Homotopy colimits in model categories. Marc Stephan

Homotopy colimits in moel categories Marc Stephan July 13, 2009

1 Introuction In [1], Dwyer an Spalinski construct the so-calle homotopy pushout functor, motivate by the following observation. In the category Top of topological spaces, one can construct the n-imensional sphere S n by glueing together two n-isks D n along their bounaries S n 1, i.e. by the pushout of i i D n, D n S n 1 where i enotes the inclusion. Let be the one point space. Observe, that one has a commutative iagram D n i S n 1 i D n, S n 1 i S n 1 where all vertical maps are homotopy equivalences, but the pushout of the bottom row is the one-point space an therefore not homotopy equivalent to S n. One probably prefers the homotopy type of S n. Having this iea of calculating the prefere homotopy type in min, they equip the functor category C D, where C is a moel category an D = a b c the category consisting out of three objects a, b, c an two non-ientity morphisms as inicate, with a suitable moel category structure. This enables them to construct out of the pushout functor colim : C D C its so-calle total left erive functor Lcolim : Ho(C D ) Ho(C) between the corresponing homotopy categories, which efines the homotopy pushout functor. Dwyer an Spalinski further inicate how to generalize this construction to efine the so-calle homotopy colimit functor as the total left erive functor for the colimit functor colim : C D C in the case, where D is a so-calle very small category. The goal of this paper is to give a proof of this generalization, since in [1] it is omitte to the reaer. The main work lies in proving our Theorem 2, which equips C D with the suitable moel category structure. For the existence of the total left erive functor for colim, we will use a result from [1]. The paper contains four sections. In section 2 an 3 we recall some efinitions an results relate to colimits an moel categories, respectively. The introuce terminology will be use in section 4, where we construct the homotopy colimit functor. I am grateful to my supervisor, Jesper Groal, an the participants of the stuent seminar TopTopics for all the helpful iscussions. 1

2 Colimits In this section let C be a category, let D be a small category an F : D C a functor. Mainly to fix notations, we recall some efinitions an results relate to colimits. Definition 2.1. The functor category C D, also calle the category of iagrams in C with the shape of D, is the category, where the objects are functors D C an the morphisms are natural transformations. Example 2.2. If D is the category a b c with three objects a, b, c an two non-ientity morphisms a b, b c, then an object X of C a b c is just a iagram X(a) X(b) X(c) in C an a morphism from X to Y in C a b c is given by a triple (s a, s b, s c ) of morphisms in C making X(a) X(b) X(c) s a Y (a) s b Y (b) s c Y (c) commute. Example 2.3. If D is the category a b with two objects a, b an one nonientity morphism a b, then an object of C a b is a morphism f : X(a) X(b) in C an a morphism from f : X(a) X(b) to g : Y (a) Y (b) in C a b is just a pair of morphisms (s a, s b ) in C making X(a) f X(b) s a Y (a) g s b Y (b) commute. We call C a b the category of morphisms in C an enote it by Mor(C). Example 2.4. If D is the category 1 consisting only out of one object 1 an one morphism, then C D is isomorphic to C via the functor given by X X(1) on objects an f f 1 on morphisms. Definition 2.5. The constant iagram functor = D : C C D is the functor, which sens an object C of C to the functor (C) given by C on objects an g i C on morphisms, an which sens a morphism f of C to the natural transformation (f) given by (f) = f in an object of D. 2

Note that a functor j : D D from a small category D to D inuces a functor ( ) D,j = ( ) D = ( ) j : CD C D, which sens an object X of C D to X j an a morphism f : X Y in C D to the natural transformation f D : X D Y D given by f j( ) in an object of D. Furthermore, if j : D D is a functor from a small category D to D, then ( ) j j = ( ) j ( ) j. (2.1) Example 2.6. The functor D 1 inuces the functor ( ) 1 : C D C D, which compose with the isomorphism C = C 1 yiels D. By (2.1), it follows that D = ( ) j D for any functor j from a small category D to D. Definition 2.7. A colimit C = (C, t) for F : D C is an object C of C together with a natural transformation t : F (C) such that for any object X of C an any natural transformation s : F (X) there exists a unique morphism s : C X in C satisfying (s )t = s. Example 2.8. If D = a b c, then a colimit for an object X of C a b c is just a pushout of the iagram X(a) X(b) X(c). Remark 2.9. If a colimit for F exists, we will sometimes speak of the colimit for F for the following reason. If (C, t) an (C, t ) are colimits for F, then the unique morphism h : C C in C such that (h)t = t hols, is an isomorphism, which will be calle the canonical isomorphism. Given furthermore an object X of C an a natural transformation s : F (X), let s : C X be the unique morphism in C satisfying (s )t = s. Then the unique morphism s : C X satisfying (s )t = s is given by s h. Remark 2.10. Assume that for any functor G : D C the colimit (colim(g), t G ) exists. Then the chosen colimits yiel a functor colim : C D C, calle colimit functor, which maps a morphism s : G G in C D to the unique morphism s : colim(g) colim(g ) in C such that (s )t G = t G s. Furthermore, we have an ajunction (colim,, α), where the natural equivalence α is given in an object (G, X) of (C D ) op C by the bijection α = α G,X : Hom C (colim(g), X) Hom C D(G, (X)), s (s )t G. (2.2) In particular, any two colimit functors C D C are naturally isomorphic an therefore, we will sometimes speak of the colimit functor. 3

