ⊂ colim I n = ∪ n≥0 I n is also a weak equivalence.
Here is our key definition: 5.2. Definition. A core of a category C is a small subcategory I ⊂ C such that, for any small subcategory J ⊂ C with I ⊂ J, there is a small subcategory K ⊂ C for which J ⊂ K and the inclusion I ⊂ K is a weak equivalence. A category is said to be essentially small if it has a core.
For example if C has a small skeleton, then this skeleton is its core.

5.3.
Proposition. Let C be a category.
(1) If I ⊂ C and J ⊂ C are cores, then I and J are weakly equivalent.
(2) A discrete essentially small category is small.
(3) If C is essentially small, then the components of C form a set. If I ⊂ C is a core, then this inclusion induces a bijection between π 0 (I) and π 0 (C).

Lemma.
(1) Let I ⊂ C be a core and I ′ ⊂ C a small subcategory containing I. Then I ′ ⊂ C is a core if and only if I ⊂ I ′ is a weak equivalence.
(2) Let J ⊂ C be a small subcategory and I ⊂ C be a core. Then there is a full subcategory K ⊂ C such that J ⊂ K ⊃ I and I ⊂ K is a weak equivalence.
Proof. (1): Assume I ⊂ I ′ is a weak equivalence. Let J ⊂ C be a small subcategory such that I ′ ⊂ J. Since I ⊂ C is a core, there is a small subcategory K ⊂ C for which J ⊂ K and I ⊂ K is a weak equivalence. By the "2 out of 3" property the inclusion I ′ ⊂ K is also a weak equivalence. This shows that I ′ ⊂ C is a core. Assume I ′ ⊂ C is a core. We define inductively a sequence of small subcategories I 0 ⊂ I ′ 0 ⊂ I 1 ⊂ I ′ 1 ⊂ · · · ⊂ C. Set I 0 = I and I ′ 0 = I ′ . Assume n > 0. Let I n ⊂ C be a small subcategory containing I ′ n−1 for which I 0 ⊂ I n is a weak equivalence. It exists since I 0 is a core in C. Similarly, let I ′ n ⊂ C be a small subcategory containing I n for which I ′ 0 ⊂ I ′ n is a weak equivalence. Note that n≥0 I n = n≥0 I ′ n . Moreover, according to 5.1, the inclusions I 0 ⊂ n≥0 I n = n≥0 I ′ n ⊃ I ′ 0 are weak equivalences. It follows that I 0 ⊂ I ′ 0 is a weak equivalence as well. (2): Define inductively a sequence of small subcategories I 0 ⊂ K 0 ⊂ I 1 ⊂ K 1 ⊂ · · · ⊂ C. Set I 0 = I and K 0 to be the full subcategory in C on objects in I 0 and J. Assume n > 0. Define I n ⊂ C to be a small subcategory such that K n−1 ⊂ I n and I 0 ⊂ I n is a weak equivalence. Define K n to be the full subcategory of C on the set of objects in I n . Set K := n≥0 K n . Since for any n, K n is a full subcategory in C, same is true for K. As K = n≥0 I n and I = I 0 ⊂ I n is a weak equivalence, for any n, I = I 0 ⊂ K is also a weak equivalence.  2) there is a small subcategory K ⊂ C such that I ⊂ K ⊃ J and I ⊂ K is a weak equivalence. Since I ⊂ C is a core we can use 5.4.(1) to conclude that K ⊂ C is also a core. By assumption J ⊂ C is a core. The inclusion J ⊂ K is thus a weak equivalence.
(2): Just note that weak equivalences of discrete categories are isomorphisms.
(3): Follows from the fact that π 0 : C → π 0 (C) maps a core to a core. By 5.3.(1) the homotopy type of a core is a well define invariant. We can thus use it to introduce various homotopy notions on essentially small categories. For example, we can define homotopy groups of an essentially small category as the homotopy groups of the nerve of its core. Usefulness of such invariants depend on how they behave under functors. For that we need to extend Definition 5.2 to: 5.5. Definition. Let F be a system of categories indexed by a small category I (see 2.3). A core of F is a subsystem F ⊂ F such that F i ⊂ F i is a core for any i. A system F is called essentially small if it has a core. 5.6. Proposition. Let F be a system of categories indexed by a small category I.
(1) F is essentially small if and only if, for any i, F i is essentially small.
(2) Assume that F ⊂ F ⊃ F ′ are cores. Then there is a core H ⊂ F such that F ⊂ H ⊃ F ′ and H i ⊂ F i is a full subcategory for any object i in I. (3) If F ⊂ F is a core, then Gr I F ⊂ Gr I F is a core.

