Obstructions in a model category and Klein and Williams' intersection invariants
aa r X i v : . [ m a t h . A T ] J u l OBSTRUCTIONS IN A MODEL CATEGORY AND KLEIN AND WILLIAMS’INTERSECTION INVARIANTS
KATE PONTOA
BSTRACT . We give an obstruction for lifts and extensions in a model category inspiredby Klein and Williams’ work on intersection theory. In contrast to the familiar obstruc-tions from algebraic topology, this approach produces a single invariant that is completein the presences of the appropriate generalizations of dimension and connectivity as-sumptions. C ONTENTS
1. Introduction 12. The Euler class 33. The cartesian gap map 44. Proof of the main result 65. Model categories 9References 111. I
NTRODUCTION
Algebraic topology has a well developed theory of obstructions to lifts and extensionsof maps that reduces these questions to the vanishing of cohomology classes [Spa95,Whi78]. For example, a diagram E p (cid:15) (cid:15) B f / / Y where p is a fibration defines classes in H j + ( B , π j ( p − ( ∗ ))). If B is a finite dimensionalCW complex the vanishing of these invariants implies the existence of a lift of f .In their work on obstructions to removing intersections [KW07], Klein and Williamsgive an alternative obstruction for lifts. Let M ( α , α ) be the homotopy pushout (doublemapping cylinder) of a map α : X → Y . There is an inclusion map Y ∐ Y → M ( α , α ).A map f : B → Y defines a map χ : B ∐ B f ∐ f −−−→ Y ∐ Y → M ( α , α ). Theorem 1.1 ([KW07]) . If a map f : B → Y has a lift to X up to homotopy, there is anextension of χ to a map Cyl( B ) → M ( α , α ). Date : 28th Jul, 2020.1991
Mathematics Subject Classification.
Suppose B is a CW complex of dimension less than n, α : X → Y is n-connected, and χ extends to a map Cyl( B ) → M ( α , α ). Then f : B → Y has a lift to X up to homotopy.
The map χ is a generalization of a classical fixed point invariant, the Reidemeistertrace [Bro71, Hus82], and is closely related to Hatcher and Quinn’s work [HQ74] onintersections of submanifolds. Remark . The dimension and connectivity hypotheses in Theorem 1.1 are reminiscentof assumptions used to imply the classical obstructions take values in trivial groups. Theranges of connectivity and dimension in Theorem 1.1 do not force this triviality.The fact that Theorem 1.1 was proven for topological spaces is an artifact of the moti-vating example. In this paper we prove a significant generalization of Theorem 1.1 whilealso making the roles of various assumptions transparent. Results are stated here in amodel category because it provides a convenient way to describe the necessary hypothe-ses, but I expect this argument can be easily adapted to closely related environments.Working in a model category with functorial good cylinders, let B ∐ B i ∐ i −−−−→ Cyl( B ) π −→ B be a factorization of the fold map through the functorial cylinder. When we need tospecify the domain of i or i we will write i B or i B . If i : A → B , let N j ( i ), j =
0, 1, bethe mapping cylinder of i and i j :(1.3) N j ( i ) : = Pushout(Cyl( A ) i j ←− A i −→ B ).Let ι j : N j ( i ) → Cyl( B ) be the map induced by Cyl( i ) : Cyl( A ) → Cyl( B ) and i j , B . Definition 1.4.
For morphisms i : A → B and α : X → Y , we write i ◊ α if for every com-mutative diagram of solid arrows as in (1.5) there are dotted arrows completing thediagram.(1.5) A i (cid:15) (cid:15) i / / Cyl( A ) (cid:15) (cid:15) H { { ①①①①①①①①① A i (cid:15) (cid:15) i o o h (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ Y X α o o B f ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ i / / Cyl( B ) K c c B i o o g _ _ This is the HELP diagram [May99, 10.3]. When A is the initial object, g is the lift of f up to homotopy and K is the homotopy in Theorem 1.1. Theorem 1.6.