Remark 2.11. Let D be a small category such that there exists an isomorphism J : C D C D with J D = D. Assume there exists a colimit functor colim : C D C. Then the composition colim J : C D C is a colimit functor. Furthermore, for any object X of C an any natural transformation s : F D (X) the inuce morphism from the colimit of F to X is given by the morphism colim(j(f )) X inuce by J(s) : J(F ) D (X). In the efinition of a moel category we will use both, the notion of colimit an limit. Definition 2.12. A limit L = (L, t) for F : D C is an object L of C together with a natural transformation t : (L) F such that for any object X of C an any natural transformation s : (X) F there exists a unique morphism s : X L in C satisfying t (s ) = s. The following result is prove on page 115 in [3]. Proposition 2.13. Let D be a small category an G : D C D a functor. Denote by ev : C D C the evaluation functor in the object of D. Assume that for all objects of D a limit for the composition ev G : D C exists. Then there exists a limit for G. The notion of colimit is ual to the notion of limit: Remark 2.14. A colimit for F is the same as a limit for the ual functor F op : D op C op. More precisely, sening a colimit (C, t) for F to (C, t ), where the natural transformation t : D op(c) F op is given by (t ) op in an object of D op, efines a one-to-one corresponence between colimits for F an limits for F op. We conclue this section with a result about pushouts. Proposition 2.15. Assume that X i W Y j i P j an Y k i P Z i Q k are pushout squares in C. Then so is X i W. kj Z i Q k j 4

Proof. From the commutativity of the first two squares one conclues that k j i = i kj, thus the thir one commutes too. Given now any commutative square X i W, Z kj s one has to show, that there is a unique morphism t : Q V such that tk j = r an ti = s. Using that P is a pushout, let t : P V be the unique morphism such that t j = r an t i = sk. Using that also Q is a pushout, efine t : Q V to be the unique morphism such that tk = t an ti = s. This gives the esire morphism t. To show the uniqueness, let t : Q V be a morphism with t k j = r an t i = s. By the universal property of P, one conclues t k = t. Hence, by the universal property of Q follows t = t. V r 3 Moel categories an homotopy categories 3.1 Moel categories The following terms will be use in the efinition of a moel category. Definition 3.1. A morphism f : X X of a category C is calle a retract of a morphism g : Y Y of C, if there exists a commutative iagram X i Y r X, f X i Y r X such that ri = i X an r i = i X. Definition 3.2. Let C be a category. i) Given a commutative iagram in C of the form g f A f X, (3.1) i B g Y a lift in the iagram is a morphism h : B X such that hi = f an ph = g. ii) Let i : A B, p : X Y be morphisms in C. We say that i has the left lifting property (LLP) with respect to p an that p has the right lifting property (RLP) with respect to i, if there exists a lift in any commutative iagram of the form (3.1). p 5

Now we are reay for the efinition of a moel category. Definition 3.3. A moel category is a category C together with three classes of morphisms of C, the class of weak equivalences W = W (C), of fibrations F ib = F ib(c) an of cofibrations Cof = Cof(C), each of which is close uner composition an contains all ientity morphisms of C, such that the following five conitions hol: MC1: Every functor from a finite category to C has a limit an a colimit. MC2: If f an g are morphisms of C such that gf is efine an if two out of the three morphisms f, g an gf are weak equivalences, then so is the thir. MC3: If f is a retract of a morphism g of C an g is a weak equivalence, a fibration or a cofibration, then so is f. MC4: i) Every cofibration has the LLP with respect to all p W F ib. ii) Every fibration has the RLP with respect to all i W Cof. MC5: Any morphism f of C can be factore as i) f = pi, where i Cof, p W F ib, an as ii) f = pi, where i Cof W, p F ib. By a moel category structure for a category C we mean a choice of three classes of morphisms of C such that C together with these classes is a moel category. Let C be a moel category until the en of this subsection. The following two remarks follow immeiately from the efinition of a moel category. Remark 3.4. Since any isomorphism f : X Y of C is a retract of i Y an since i Y W F ib Cof, it follows by MC3 that also f W F ib Cof. Remark 3.5. The opposite category C op is a moel category by efining a morphism f op in C op to be in W (C op ) if f is in W (C), to be in F ib(c op ) if f is in Cof(C) an to be in Cof(C op ) if f is in F ib(c). The proofs of the following two propositions can be foun on page 87 an 88 of [1]. Proposition 3.6. A morphism i of C is i) in Cof, if an only if it has the LLP with respect to all p W F ib, ii) in W Cof, if an only if it has the LLP with respect to all p F ib. 6