5.7.
Lemma. Let G i ⊂ F i be a small subcategory for any object i in I. Then there is a core H ⊂ F such that G i ⊂ H i and H i ⊂ F i is a full subcategory for any i.
Proof. We construct inductively a sequence of small subcategories, for any object Assume n > 0. Let (H i ) n ⊂ F i to be a core such that (G i ) n−1 ⊂ (H i ) n . It exists by 5.4. (2). Define (G i ) n to be the full subcategory of F i on the set of objects α : i→j F α ((H j ) n ) where the index α : i → j runs over all possible morphisms in I with domain i. The purpose of this definition is to ensure that, for any n > 0, a. (H i ) n ⊂ F i is a core for any object i in I; b. (G i ) n ⊂ F i is a full subcategory for any object i in I; c. F α : F j → F i takes (H j ) n to (G i ) n for any morphism α : i → j in I.
Proof. Choose a sequence of natural transformations rf = h 0 → · · · ← h m = id B . For small subcategories I ⊂ A and K ⊂ B, define inductively a sequence of small subcategories of A and B that fit into the following commutative diagram: Let n > 0. Using 5.4.(2), define I n ⊂ A to be a core which is a full subcategory and contains the set of objects that either belong to I n−1 or are of the form f (b) where b is in K n−1 . Set K n to be the full subcategory of B on the set of objects that either belong to K n−1 , or are of the form r(a) where a is in I n , or are of the form h k (b) where b is in K n−1 . The purpose of this definition is to ensure: a. f : B → A takes K n−1 to I n and r : A → B takes I n to K n ; b. I n ⊂ A is a core for any n > 0; c. h k : B → B takes K n−1 to K n for any 0 ≤ k ≤ m; d. K n ⊂ B is a full subcategory. Define I ∞ := ∪ n≥0 I n and K ∞ := ∪ n≥0 K l . The above requirements imply: is homotopic to the identity functor. The appropriate "zig-zag" is obtained by restriction using fulness of K ∞ in B. We need to prove that K ∞ ⊂ B is a core. Let J ⊂ B be a small subcategory containing K ∞ . The above construction applied to I ∞ ⊂ A and J ⊂ B yields: Since rf : J ∞ → J ∞ and rf : K ∞ → K ∞ are homotopic to the identity functors, K ∞ ⊂ J ∞ , as a homotopy retract of a weak equivalence, is a weak equivalence.

5.9.
Corollary. Let f : B → A be a homotopy equivalence. Then A is essentially small if and only if B is.

Weak homotopy notions for essentially small categories
Our aim is to extend the dictionary from Section 4 to essentially small categories. A functor f : F 1 → F 0 is a system of categories indexed by the poset [1]. Thus its core consist of cores F 1 ⊂ F 1 and F 0 ⊂ F 0 such that f takes F 1 to F 0 . The restricted functor f : F 1 → F 0 fits into a commutative diagram: By the "2 out of 3" property of weak equivalences and 5.6.(2), if f : Similarly, a commutative square on the left below is a system of categories indexed by the poset of all the subsets of {0, 1}. Its core consists of cores F ∅ ⊂ F ∅ , F 0 ⊂ F 0 , F 1 ⊂ F 1 , and F 0,1 ⊂ F 0,1 making the right cube commutative: Again, by the "2 out of 3" property and 5.6.(2), if one core of a commutative square is homotopy pull-back then so is any other. This justifies: • A functor is called a weak equivalence if it has a core which is a weak equivalence.
a core which is homotopy pull-back.
Note that a weak equivalence can only be between essentially small categories.

6.2.
Proposition. Let f : B → A be a homotopy equivalence of essentially small categories. Then f is a weak equivalence.
Proof. Let g be a homotopy inverse to f . Consider a system of categories indexed by the free category on the graph on the left given by the diagram on the right: This system is essentially small (see 5.6.(1)) and hence has a core. It consists of small subcategories A ⊂ A and B ⊂ B that fit into a commutative diagram: To show that f : A → B is a weak equivalence, it is enough to prove that the compositions f g : A → A and gf : B → B are weak equivalences. Choose a sequence of natural transformations f g = h 0 → · · · ← h m = id A . The functors {h k : A → A} 0≤k≤m form a system of categories indexed by: It has a core given by full subcategories A 0 ⊂ A and A 1 ⊂ A containing A (see 5.7). Take the restrictions h k : A 0 → A 1 . Since A 0 and A 1 are cores of A, h m : A 0 → A 1 is a weak equivalence as it is the restriction of the identity. Use fullness of A 1 in A to get a sequence of natural transformations h 0 → · · · ← h m between these restrictions. Thus h 0 : A 0 → A 1 is a weak equivalence too and hence, by the "2 out of 3" property, so is f g : A → A. By symmetry gf is also a weak equivalence.
If F is a system of essentially small categories indexed by a small category I, then according to 5.6.(3), Gr I F is essentially small. This is probably not true in general if we just assume that I is essentially small. However, we have the following lemma, which will be an important tool for us later in this paper. 6.3. Lemma. Let F be a system of essentially small categories indexed by an essentially small category A. If, for any morphism α : a 1 → a 0 in A, the functor F α : F a0 → F a1 is a weak equivalence, then Gr A F is essentially small.
Proof. Choose a core A ⊂ A. Let F be a core of the restriction of F to A (see 5.6.(1)). We claim Gr A F ⊂ Gr A F is a core. Let J ⊂ Gr A F be a small subcategory containing Gr A F and A ′ ⊂ A be a core containing the full subcategory on all the objects of the form π(x) where x is in J and π : Gr A F → A is the projection (see 2.4). Let F ′ be a core of the restriction of F to A ′ such that . According to 4.1. (3), the homotopy fibers of the nerves of the projections π : Gr A F → A and π : Gr A ′ F ′ → A ′ over a vertex given by an object a in A are weakly equivalent to the nerves of F a and F ′ a . As these categories are the cores of F a , they are weakly equivalent and consequently the following square is a homotopy pull-back: We can now apply 6.3.