If i is a cofibration, X is fibrant and ι is an acyclic cofibration there is amap χ generalizing the map χ of Theorem 1.1.i . If i ◊ α then χ is trivial.ii . Suppose A is cofibrant, ι and ι are an acyclic cofibrations, and i ◊ ⌈ α , α ⌉ (seeDefinition 3.3). If χ is trivial then i ◊ α . Comparing Theorem 1.6 to Theorem 1.1, if A is empty the conditions that i is a cofi-bration and ι and ι are acyclic cofibrations become B is cofibrant. The condition that X is fibrant is invisible in the category of spaces. The condition i ◊ ⌈ α , α ⌉ becomes thedimension and connectivity conditions of Theorem 1.1. See Proposition 3.5. BSTRUCTIONS IN A MODEL CATEGORY 3
I came to this project from the paper of Klein and Williams [KW07] and related workfocused on fixed point theory [Cou09, Sun13] and have not found much existing liter-ature related to obstructions in this generality. The closest seems to be the paper ofChristensen, Dwyer and Isaksen [CDI04] but there does not seem to be an immediatetranslation between perspectives.
For the reader familiar with Klein and Williams’s work.
After defining the map χ for topological spaces, Klein and Williams [KW07] show the corresponding sable invari-ant is only trivial when χ is trivial and they use duality to give an alternative descriptionof their stable invariant. Those steps don’t make sense for the generality considered hereand so are omitted. Organization.
In §2 we define the map χ and prove a more general version of Theo-rem 1.6 i . In §3 we fix notation and recall the cartesian gap map and the Blakers-Masseytheorem. In §4 we prove the more general version of Theorem 1.6 ii . In §5 we completethe proof of Theorem 1.6, Acknowledgements.
This paper evolved from conversations with Inbar Klang, SarahYeakel, and Cary Malkiewich about generalizations of fixed point invariants. Manythanks to Inbar and Sarah for comments on an earlier version of this paper. Also thanksto Nima Rasekh and Dan Duggar for helping untangle confusions.The author was partially supported by NSF grant DMS-1810779 and the RoysterResearch Professorship at the University of Kentucky.2. T HE E ULER CLASS
For a map α : X → Y , let(2.1) X c ( α ) −−−→ F ( α ) ˜ f ( α ) −−−→ Y be a factorization of α as a cofibration and acyclic fibration For maps α : X → Y and β : X → Z let M ( α , β ) be the pushout in (2.2).(2.2) X c ( α ) / / β (cid:15) (cid:15) F ( α ) j (cid:15) (cid:15) Z j ′ / / M ( α , β )Using the factorization in (2.1), we replace the diagram in (1.5) with the diagram in(2.3).(2.3) A i (cid:15) (cid:15) i / / Cyl( A ) (cid:15) (cid:15) H | | ①①①①①①①①① A i (cid:15) (cid:15) i o o c ( α ) ◦ h } } ④④④④④④④④ Y F ( α ) ˜ f ( α ) o o B f ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) i / / Cyl( B ) J b b B i o o ˆ h a a Let N ( i ) be the double mapping cylinder on i : N ( i ) : = Pushout(Cyl( A ) i ∐ i ←−−−− A ∐ A i ∐ i −−→ B ∐ B ).If i ◊ ˜ f ( α ), then the map h from (2.3) makes the top square in (2.5a) commute. Theremaining squares of (2.5a) commute by definition or assumption. BSTRUCTIONS IN A MODEL CATEGORY 4