Proposition 3.7. Let X j Z be a pushout square in C. Y i j P i) If i is in Cof(C), then so is i. ii) If i is in Cof(C) W (C), then so is i. i The next two results are concerne with cofibrations an pushouts an will be use in the proof of Theorem 2. Lemma 3.8. Given a commutative square in C of the form A C B. (3.2) Let P, Q be pushouts of C A B an let i P : P D, i Q : Q D be the morphisms inuce by (3.2). Then i P is a cofibration [resp. weak equivalence] if an only if i Q is a cofibration [resp. weak equivalence]. Proof. Let j : P Q be the canonical isomorphism, then i P = i Q j by Remark 2.9. Now, since Cof(C) [resp. W (C)] is close uner composition, Lemma 3.8 follows from Remark 3.4. Proposition 3.9. Given a commutative cube in C of the form A D B i B, (3.3) A B C D i D C D where the back face an the front face are pushout squares. Let P enote the pushout of the iagram C A A an let i P : P C be the morphism inuce by the left-han face of (3.3). If i B an i P are cofibrations, then so is i D. Proof. By Proposition 3.6i), it s enough to fin a lift in any given commutative iagram D X, D i D Y p 7

where p is in W F ib. Since i B is a cofibration, there exists a lift h B : B X in B D X. i B B D Y Defining h A as the composition A B h B X yiels a commutative iagram A A C D X an hence an inuce morphism P X. This morphism makes the iagram P i P p h A X C D Y p (3.4) commute, as one checks using the universal property of the pushout P. Since i P is a cofibration by assumption, there exists a lift h C : C X in (3.4). It makes the square A C h C commute. This square inuces a morphism h from the pushout D to X. One checks that h is the esire lift, using the universal property of the pushouts D an D. B X h B 3.2 The homotopy category of a moel category Definition 3.10. Let C be a category an W a class of morphisms of C. A localization of C with respect to W is a category D together with a functor F : C D such that the following two conitions hol: i) F (f) is an isomorphism for every f in W. ii) If G is a functor from C to a category D, such that G(f) is an isomorphism for every f in W, then there exists a unique functor G : D D with G F = G. Remark 3.11. Let C be a category an W a class of morphisms of C. If (D, F ) an (D, F ) are localizations of C with respect to W, then the unique functor G such that G F = F is an isomorphism. Therefore, if a localization exists, we will sometimes speak of the localization. 8

Let C be a moel category until the en of this subsection. The localization (Ho(C), γ C ) of C with respect to the class of weak equivalences W exists by Theorem 6.2 of [1]. This fact makes the following efinition possible. Definition 3.12. The homotopy category of the moel category C is the localization of C with respect to W. Until the en of this subsection let F be a functor from C to another moel category D. The homotopy colimit functor will be efine as a total left erive functor, which is efine as follows. Definition 3.13. A total left erive functor LF for the functor F is a functor LF : Ho(C) Ho(D) together with a natural transformation t : (LF )γ C γ D F such that for any pair (G, s) of a functor G : Ho(C) Ho(D) an a natural transformation s : Gγ C γ D F, there exists a unique natural transformation s : G LF satisfying t s γc = s, where the natural transformation s γc : Gγ C (LF )γ C is given by s γ C (X) in an object X of C. Remark 3.14. Assume that (LF, t) an (L F, t ) are total left erive functors for F. Then the unique natural transformation s : L F LF such that t s γc = t is a natural equivalence. Therefore, if a total left erive functor for F exists, we will sometimes speak of the left erive functor. The proof of the following theorem is given on page 114 in [1]. Theorem 1. Assume G : D C is a functor, which is right ajoint to F an which carries morphisms of F ib(d) to F ib(c) an morphisms of F ib(d) W (D) to F ib(c) W (C). Then the total left erive functor LF : Ho(C) Ho(D) for F exists. 4 Homotopy colimits In this section let C be a moel category. We want to efine the homotopy colimit functor as the total left erive functor for the functor colim : C D C. Therefore, we have to equip the functor category C D with a suitable moel category structure, which can be one uner the assumption that D is as in the next efinition. Definition 4.1. A non-empty, finite category D is sai to be very small if there exists an integer N 1 such that for any composition f N...f 2 f 1 of morphisms (f i ) 1 i N in D, at least one morphism f i is an ientity morphism. 9