Proposition. Let f : B → A be a strong fibration between essentially small categories. Then f is a quasi-fibration.
Proof. We claim: for small subcategories A ′ ⊂ A and B ′ ⊂ B, there is a core f : B → A of f which is a quasi-fibration and such that A ′ ⊂ A and B ′ ⊂ B. Assume the claim. To prove the proposition we need to show a ↑ f is essentially small for any a in A. The under category of the restriction of f to C ⊂ B is denoted by a ↑ C. Use the claim to get a quasi-fibration core f : B → A of f : B → A such that a is in A. We will show that a ↑ B ⊂ a ↑ f is a core. Let J ⊂ a ↑ f be a small subcategory containing a ↑ B and B ′ be the full subcategory of B on the set of objects b for which there is α : Use the claim again to get a quasi-fibration core f :B →Â of f : A → B such that A ⊂Â and B ′ ⊂B. All this fits into a commutative diagram: The inclusions A ⊂Â and B ⊂B are weak equivalences (see 5.4. (1)). Since f : B → A and f :B →Â are quasi-fibrations, the homotopy fibers of their nerves over the components containing a are given by N (a ↑ B) and N (a ↑B) (see 4.1.(4)). These two observations imply that a ↑ B ⊂ a ↑B is a weak equivalence.
It remains to show the claim. Since f : B → A is a strong fibration, for any α : It exists by 5.7. Assume n > 0 and the sequence is defined for indices smaller than n. Let D n to be the full subcategory of B on the set of objects b such that: • b either belongs to B n−1 or • there is an object b ′ in B n−1 and morphisms α : Define C n to be the full subcategory of A on the set of objects that belong either The purpose is to ensure that for any α : Let f : B n → A n to be a core of f : B → A such that C n ⊂ A n and D n ⊂ B n . Define A := ∪ n≥0 A n and B := ∪ n≥0 B n . According to 5.1 and 5.4.
(1) f : B → A is also a core of f : B → A. We are going to show that f : B → A is a strong fibration. The requirements above imply that, for any α : a 1 → a 0 in A: Since B is a full subcategory in B, for any α : a 1 → a 0 in A, by restricting to a 0 ↑ B and a 1 ↑ B we have two sequences of natural transformations is therefore a homotopy equivalence and hence a weak equivalence. Which shows the claim.
6.6. Corollary. Let the following be a strong homotopy pull-back square: Assume that A, B, and C are essentially small categories. Then D is also essentially small and the above square is homotopy pull-back.
Proof. The strong homotopy pull-back assumption implies (e, h) : c ↑ g → h(c) ↑ f is a homotopy equivalence for any c in C, and f : B → A and g : D → C are a strong fibrations. Since A and B are essentially small, by 6.5, f : B → A is a quasi-fibration and thus a ↑ f is essentially small for any a in A. By 5.9, c ↑ g is then also essentially small for any c in C. The functor g : D → C is therefore a quasi-fibration. As C is essentially small, 6.4 implies that so is D.
By the claim in the proof of 6.5, there is a core g : D → C of g which is a quasifibration. Let B ′ ⊂ B be the full subcategory on the set of all objects e(d) where d is in D and the full subcategory A ′ ⊂ A on the set of all objects h(c) where c is in C. The same claim yields a core f : B → A of f which is a quasi-fibration and such that A ′ ⊂ A and B ′ ⊂ B. This leads to a commutative diagram of categories: In the proof of 6.5 it was also shown that a ↑ B ⊂ a ↑ f and c ↑ D ⊂ c ↑ g are cores for any a in A and c in C. By the "2 out of 3" property (e, h) : The following square is a therefore a homotopy-pull-back: The aim of this part is to present a construction of the spaces of weak equivalences and their deloopings in an arbitrary model category based on [4].

Simplex categories
Let A be a simplicial set. Its simplex category (see [3,Section 6]), denoted by the same symbol A, is a category whose objects are simplices of A i.e., maps of the form σ : ∆ The subdivision has the following properties: , it maps non-degenerate simplices to non-degenerate simplices (see [3,Definition 12.9]).
(4) The subdivision is a left adjoint and hence it commutes with colimits. In particular if the left square below is a push-out, then so is the right square: The symbols Fun(−, C) and Fun(N (−), C) denote systems of categories indexed by Spaces (see 2.3) given by the assignments 7.1. Clutching construction. To construct and analyze functors indexed simplex categories one can use the geometry of the underlying spaces. For example the clutching construction can be described as follows. An initial data consists of a push-out square of spaces where the indicated maps are inclusions: Out of this data we are going to construct a functor H : D → C and a natural transformation ψ : g * H → G in Fun(C, C). This functor is called the clutching of F and G along ψ and is denoted by H(ψ, F, G). The functor H and the natural transformation ψ are supposed to satisfy the following properties: (1) j * H = F ; (2) the following diagram commutes: is an isomorphism for any simplex σ in C which is not in the image of i. We use the following diagrams to depict an initial data and its clutching: By the push-out assumption in the initial data, there is a bijective correspondence between the set D n and the disjoint union j(B n ) g(C n \ i(A n )). Furthermore, j : B n → j(B n ), g : C n \ i(A n ) → g(C n \ i(A n )), and i : A n → i(A n ) are bijections. This justifies the use of the following notation. If σ : ∆[n] → C belongs to i(A), then σ ′ : ∆[n] → A denotes the unique simplex for which iσ ′ = σ. If σ : ∆[n] → D belongs to j(B), then σ ′ : ∆[n] → B denotes the unique simplex for which jσ ′ = σ. If σ : ∆[n] → D does not belong to j(B), then σ : ∆[n] → C denotes the unique simplex in C for which gσ = σ. We can now define: We are going to define H(α) : H(τ ) → H(σ). Note that if σ belongs to j(B), then so does τ . Thus there are no morphisms in D between any simplex that does not belong to j(B) and a simplex that belongs to j(B). Three possibilities remain.
• If τ ∈ j(B) and σ ∈ j(B), then • If τ ∈ j(B) and σ ∈ j(B), then • If τ ∈ j(B) and σ ∈ j(B), then we have a commutative diagram of spaces: as the following composition: This procedure indeed defines a functor H : D → M such that Hj = F , which is the requirement (1). It remains to construct ψ : Hg → G.
• If σ : ∆[n] → C does not belong to i(A), then ψ σ is given by the identity id : The morphisms {ψ σ } σ∈C form a natural transformation between g * H and G that fulfills the requirements (2) and (3).
Assume now that we have a push-out square of spaces, where the indicated maps are inclusions: functors F : B → C and G : D → C, and a natural transformation ψ : F → j * G in Fun(B, C). This data induces an initial data for the clutching that consists of the above push-out, functors F : A → C and g * G : C → C, and a natural transforma- Its clutching is a functor H(f * ψ, F, g * G) : D → C and a natural transformation f * ψ : g * H(f * ψ, F, g * G) → g * G.
Proof. Let σ be a simplex in D. If it belongs to B, define ψ σ to be ψ σ . If it does not, define ψ σ to be the identity morphism id : H(f * ψ, F, g * G)(σ) = g * G(σ) = G(σ). These morphisms define the desired natural transformation ψ.