Definition 2.4. If i ◊ ˜ f ( α ) then χ : N ( i ) → M ( α , α )is the map defined by (2.5a) using the universal property of the pushout. We say χ is trivial if there is a map X as in (2.5b)(2.5a) A i / / i (cid:15) (cid:15) g " " ❊❊❊❊❊❊❊❊❊ B h ●●●●●●●●● N ( i ) f ∐ H " " ❉❉❉❉❉❉❉❉ X c ( α ) / / α (cid:15) (cid:15) F ( α ) j (cid:15) (cid:15) Y j ′ / / M ( α , α ) (2.5b) N ( i ) (cid:15) (cid:15) χ / / M ( α , α )Cyl( B ) X Remark . The hypothesis that i ◊ ˜ f ( α ) is excessive. It would be enough to assume thatthe maps h and J exist for a particular choice of f , H and g . We stick with the strongerhypothesis for clarity, but note that at many later points similar changes could be madeto weaken assumptions. Proposition 2.7 (Theorem 1.6. i ) . If i ◊ α then i ◊ ˜ f ( α ) and χ is trivial.Proof. Take J in (2.3) to be K in (1.5) and h in (2.3) to be composite B g −→ X c ( α ) −−−→ F ( α ).The diagram in (2.8) defines a map Cyl( B ) → M ( α , α ) that extends χ . (cid:3) (2.8) B id / / i (cid:15) (cid:15) ❋❋❋❋❋❋❋❋❋ B h ●●●●●●●●● Cyl( B ) J " " ❋❋❋❋❋❋❋❋❋ X c ( α ) / / α (cid:15) (cid:15) F ( α ) j (cid:15) (cid:15) Y j ′ / / M ( α , α ) Remark . The choice to make N ( i ) the double mapping cylinder and M ( α , β ) the ho-motopy pushout is intentional. In what follows we will need to have very explicit accessto the cylinder in N ( i ), while there will not be similar requirements for M ( α , β ).3. T HE CARTESIAN GAP MAP
For a map α : X → Y , let(3.1) X ˜ c ( α ) −−−→ C ( α ) f ( α ) −−−→ Y be a factorization of α as an acyclic cofibration and a fibration. For maps α : X → Y and β : X → Z let P ( α , β ) be the pullback in (3.2a). BSTRUCTIONS IN A MODEL CATEGORY 5 (3.2a) P ( α , β ) k / / k ′ (cid:15) (cid:15) C ( j ) f ( j ) (cid:15) (cid:15) Z j ′ / / M ( α , β ) (3.2b) X ⌈ α , β ⌉ β (cid:30) (cid:30) c ( α ) / / F ( α ) ˜ c ( j ) (cid:31) (cid:31) P ( α , β ) k / / k ′ (cid:15) (cid:15) C ( j ) f ( j ) (cid:15) (cid:15) Z j ′ / / M ( α , β ) Definition 3.3.
The cartesian gap map of α and β is the dotted map in (3.2b) inducedby the universal property of the pullback. Example 3.4.
There are several important connectivity results for the cartesian gapmap. i . In a stable model category, such as chain complexes of R -modules or orthogonalspectra, the homotopy cartesian and cocartesian squares agree and the cartesiangap map is a weak equivalence [Hov99, 7.1.12]. ii . In the category of topological spaces, if α : X → Y is n -connected and β : X → Z is n ′ -connected the classical Blakers-Massey theorem [tD08, 6.9] asserts ⌈ α , β ⌉ is( n + n ′ − iii . The recent papers [ABFJ, CSW16] prove generalizations of the classical Blakers-Massey theorem. [ABFJ] proves a version for higher topoi.The essential hypothesis in the converse of Proposition 2.7 is a lifting condition forthe cartesian gap map. The first two examples in Example 3.4 imply relevant liftingconditions. Proposition 3.5. i . In a right proper stable model category, if i : A → B is a cofibration, A is cofibrant,the induced map N ( i ) → Cyl( B ) is acyclic cofibration, and the source and target of ⌈ α , β ⌉ are fibrant then i ◊ ⌈ α , β ⌉ . ii . If i : A → B is a relative CW complex of dimension m, α : X → Y is n-connected, β : X → Z is n ′ -connected and m ≤ ( n + n ′ − theni ◊ ⌈ α , β ⌉ . Proof.
We postpone the proof of the first statement to the end of Section 5 since it is com-paratively lengthy and not illuminating for the ideas considered here. See Lemma 5.5with θ = ⌈ α , β ⌉ .For the second, the classical Blakers-Massey theorem implies the map X → P ( α , β )is ( n + n ′ − (cid:3) Proposition 3.5 ii should have an equivariant generalization following [Dot16] and thisshould allow for an alternative approach to the main result in [KW10]. BSTRUCTIONS IN A MODEL CATEGORY 6
4. P
ROOF OF THE MAIN RESULT
Recall the mapping cylinder N j ( i ) and maps ι j from the introduction. The maps in(2.1), (3.1), and (1.5) define the commutative diagram in (4.1).(4.1) A h / / i (cid:15) (cid:15) X c ( α ) / / F ( α ) ˜ c ( j ) / / C ( j ) f ( j ) (cid:15) (cid:15) N ( i ) H ∐ f / / Y j ′ / / M ( α , α ) Proposition 4.2.