More geometrically, note that a non-empty, finite category D is very small if an only if it has no cycles, i.e. given any integer n 1 an any composition f n...f 2 f 1 : of morphisms (f i ) 1 i n in D, then each morphism f i is the ientity morphism i. The avantage of a very small category is that it enables us to o inuction involving the egree, which is efine as follows. Definition 4.2. Let D be a very small category an any object of D. The egree eg() of is efine by eg() := max({0} {n 1; there exists a composition f n...f 2 f 1 : e of morphisms (f i i ) 1 i n in D}), the total egree eg(d) of D by eg(d) := max({eg(); an object of D}). Remark 4.3. Let e, be objects of a very small category D an assume, there exists a non-ientity morphism e. Then eg(e) < eg(). 4.1 A moel category structure for C D Let D be a very small category. The functor category C D can be given a moel category structure in the following way. Let be an object of D. Recall that an object m of the over category D is given by a morphism m : e in D an that a morphism k in D from m : e to m : e is a morphism k : e e in D such that m k = m. Denote by the full subcategory of D which contains all objects of D except i. Let the functor j : D be given by (m : e ) e on objects an k k on morphisms. Composing the inuce functor ( ) : C D C with colim : C C gives a functor := colim ( ) : C D C. Let X be an object of C D. The natural transformation s X : X (X()) given in an object m : e of by X(m), inuces the morphism α 1 (s X ) : (X) X(), where the bijection α comes from the ajunction (colim,, α). This inuce morphism is natural. Inee, given any morphism f : X Y in C D, we show that the iagram (X) (f) (Y ) X() f Y () (4.1) 10

commutes or equivalently, that α(α 1 (s Y ) (f)) = α(f α 1 (s X )) (4.2) hols. Using the naturality of α, one conclues that the left-han sie of (4.2) equals s Y f, which is Y (m) f e in an object m : e, an that the right-han sie equals (f ) s X, which is f X(m) in m : e. Finally, the equation Y (m) f e = f X(m) hols, since f is a natural transformation by assumption. Define the functor δ as the composition where the first functor is given by δ : Mor(C D ) C a b c colim C, (f : X Y ) ( (Y ) (f) (X) X()) on objects, (s, s ) ( (s ), (s), s ) on morphisms. Given any morphism f : X Y in C D, consier that δ (f) is a pushout by efinition an that therefore the commutative square (4.1) inuces a morphism i (f) : δ (f) Y (), which is natural. Inee, given any morphism (s, s ) in Mor(C D ) from f : X X to g : Y Y, one checks that δ (s,s ) δ (f) i (f) X () s δ (g) i (g) Y () commutes using the universal property of pushouts, the equation s f = g s which hols by assumption an the naturality of the morphism (X ) X (). Using the above constructe morphisms (i (f)) for a morphism f in C D, we give C D a moel category structure. Theorem 2. Define a morphism f of C D to be in i) W (C D ), if f is in W (C) for all objects of D, ii) F ib(c D ), if f is in F ib(c) for all objects of D, an to be in iii) Cof(C D ), if i (f) is in Cof(C) for all objects of D. Then C D together with these three classes is a moel category. Remark 4.4. One can check irectly, that the property of a morphism f in C D to be in Cof(C D ) oesn t epen on the choices of colimits involve in the construction of the morphisms (i (f)). However, this fact follows from Proposition 3.6i) after having prove Theorem 2. 11

Proof of Theorem 2. Since W (C) an F ib(c) are close uner composition an contain all ientity morphisms, it follows immeiately that W (C D ) an F ib(c D ) share the same properties. To show that Cof(C D ) is close uner composition, let two morphisms f : X Y, g : Y Z in Cof(C D ) be given. We have to prove, that i (gf) is in Cof(C) for all objects of D. By Lemma 3.8, it s enough to show that for any pushout Q of (Z) (X) X(), the morphism i : Q Z() inuce by the iagram (g) (f)= (gf) (X) (Z) X() Z() (gf) =g f is a cofibration in C. Using Proposition 2.15, efine such a Q as the pushout of (Z) (Y ) δ (f). To show that the inuce morphism i : Q Z() is in Cof(C), let Q δ (g) be the unique morphism such that (Y ) = (Z) (Y ) = (Z) δ (f) i (f) Q Y () δ (g) (4.3) commutes. The pushout P of (Z) (Y ) = (Y ) is just (Z) an the morphism P (Z) inuce by the left-han face of (4.3) is i (Z) an therefore in Cof(C). Applying Proposition 3.9, we get that Q δ (g) is in Cof(C). Finally, using the universal property of the pushout Q of (Z) (X) X(), one checks that i equals the composition Q δ (g) i (g) Z() an therefore is in Cof(C) as a composition of two cofibrations in C. To prove that Cof(C D ) contains all ientity morphisms of C D, one has to show the following claim. For any object of D an any object X of C D, the morphism i (i X ) : δ (i X ) X() is a cofibration in C. Since (i X ) = i (X), it follows that X() is a pushout of (X) (X) X(). The claim is now a consequence of Lemma 3.8. 12