Bounded functors
In a simplex category A the face and degeneracy morphisms: Since ǫ maps all the degeneracy morphisms to identities, it is a bounded functor. Proof. This is a consequence of properties (1) and (3) given in Section 7.1.
Let M be a model category and A a simplicial set. A natural transforma- It is called a cofibration if, for any non-degenerate σ : ∆[n] → A, the morphism: induced commutativity of the following square is a cofibration:  Let j be an object in J and g : f ↓ j → I be the forgetful functor. The value of the derived left Kan extension f k P F (j) = colim f ↓j (P F )g is weakly equivalent to the homotopy colimit of the composition F g : f ↓ j → M.

Mapping spaces in model categories
The homotopy colimits of constant functors are homotopy invariant with respect to the indexing categories and thus the functor Spaces ∋ A → hocolim A X ∈ Ho(M) is a composition of the localization Spaces → Ho(Spaces) and a functor denoted by X ⊗ l − : Ho(Spaces) → Ho(M) ([4, Proposition 7.1]). A key result in [4] states that X ⊗ l − has a right adjoint map(X, −) : Ho(M) → Ho(Spaces). Its value map(X, Y ) is what we take to be the homotopy type of the mapping space between X and Y in M. If M is a simplicial model category, then the mapping space between a cofibrant and a fibrant objects given by the simplicial structure on M is weakly equivalent to the value of this right adjoint. We now recall the construction of mapping spaces in a model category from [4]. In this way we obtain a simplicial set map(F, G).
Step 2. (1) The following diagrams commute (• is associative and has a unit): Compositions of weak equivalences are weak equivalences and hence the restriction of the composition operation • fits into a commutative diagram: This together with 11. (1) The following diagrams commute (• is associative and has a unit):

Deloopings of the spaces of weak equivalences
In this section we describe the standard delooping of the spaces of weak equivalences with the monoid structure given by the composition. The proof of the main result and needed techniques are places in the appendix (Section 19).
13.1. Notation. Let X be an object in M. The symbol Cons (N (∆[0]), X we ) denotes the full subcategory of Fun b (N (∆[0]), X we ) whose objects are functors whose composition with the inclusion X we ⊂ M is cofibrant, fibrant, and weakly equivalent to the constant functor X in Fun b (N (∆[0]), M) (compare with 11.1). The set of morphisms between two such functors F and G is given by Natwe (F, G).
Let S be a collection of objects in Cons (N (∆[0]), X we ). Define S n to be the full subcategory of (1) Bwe(S) is a connected space.
(2) The loop space ΩBwe(S) is weakly equivalent to we(X, X). Part III. In this part we show that Fun(I, X we ) is essentially small and the nerve of its core is weakly equivalent to map(N (I), Bwe(X, X)). This will be applied to prove Theorem A.
14. The category of weak equivalences 14.1. Theorem. Let X be an object in M. The category X we is essentially small and the nerve of its core is weakly equivalent to Bwe(X, X). (N (∆[0]), X we ). Consider the system T − indexed by ∆ (see 13.1). The objects in its Grothendieck construction Gr ∆ T − are given by functors F [n] where n ≥ 0 and F is an object in Cons (N (∆[0]), X we ). A morphism in Gr ∆ T − between two such functors F [n] and G[m] is a pair (α :