4.1.1 Proof of MC1-MC3 By MC1 in C an Proposition 2.13, it follows that any functor F from a finite category to C D has a limit. We want to show that F has a colimit or equivalently by Remark 2.14 that F op has a limit. Note that (C D ) op is isomorphic to (C op ) Dop an conclue from Proposition 2.13 an MC1 in C op, that F op has a limit. Hence, MC1 hols in C D. From MC2 for C, we will euce MC2 for C D. Let f, g be morphisms in C D such that gf is efine an such that two of the three morphisms f, g, gf are in W (C D ). Then for all objects of D, two of the three morphisms f, g, g f are in W (C). Thus by MC2 for C an the equality g f = (gf), all three morphisms f, g, (gf) are in W (C) an hence f, g, gf are in W (C D ). To show MC3, let f : X X, g : Y Y be morphisms in C D such that f is a retract of g, i.e. there exists a commutative iagram X i Y f g r X X i Y r X such that ri = i X an r i = i X. Note that therefore f is a retract of g for all objects of D. Thus the part of MC3 ealing with fibrations [resp. weak equivalences] is a irect consequence of MC3 for C. Assume that g is a cofibration, i.e. for all objects of D the morphism i (g) is in Cof(C). We will euce that f is a cofibration by showing that i (f) is a retract of i (g) an hence is in Cof(C) by MC3 for C. Consier the iagram f, δ (f) δ (i,i ) δ (g) δ (r,r ) δ (f). i (f) X () i i (g) Y () r i (f) X () The morphism i (f) is natural as alreay shown, so the iagram commutes. By functoriality an since ri = i X, r i = i X, it follows that the composition in the top row is the ientity morphism. The equation r i = i X implies that the composition in the bottom row equals i X (). Hence, i (f) is a retract of i (g). 4.1.2 Proof of MC4 an MC5 We will use inuction over the total egree eg(d) of D to prove MC4, MC5 an the following proposition. Proposition 4.5. Let i : A B be a morphism in C D. Then i is in Cof(C D ) W (C D ), if an only if i (i) is in Cof(C) W (C) for all objects of D. 13

Remark 4.6. Note that to prove any irection of Proposition 4.5, it s enough to show that (i) is in Cof(C) W (C) for all objects of D by Proposition 3.7ii) an MC2 in C. For the proof of the initial case eg(d) = 0, we will use the following lemma. Lemma 4.7. Assume eg(d) = 0. Let be any object of D an i : A B a morphism in C D. Then (i) is an isomorphism in C. Furthermore, the morphism i (i) is in Cof(C), if an only if i is in Cof(C). Proof of Lemma 4.7. Since eg(d) = 0 implies eg() = 0, it follows by Remark 4.3 that is the empty category. Hence, the colimits (A) an (B) are initial objects an (i) is an isomorphism. A pushout of (B) (A) A() is given by A() an the morphism A() B() inuce by the commutative square (A) A() (B) i B() is i. Hence, the secon statement of Lemma 4.7 follows from Lemma 3.8. We show Proposition 4.5, MC4 an MC5 in the initial case. By Remark 4.6, Lemma 4.7 implies Proposition 4.5, since any isomorphism in C is in Cof(C) W (C) by Remark 3.4. To prove MC4, let a commutative iagram A f X i B g Y in C D be given, such that i is in Cof(C D ) an p is in F ib(c D ). We have to fin a lift h : B X, whenever p is in W (C D ) [resp. i is in W (C D )]. By Lemma 4.7, it follows that i is in Cof(C) for all objects of D. If p is in W (C D ) [resp. if i is in W (C D )], then applying MC4i) [resp. MC4ii)] in C yiels an objectwise lift h : B() X(). These objectwise lifts (h ) fit together to give the esire lift h : B X. To prove MC5i) [resp. MC5ii)], let f : A B be a morphism in C D. Using MC5i) [resp. MC5ii)] in C, factor for every object of D the morphism f as f = p i, where i is in Cof(C) [resp. Cof(C) W (C)] an p is in W (C) F ib(c) [resp. F ib(c)]. By construction an Lemma 4.7, the morphisms (i ) fit together to give a morphism i in Cof(C D ) p 14