Proof. Let T be the collection of all the objects in Cons
where α is a map and φ is a natural weak equivalence. Define a functor Ψ : Gr ∆ T − → M as follows: Since ∆[n] is contractible, by 10.2, Ψ has values in X we . We claim the induced functor Ψ : Set QF = P RF (see the end of Section 11). Let us denote by the same symbol Q : X we → Cons(N (∆[0]), X we ) the restriction of Q to the constant functors. Define Φ : X we → Gr ∆ T − to be the following composition: We will show that ΨΦ and ΦΨ are homotopic to the identity functors. The composition ΨΦ : X we → X we assigns to Y the object colim Consider next the following sequence of morphisms in Gr ∆ T − : ) ֒→ P (π), and P (π) ← P R(colim N (∆[n]) F [n]) is the vertical morphism in the above commutative diagram. These morphisms give a "zig-zag" of natural weak equivalences between id Gr ∆ T− and ΦΨ.
Since X we and Gr ∆ T − are homotopy equivalent, one of them is essentially small if and only if the other one is (see 5.9). Consider the one element set {QX}. We claim that Gr ∆ {QX} − ⊂ Gr ∆ T − is a core. Let J ⊂ Gr ∆ T − be a small subcategory containing Gr ∆ {QX} − and S be the set of all F in Cons(N (∆[0]), X we ) such that, for some n ≥ 0, F [n] is in J. We have a sequence of inclusions Gr ∆ {QX} − ⊂ J ⊂ Gr ∆ S. Its composition is a weak equivalence by 13.3. (3), which shows the claim. By Thomason's theorem 4.1. (1)) and the fact that homotopy colimit of a simplicial space is weakly equivalent to its diagonal we get that the following spaces are weakly equivalent to each other: the nerve N (Gr ∆ {QX} − ), the homotopy colimit hocolim ∆ op N ({QX} − ) and the diagonal Bwe(X, X) of N ({QX} − ) (see 13.2).

15.1.
Proposition. Let f : I → J be a functor between small categories. Assume that, for any object j in J, the over category f ↓ j is contractible. Then the functor f * : Fun(J, X we ) → Fun(I, X we ) is a homotopy equivalence.
Proof. Let P be a cofibrant replacement in Fun(I, M) (see Section 9). Recall that the derived left Kan extension f k P : Fun(I, M) → Fun(J, M) maps the subcategory Fun(I, X we ) ⊂ Fun(I, M) into Fun(J, X we ) (see Section 10). We claimed that the functor f k P : Fun(I, X we ) → Fun(J, X we ) is a homotopy inverse to f * .
Let F : J → X we be a functor, P f * F → f * F the cofibrant replacement, and φ F : f k P f * F → F its adjoint. Choose an object a = (i, α : f (i) → j) in f ↓ j and consider the following commutative diagram: where the left vertical morphism is induced by the inclusion of the object a in f ↓ j. According to 10.1 this morphism is a weak equivalence and hence so is (φ F ) j . The morphisms {φ F : f k P f * F → F } F form therefore a natural transformation between f k P f * : Fun(J, X we ) → Fun(J, X we ) and the identity functor. By the same argument the morphism ψ F : P F → f * f k P F , which is adjoint to id f k P F , is also a natural weak equivalence. Thus the natural transformations F ← P F and ψ F form a "zig-zag" connecting f * f k P with id Fun(I,Xwe) .
Using the above proposition we can extend Theorem 14.1 to: 15.2. Corollary. If I is a small contractible category, then Fun(I, X we ) is essentially small and the nerve of its core is weakly equivalent to Bwe(X, X).
The cofinality result 15.1 can also be used to translate between categories of functors indexed by arbitrary small categories and simplex categories: 15.3. Corollary. Let I be a small category and ǫ : N (I) → I be the functor defined in Section 8. Then both functors ǫ * : Fun(I, X we ) → Fun(N (I), X we ) and ǫ * : Fun(I, X we ) → Fun b (N (I), X we ) are homotopy equivalences.
Proof. According to the proof of 15.1, ǫ k P : Fun(N (I), X we ) → Fun(I, X we ) is a homotopy inverse to the first functor in the statement of the corollary. Its restriction to Fun b (N (I), X we ) is a homotopy inverse to the second functor.
The following special cases of 15.1 are of particular interest to us: is contractible for any σ in B, then f * : Fun(B, X we ) → Fun(A, X we ) is a homotopy equivalence.
(2) Let K be a contractible simplicial set and pr : B×K → B be the projection.
Then pr * : Fun(B, X we ) → Fun(B × K, X we ) is a homotopy equivalence.
Define f : Tel → A to be the following map: where on each component: . Then f * : Fun(A, X we ) → Fun(Tel, X we ) is a homotopy equivalence.

Clutching
Recall that Cof(A, X we ) denotes the full subcategory of Fun b (A, X we ) whose objects are bounded functors F : A → X we whose composition with X we ⊂ M is cofibrant in Fun b (A, M) (see Section 8).

Proof. A homotopy inverse is given by a cofibrant replacement.
Recall that if f : A → B is reduced (see Section 7), then f * : Fun b (B, X we ) → Fun b (A, X we ) maps the subcategory Cof(B, X we ) into Cof(A, X we ).