[resp. Cof(C D ) W (C D )] an the morphisms (p ) efine a morphism p in W (C D ) F ib(c D ) [resp. F ib(c D )]. The factorization f = pi shows that MC5i) [resp. MC5ii)] hols. To show the inuction step, assume that n := eg(d) 1. We prove Proposition 4.5. Let a morphism i : A B in C D an any object of D be given. For the proof of any irection, it s enough to show that (i) is in Cof(C) W (C) or equivalently by Proposition 3.6ii), to fin a lift in any commutative iagram (A) f C (4.4) (i) (B) g in C, where p is in F ib(c). From the commutativity of the iagram (4.4) an the efinition of (i), it follows that also the iagram D p A ( (A)) (f) (C) (4.5) i ( (i)) (p) B ( (B)) (g) (D) in C commutes. In any of the two irections of Proposition 4.5 we will apply the inuction hypothesis to fin a lift h : B (C) in (4.5), which will inuce the esire lift (B) C in (4.4). Inee, assume h is a lift in (4.5) an let h : (B) C be the inuce morphism, i.e. h = α 1 (h), where the bijection α comes from the ajunction (colim,, α). Note that by the naturality of α the equations α(h (i)) = α(h )i an α(ph ) = (p)α(h ) hol. Since h = α(h ) is a lift, euce that α(f) = α(h (i)) an α(g) = α(ph ). This shows that h is a lift in (4.4). To fin a lift h in (4.5), note that the category is very small with eg( ) < n an that (p) is in F ib(c ). Hence, using the inuction hypothesis to apply MC4ii) in C, it s enough to show that i is in Cof(C ) W (C ), that is by efinition that i m (i ) is in Cof(C) an that (i ) m is in W (C) for every object m : e of. We calculate i m (i ). Recall that an object of the subcategory m of m is given by a non-ientity morphism k : (m : e ) m in, which by efinition of is a non-ientity morphism k : e e in D such that mk = m. Furthermore, a morphism g : k l in m from k : (m : e ) m to l : (m : e ) m is a morphism g : m m in such that lg = k, which is a morphism g : e e in D with m g = m an lg = k : e e. Hence, we can efine a functor j : e m by (k : e e) (k : mk m) on objects an (g : k l) (g : j (k) j (l)) on morphisms, 15

which is an isomorphism. Using (2.1), one conclues that the inuce functor ( ) j : C m C e is an isomorphism. Since e = ( ) j m by Example 2.6, it follows by Remark 2.11 that the composition colim ( ) j of colim : C e C with ( ) j is the colimit functor C m C. Hence, the functor m : C C is the composition colim ( ) j ( ) m. The composition j j m j of the functors j : e m, j m : m, j : D equals j e. It follows by (2.1) that ( ) j ( ) m ( ) = ( ) e an hence m (A ) = colim(a e ) = e (A), m (B ) = e (B) an m (i ) = e (i). Note that i : A B is just i e in the object m : e of. It follows that the iagram is just m (A ) m(i ) m (B ) e (A) e(i) e (B) (A )(m) (i ) m (B )(m) A(e) i e B(e) by Remark 2.11, since for any Y in C D, the natural transformation s Y m : (Y ) m m (Y (m)) is in an object k : m m of m given by Y (k) = Y (j (k)) = Y (k) an hence the equation (s Y m ) j = s Y e hols. It follows that i m (i ) equals i e (i). Assuming now first, that i is in Cof(C D ) W (C D ), one conclues that for every object m : e of the morphism i m (i ) is in Cof(C) an that (i ) m = i e is in W (C). This shows one irection of Proposition 4.5. For the other irection, assume that i (i) is in Cof(C) W (C) for all objects of D. It follows that for any object m : e in the morphism i m (i ) is in Cof(C) W (C). Using the inuction hypothesis to apply Proposition 4.5 one euces that i is in Cof(C ) W (C ), which completes the proof of Proposition 4.5. In the proof of MC4 an MC5 we will use the following notation. Let D n 1 be the full subcategory of D which contains precisely the objects e of D with eg(e) n 1. It is very small an has total egree n 1. The inclusion functor j : D n 1 D inuces the functor ( ) D n 1 : C D C Dn 1, which carries weak equivalences [resp. fibrations] in C D to weak equivalences [resp. fibrations] in C Dn 1. Note that D n 1 e = D e for any object e of D with eg(e) < n. One conclues that for any morphism f of C D the equation i e (f D n 1) = i e (f) (4.6) 16