16.2.
Proposition. If f : A → B is reduced, then f * : Cof(B, X we ) → Cof(A, X we ) is a strong fibration (see Section 3).
Proof. We need to show that ψ ↑ f * : F 0 ↑ f * → F 1 ↑ f * is a homotopy equivalence for any ψ : F 1 → F 0 in Cof(A, X we ). Let (G, α 1 ) be an object in F 1 ↑ f * . It consists of a functor G : B → X we in Cof(B, X we ) and a natural weak equivalence α 1 : F 1 → f * G. Consider the following commutative diagram in Fun b (B, M): ։ G is the functorial factorization given in 2.2, and the left hand square is a push-out. Since ψ is a week equivalence between cofibrant objects, its left Kan extension f k ψ is also a weak equivalence between cofibrant objects (see 8.3). Thus the natural transformation P ( α 1 ) → Φ(G, α 1 ) is a weak equivalence and α 0 is a cofibration. The functor Φ(G, α 1 ) is therefore cofibrant and has values in X we . Define α 0 : F 0 → f * Φ(G, α 1 ) to be the adjoint to α 0 . In this way out of an object (G, α 1 ) in F 1 ↑ f * we have constructed an object (Φ(G, α 1 ), α 0 ) in F 0 ↑ f * . This whole procedure is functorial. The induced functor is denoted by Φ : give a "zig-zag" of natural transformations between the composition (ψ ↑ f * )Φ and id F1↑f * . Let (G : B → X we , λ : F 0 → f * G) be an object in F 0 ↑ f * and consider the following commutative diagram in Fun b (C, M): − → G is given by the universal property of a push-out. These week equivalences form a natural transformation between Φ(ψ ↑ f * ) to id F0↑f * .

Proposition.
Assume that the square on the left below is a push-out with i an inclusion and f reduced (see Section 7). Then g is also reduced and the square on the right is a strong homotopy pull-back (see Section 3): Proof. The proof of the fact that g is reduced is left for the reader.
By definition, the claimed square is a strong homotopy pull-back if two requirements are satisfied: i * is a strong fibration and, for any object F in Cof(B, X we ), the functor (g * , f * ) : F ↑ j * → f * F ↑ i * is a homotopy equivalence (see Section 3). The first requirement is the content of 16.2. It remains to prove the second one. A short argument for (g * , f * ) being a homotopy equivalence is that the clutching construction is its homotopy inverse. Here is a more detailed explanation of why this is so. An object (G, ψ) in f * F ↑ i * consists of a functor G in Cof(C, X we ) and a natural weak equivalence ψ : f * F → i * G. This is an example of a clutching data (see Section 7.1). Its clutching is a functor H(ψ, F, G) : D → M and a natural transformation ψ : g * H(ψ, F, G) → G. The functor H(ψ, F, G) is bounded (see 8.1) and ψ is a weak equivalence. However H(ψ, F, G) may not be cofibrant in Fun b (D, M). Let α : j k F → H(ψ, F, G) be the adjoint to the identity functor F = j * H(ψ, F, G). Define H(ψ, F, G) to be the functor that fits into the following functorial factorization in Fun b (D, M) (see 2.2): By applying g * to the acyclic fibration on the right and composing it with ψ we get a natural transformation denoted by the same symbol ψ : g * H(ψ, F, G) → G: be the adjoint to j k F ֒→ H(ψ, F, G). Since F is cofibrant, then so are j k F and H(ψ, F, G). Thus H(ψ, F, G) is an object in Cof(D, X we ). This data can be arranged into a commutative diagram: Out of an object (G, ψ) in f * F ↑ i * , we have constructed an object (H(ψ, F, G), α) in F ↑ j * . All the steps in this construction are functorial. The obtained functor is denoted by Φ : f * F ↑ i * → F ↑ j * . We claim that (g * , f * )Φ and Φ(g * , f * ) are homotopic to the identity functors. The morphisms ψ : g * H(Ψ, F, G) → G in f * F ↑ i * form a natural transformation between (g * , f * )Φ and id f * F ↑i * . Let (G, ψ) be an object in F ↑ j * consisting of a functor G in Cof(D, X we ) and a weak equivalence ψ : F → j * G. The composition of ψ : H(f * ψ, F, g * G) → G given in 7.2 with the cofibrant replacement H(f * ψ, F, g * G) → H(f * ψ, F, g * G) is a natural transformation between Φ(g * , f * ) and id F ↑j * .
17. The category of functors with values in X we 17.1. Theorem. Let I be a small category. Then Fun(I, X we ) is essentially small and the nerve of its core is weakly equivalent to map(N (I), Bwe(X, X)).
Our strategy to prove 17.1 is to show that the nerve of a core of Fun(I, X we ) is weakly equivalent to the homotopy limit of the constant functor indexed by I with value Bwe(X, X). For this strategy to work we need to choose these cores in a certain functorial way with respect to I. We set this functoriality first.
Let φ : I → Spaces be a functor. For any morphism λ : i → j in I and any commutative diagram on the left below we have the following commutative diagram of functors on the right: The symbol F (φ, X) denotes the system of categories given by all the functors in the right diagram above for all the morphisms λ in I. To describe this system we use the following notation. Let i be an object in I and σ : ∆[n] → φ(i) a simplex: For example let A be a space and A : [0] → Spaces be the constant functor with value A. The corresponding system of categories F (A, X) is given by the following functors indexed by morphism α : σ → τ in A: 17.2. Proposition. The system F (A, X) is essentially small and, for any of its cores F ⊂ F (A, X), the maps N (F tA ) → holim σ∈A N (F σ ) ← holim σ∈A N (F eA ), induced by F π : F tA → F σ and F σ ← F eA : F p , are weak equivalences.
Proof. If K is a contractible space, then Fun(K, X we ) is essentially small (see 15.2). Thus to prove that F (A; X) is an essentially small system, we need to show that Fun(A, X we ) is an essentially small category (see 5.6.(1)).
Let T to be the collection of all spaces for which the proposition is true. To show that any space belongs to T , we prove that T satisfies the following properties: (1) If in the following push square A, B, and C belong to T , then so does D.
is contractible for any σ in B. Then A belongs to T if and only if B does.
(3) If A is contractible, then A belongs to T . (4) Let S be a set. If A s ∈ T for any s ∈ S, then s∈S A s ∈ T .
A i ∈ T for any i, then to A ∈ T . In each of the following steps we prove the corresponding property.
Step (1). By 15.3 and 16.1 the functors Cof(N (I), X we ) ⊂ Fun b (N (I), X we ) and ǫ * : Fun(I, X we ) → Fun b (N (I), X we ) are homotopy equivalences. Thus if one of those categories is essentially small, then so is any other (see 5.9) in which case these functors are weak equivalences (see 6.2).
Since the following square on the left is a push-out with all the maps being reduced, according to 16.3 we have a strong homotopy pull-back on the right: essentially small and it has a core F . Consider the following commutative diagram induced by the functors in such a core: The right vertical map is a weak equivalence since the functors involved are constant. The right horizontal maps are weak equivalences as the homotopy limit preserves weak equivalences. It follows that so is the middle vertical map. The left vertical map is a weak equivalence by 15.4.(1). We can then conclude that if one of the left horizontal maps is a weak equivalence then so is the other. This means that A belongs to T if and only if B does which proves step 2.
Step (3). According to 15.2 the category Fun(A, X we ) is essentially small and consequently F (A, X) is an essentially small system (see 5.6. (1)). Let F ⊂ F (A, X) be its core and consider the following commutative diagram: where a is the diagonal map from N (F eA ) to the homotopy limit of the constant functor N (F eA ) and the rest of the maps are induced by functors in the system F . Since A is contractible, the vertical maps and a are weak equivalences (see 15.2). Thus so is the bottom map and A belongs to T .
Step (4). Consequence of the fact that products preserve weak equivalences.
Step (5). The map f : Tel → A (see 15.4. (3)) satisfies the requirements in step (2). Thus we need to prove that Tel belongs to T . Again by step (2) the products A i ×∆ [1] belong to T and so by step (4) the components of the push-out describing Tel also belong to T . We can now use step (1).
Proof of Theorem 17.1. Apply 17.2 to F (N (I), X) to see that the nerve of a core of F (N (I), X) t N (I) = Fun(N (I), X we ) is weakly equivalent to the homotopy limit of the constant functor holim σ∈N (I) N (F e N (I) ) ≃ map(N (I), N (F e N (I) )). By 15.2 the nerve of F e N (I) , which is a core of F (N (I), X) e N (I) = Fun(∆[0], X we ), is weakly equivalent to Bwe(X, X). To finish the proof recall that according to 15.3, the functor ǫ * : Fun(I, X we ) → Fun(N (I), X we ) is a homotopy equivalence.
As a direct consequence of 17.2 we also get: Corollary. If f : I → J be a weak equivalence of small categories, then f * : Fun(J, X we ) → Fun(I, X we ) is a weak equivalence.

Theorem A
The aim of this final section is to prove Theorem A from the introduction.
18.1. Definition. Let X and F be spaces. Ext(X, F ) is a subcategory of Fib(X, F ) whose objects are maps f : E → X with a fixed codomain X. The set of morphisms in Ext(X, F ) between f : E → X and f ′ : E ′ → X consists of weak equivalences g : E → E ′ for which f ′ g = f .
To prove Theorem A we are going to use the following constructions.
18.2. Fibrant replacement. In addition to the factorization given in 2.2, the category Spaces has the following factorization: any commutative square on the left below can be extended functorially to a commutative diagram on the right with the indicated morphisms being acyclic cofibrations and fibrations: The commutativity of the right diagram means that the morphisms γ f : X → R(f ) and µ f : R(f ) → Y form natural transformations. Define µ : Ext(X, F ) → Ext(X, F ) to be the functor assigning to an object f : E → X the fibration µ f : R(f ) → X and to a morphism g : E → E ′ , between f : E → X and f ′ : . For any such σ, define df σ : df (σ) → ∆ X (σ) to be the map that fits into the pull-back square on the left below. These maps form a natural transformation that we denote by df : df → ∆ X . The horizontal maps in this square satisfy the universal property of the colimit inducing canonical isomorphisms on the right: In this way we obtain a functor d : Spaces ↓ X → Fun(X, Spaces) ↓ ∆ X which we call the decomposition. Its composition with the fibrant replacement is called the derived decomposition and is denoted by ∂: Assembly. Let colim : Fun(X, Spaces) ↓ ∆ X → Spaces ↓ X be the functor whose value on an object g : G → ∆ X is the composition of the colimit colim X g with the canonical isomorphism colim X ∆ X ∼ = X. It maps a morphism ψ : g → g ′ to colim X ψ. Let us choose a cofibrant replacement P in Fun(X, Spaces). For an object g : G → ∆ X in Fun(X, F ) ↓ ∆ X , take the cofibrant replacement P G → G and define g to be the composition: Note that if g belongs to Fun(X, F we ) ↓ ∆ X , then by 4.1, the homotopy fiber of g is weakly equivalent to F and hence g is an object in Ext(X, F ). We call this composition of P and the colimit the assembly functor and denote it by Proof. We are going to show that the assembly functor is a homotopy inverse to ∂.
To understand Fib(X, F ) we will use the range functor R : Fib(X, F ) → X we that assigns to a morphism (φ, ψ) in Fib(X, F ) (see 1.1) the map ψ.
Proof. Let B be in X we . An object in B ↑ R is a pair of maps E p → B ′ φ ← B where φ is a weak equivalence and p is in Fib(X, F ). A morphism between such objects is a pair of weak equivalences (g, h) making the following diagram commutative: Define F B : Ext(B, F ) → B ↑ R to be the functor given by the assignment: Let G B : B ↑ R → Ext(B, F ) be the functor that maps a morphism (*) in B ↑ R to a morphism in Ext(B, F ) described as follows. Use the fibrant replacement R to extend (*) to a diagram on the left below (see 18.2) and set G B (g, h) : G B (p 1 , φ 1 ) → G B (p 0 , φ 0 ) to be the unique morphism in Ext(B, F ) that fits into the following commutative cube on the right where the front and back squares are pull-backs: ← B in B ↑ R the following morphisms give a "zig-zag" of natural transformations between F B G B and the identity functor: Instead of showing directly that the other composition G B F B is homotopic to the identity, we prove slightly more. Let f : C → B be a morphism in X we and f * : Ext(C, F ) → Ext(B, F ) by the composition with f functor: We claim that Ext(B, F ) FB / / B ↑ R f ↑R / / C ↑ R GC / / Ext(C, F ) is a homotopy inverse to f * . The action of f * G C (f ↑ R)F B on an object p : E → B in Ext(B, F ) can be understood through the following commutative diagram: The maps E ֒→ R(p) ← f * R(p) give a "zig-zag" of natural transformations between f * G C (f ↑ π)F B and id Ext(B,F ) . Conversely, by applying G C (f ↑ π)F B f * to an object q : E → C in Ext(C, F ), we get a commutative diagram: where E → f * R(f q) is the unique map into the pull-back given by the commutativity of the inner square. These maps give a natural transformation between G C (f ↑ R)F B f * and id Ext(C,F ) . For example if f = id B , then we see that G B F B is homotopic to id Ext(B,F ) . Thus according to 18.6 and 6.2, f ↑ R : B ↑ R → C ↑ R is a weak equivalence of essentially small categories, which shows the proposition. With 18.6 and 18.7 we have the tools necessary to prove: Proof of Theorem A. Consider the range functor R : Fib(X, F ) → X we . The categories Fib(X, F ) and Gr Xwe (− ↑ R) are homotopy equivalent. (see 2.5). According to 14.1, the category X we is essentially small and by 18.7 the system of categories − ↑ R satisfies the requirement of 6.3. The category Gr Xwe (− ↑ R) is therefore essentially small. Moreover, there is a core Ξ of X we and a core Ψ of the restriction of the system (− ↑ R) to Ξ for which Gr Ξ Ψ is a core of Gr Xwe (− ↑ R). We can now use Thomason's theorem 4.1.(3) to get that the homotopy fiber of the the nerve of the projection Gr Ξ Ψ → Ξ is weakly equivalent to the nerve of Ψ i , which is weakly equivalent to the mapping space map(X, Bwe(F, F )) (see the proof of 18.7 and 18.6). The theorem follows as the projection Gr Ξ Ψ → Ξ has a section and the nerve of Ξ is weakly equivalent to Bwe(X, X) (see 14.1).