hols an in particular that ( ) D n 1 carries cofibrations in C D to cofibrations in C Dn 1. To show MC4i) [resp. MC4ii)], let a commutative iagram A f X i B g Y in C D be given, such that i is in Cof(C D ) an p is in F ib(c D ). Assuming that p is in W (C D ) [resp. i is in W (C D )], one has to fin a lift h : B X. Use the inuction hypothesis to apply MC4i) [resp. MC4ii)] to fin a lift h n 1 : B D n 1 X D n 1 in the commutative square p f D A n 1 D n 1 X D n 1. i D n 1 p D n 1 g D B n 1 D n 1 Y D n 1 Now, the strategy is to fin for each object of egree n of D an objectwise lift B() X(), such that these lifts an h n 1 fit together to give the esire lift h : B X. Let the natural transformation B (X()) be given by the composition B(e) (hn 1 ) e X(e) X(m) X() in an object m : e of. The inuce morphism (B) X() makes the iagram (A) α 1 (s A ) A() (4.7) (i) (B) f X() commute. Inee, by the naturality of α, one conlues that α(f α 1 (s A )) = (f )s A, which is f A(m) in an object m : e of, an that α of the composition (A) (B) X() equals the composition A B (X()), which is X(m)(h n 1 ) e i e in m : e. Finally, the equation f A(m) = X(m)(h n 1 ) e i e hols, since h n 1 i D n 1 = f D n 1 an since f is a natural transformation by assumption. Let δ (i) X() be the morphism inuce by the commutative iagram (4.7). Using the universal property of pushouts, one checks that it makes δ (i) X() (4.8) i (i) B() g p Y () 17

commute. Apply MC4i) in C [resp. apply MC4ii) in C an Proposition 4.5] to fin a lift h : B() X() in (4.8). One checks that any lift B() X() in (4.8) is also a lift in A() i B() f g X(). p Y () The esire lift h : B X can now be efine as (h n 1 ) e in an object e of D with eg(e) < n an as the constructe lift h of (4.8) in an object of egree n of D. To show that h is a natural transformation, note that by Remark 4.3 an since h n 1 is a natural transformation, one only has to consier morphisms m : e in D, where eg(e) < n, eg() = n, an to check that X(m)(h n 1 ) e = h B(m) hols. Denoting colim(b ) = ( (B), t), this is one by the following sequence of equations of compositions of morphisms: B(e) B(m) B() h X() = B(e) tm (B) B() h X() = B(e) tm (B) δ (i) i (i) B() h X() = B(e) tm (B) δ (i) X() = B(e) tm (B) X() = B(e) (hn 1 ) e X(e) X(m) X(). To prove MC5, we have to factor a given morphism f : A B in C D in the two ways i) an ii). Use the inuction hypothesis to factor f D n 1 as A D n 1 h n 1 n 1 gn 1 X B D n 1 (4.9) as in MC5i) [resp. MC5ii)]. Let be an object of egree n of D. Note that the functor j : D carries any object m of to an object j (m) with eg(j (m)) < n. It therefore inuces a functor j : Dn 1, which compose with the inclusion functor j : D n 1 D equals j. For the inuce functors follows ( ) = ( ) j ( ) D n 1 (4.10) by (2.1). Hence, applying the composition colim ( ) j with ( ) j to (4.9) yiels a factorization of colim : C D C (A) colim(x n 1 j ) (B) (4.11) 18

of (f). Now efine X() an h through the pushout square (A) A(). (4.12) colim(x n 1 j ) h X() Denote colim(x n 1 j ) = (colim(x n 1 j ), t ) an for a non-ientity morphism m : e efine the morphism X(m) : X n 1 (e) X() as the composition X n 1 (e) (t ) m colim(x n 1 j ) X(). (4.13) Furthermore, set X(i ) = i X(). Then these choices, X n 1 an h n 1 fit together to give a functor X : D C an a natural transformation h : A X. Inee, let m : e be a non-ientity morphism in D with eg() = n. If m : e e is another non-ientity morphism in D, then one euces by the naturality of t that (t ) m X n 1 (m ) = (t ) mm hols an therefore X(mm ) = X(m)X(m ). To prove the naturality of h, check that A(e) X(e) (h n 1 ) e A(m) A() h X() X(m) commutes by noting that (h n 1 ) e = (h n 1 j ) m an by using the commutativity of the iagram (4.12) an of the iagram A(e) (h n 1 j ) m (X n 1 j )(m). (t ) m (A) colim(h n 1 j ) colim(x n 1 j ) We want to show that h is in Cof(C D ) [resp. h is in Cof(C D ) W (C D )]. Note that X D n 1 = X n 1 an h D n 1 = h n 1 by efinition. In case i), since by construction h n 1 is in Cof(C Dn 1 ), it follows by (4.6) that i e (h) is in Cof(C) for every object e of D with eg(e) < n. Similarly, in case ii), since by construction h n 1 is in Cof(C Dn 1 ) W (C Dn 1 ), it follows by Proposition 4.5 an by (4.6) that i e (h) is in Cof(C) W (C) for every object e of D with eg(e) < n. Let be an object of egree n of D. By (4.10), euce the equations X = X n 1 j, h = h n 1 j an therefore (X) = colim(x n 1 j ), (h) = colim(h n 1 j ). Recall that the morphism 19

(X) X() is inuce by the natural transformation s X which is given in an object m : e of by X(m). Since X(m) is the composition (4.13), it follows that (X) X() is just colim(x n 1 j ) X(). Hence, δ (h) = X() an i (h) equals i X() an in particular is in Cof(C) W (C). Thus in case i) we have shown that h is in Cof(C D ) an by Proposition 4.5, it follows in case ii) that h is in Cof(C D ) W (C D ). Using g n 1, we want to efine a natural transformation g : X B such that f = gh. Let be an object of egree n of D. The natural transformation X (B()) given by B(m)(g n 1 ) e in an object m : e of, inuces a morphism (X) B() in C. One checks that this morphism makes the square (A) A() (h) (X) f B() commute. This gives an inuce morphism g from the pushout X() to B() such that f = g h. The constructe morphisms (g ) an g n 1 fit together to give a natural transformation g : X B. Inee, if m : e is a non-ientity morphism in D with eg() = n, then B(m)g e = g X(m) hols, as is shown by the following sequence of equations of compositions of morphisms: X(e) (gn 1 ) e B(e) B(m) = X(e) (t ) m (X) B() = X(e) (t ) m (X) X() g B() = X(e) X(m) X() g B(). We have factore f as gh, such that h is in Cof(C D ) [resp. Cof(C D ) W (C D )]. Thus to prove MC5i) an MC5ii), it s enough to factor g as pi in C D such that respectively, i Cof(C D ), p W (C D ) F ib(c D ) an i Cof(C D ) W (C D ), p F ib(c D ). Set Z n 1 := X D n 1 : D n 1 C, let i n 1 : X D n 1 Z n 1 be the ientity morphism an efine the natural transformation p n 1 := g n 1 : Z n 1 B D n 1. Let be any object of egree n of D. Using MC5 in C, factor g as X() i Z() p B(), as in MC5i) [resp. MC5ii)]. For any non-ientity morphism m : e in D set Z(m) := i X(m) : Z(e) Z() an let the morphism Z(i ) in C be given by i Z(). Then these choices an Z n 1 fit together to give a functor Z : D C. Inee, if m : e e, m : e are non-ientity morphisms in D such that eg() = n, then Z(mm ) = i X(mm ) = i X(m)X(m ) = Z(m)Z(m ). 20

The constructe morphisms (i ) an i n 1 fit together to give a natural transformation i : X Z. Inee, given a non-ientity morphism m : e in D with eg() = n, one has i e = i Z(e) an thus Z(m)i e = i X(m). The morphisms (p ) an p n 1 yiel a natural transformation p : Z B. Inee, for a non-ientity morphism m : e in D with eg() = n hols B(m)p e = B(m)g e = g X(m) = p i X(m) = p Z(m). Note that for an object of arbitrary egree of D, the functor X equals Z an that i = i X. Hence a pushout of (Z) = (X) X() is given by X() an the morphism X() Z() inuce by the commutative square (X) X() (Z) i Z() is just i. By Lemma 3.8, it follows that i (i) is a cofibration [resp. weak equivalence] if an only if i is a cofibration [resp. weak equivalence]. Furthermore, note that if eg() < n, then i = i Z() an hence is in Cof(C) W (C). In case i), since by construction for any object of egree n of D the morphism i is in Cof(C), it follows that i is in Cof(C D ). One checks that by construction p is in F ib(c D ) W (C D ). Similarly, in case ii), the morphism i is in Cof(C) W (C) for every object of egree n of D. By Proposition 4.5, it follows that i Cof(C D ) W (C D ). One checks that p F ib(c D ). This completes the proof of MC5 an hence the inuction step. We have shown Theorem 2. 4.2 The homotopy colimit functor Let D be a very small category. Let C D be the moel category of Theorem 2. Recall that by Remark 2.10 the functor : C C D is right ajoint to colim : C D C an note that it carries morphisms of F ib(c) an morphisms of F ib(c) W (C) to F ib(c D ) an F ib(c D ) W (C D ) respectively. By Theorem 1 we are finally in position to efine homotopy colimits or more precisely, the homotopy colimit functor. Definition 4.8. The homotopy colimit functor is the total left erive functor Lcolim : Ho(C D ) Ho(C) for the functor colim : C D C. 21

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