Appendix: delooping of homotopy groupoids
In this appendix we recall the standard delooping machinery and prove 13.3.
19.1. Definition. Let S be a set. A homotopy groupoid indexed by S consists of: π t : EG t = G(t, t −1 ) → ∆[0] = BG t is the unique map. By assembling all these spaces we define: For any φ : [m] → [n] in ∆, define BG φ : BG n → BG m and EG φ : EG n → EG m to be the maps which on components are given by BG tn,...,t0,φ and EG tn,...,t0,φ . The requirements (1) and (2) of 19.1 are exactly what is needed for BG and EG to be simplicial spaces (see for example [15,18]). From the above definition it is also clear that π : EG → BG is a map of simplicial spaces. Note further that if S ′ ⊂ S, then EG S ′ and BG S ′ are simplicial subspaces of EG S and BG S .

19.2.
Proposition. Let S ′ ⊂ S be non-empty sets and G S be a connected homotopy groupoid indexed by S. Let G S ′ be the restriction of G S to S ′ . Then: (1) hocolim ∆ op BG S is connected and hocolim ∆ op EG S is contractible.
(2) G S has the homotopy tope of the the loop space Ωhocolim ∆ op BG S .
(3) the map hocolim ∆ op BG S ′ → hocolim ∆ op BG S , induced by the inclusion of simplicial spaces BG S ′ ⊂ BG S , is a weak equivalence.
In the rest of this section we prove 19.2. We start with two classical lemmas whose proves can be found for example in [1] and [2]. 19.3. Lemma. Let X be a simplicial space. Set d 0 : X 0 → ∆[0] =: X −1 to be the unique map. Assume that there are maps s : X n → X n+1 for n ≥ −1 such that d 0 s = id and d i s = sd i−1 for i > 0. Then hocolim ∆ op X is contractible.
19.4. Lemma. Let ψ : F → G be a natural transformation in Fun(I, Spaces). Assume the commutative diagram on the left below is a homotopy pull-back for any α : i → j in I. Then, for any k in I, the diagram on the right is a homotopy pull-back, where the horizontal maps are induced by the inclusion of k in I.