If i ◊ ⌈ α , α ⌉ and there is a lift in (4.1) then i ◊ α .Proof. A lift in (4.1) defines maps φ : B → C ( j ) and ψ : Cyl( A ) → C ( j )so that the diagrams in (4.3a), (4.3b), (4.3c), and (4.3d) commute.(4.3a) B φ / / f (cid:15) (cid:15) C ( j ) f ( j ) (cid:15) (cid:15) Y j ′ / / M ( α , α ) (4.3b) Cyl( A ) ψ / / H (cid:15) (cid:15) C ( j ) f ( j ) (cid:15) (cid:15) Y ′ j ′ / / M ( α , β )(4.3c) A i (cid:15) (cid:15) h / / X c ( α ) / / F ( α ) ˜ c ( j ) (cid:15) (cid:15) Cyl( A ) ψ / / C ( j ) (4.3d) A i / / i (cid:15) (cid:15) Cyl( A ) ψ (cid:15) (cid:15) B φ / / C ( j )Then (4.4) commutes by assumption.(4.4) A i / / h (cid:15) (cid:15) Cyl( A ) H (cid:15) (cid:15) X α / / Y The diagram in (4.3a) defines a map Φ : B → P ( α , β )and (4.3b) defies a map Ψ : Cyl( A ) → P ( α , β ).By definition and (4.3c), (4.5a) commutes. Similarly, definition and (4.4) imply (4.5b)commutes. Finally, (4.3d) implies (4.5c) commutes. The diagram in (4.5d) commutes byassumption.(4.5a) A i / / h (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) ψ (cid:22) (cid:22) X c ( α ) ' ' ⌈ α , α ⌉ / / P ( α , α ) k $ $ ❍❍❍❍❍❍❍❍❍ F ( α ) ˜ c ( j ) / / C ( j ) (4.5b) A i / / h (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) H (cid:21) (cid:21) X ⌈ α , α ⌉ / / α , , P ( α , α ) k ′ ●●●●●●●●● Y BSTRUCTIONS IN A MODEL CATEGORY 7 (4.5c) A i / / i (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) ψ (cid:22) (cid:22) B Φ / / φ , , P ( α , α ) k $ $ ❍❍❍❍❍❍❍❍❍ C ( j ) (4.5d) A i / / i (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) H (cid:21) (cid:21) B Φ / / α , , P ( α , α ) k ′ ●●●●●●●●● Y The diagrams in (4.5b) and (4.5a) imply (4.6a) commutes and (4.5c) and (4.5d) imply(4.6b) commutes.(4.6a) A i / / g (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) X ⌈ α , α ⌉ / / P ( α , α ) (4.6b) A i / / i (cid:15) (cid:15) Cyl( A ) Ψ (cid:15) (cid:15) B Φ / / P ( α , α )The diagrams in (4.6a) and (4.6b) imply (4.7a) commutes. The diagram in (4.7b) com-mutes and if i ◊ ⌈ α , α ⌉ , maps ˆ Ψ and ˆ g completing (4.7b) define lifts in (1.5).(4.7a) A i / / i (cid:15) (cid:15) Cyl( A ) (cid:15) (cid:15) Ψ y y ttttttttt A i o o i (cid:15) (cid:15) h (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) P ( α , α ) X ⌈ α , α ⌉ o o B i / / Φ < < ①①①①①①①①① Cyl( B ) ˆ Ψ e e B i o o ˆ g _ _ (4.7b) A i / / i (cid:15) (cid:15) Cyl( A ) (cid:15) (cid:15) Ψ y y ttttttttt H x x A i o o i (cid:15) (cid:15) h (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Y P ( α , α ) k ′ o o X ⌈ α , α ⌉ o o B i / / f ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) Φ : : Cyl( B ) ˆ Ψ e e B i o o ˆ g _ _ (cid:3) We now compare a lift in (4.1) to the vanishing of χ . This requires an additionaldefinition. Definition 4.8.
We write i (cid:3) α if every commutative diagram as in (4.9) has a lift.(4.9) A / / i (cid:15) (cid:15) X α (cid:15) (cid:15) B / / Y Lemma 4.10.
Suppose i A (cid:3) f ( j ) , ι (cid:3) f ( j ) , and i ◊ f ( α ) . If χ is trivial then (4.1) has a lift. BSTRUCTIONS IN A MODEL CATEGORY 8
Proof.
Since i A (cid:3) f ( j ), there is a lift λ in (4.11).(4.11) A h / / i (cid:15) (cid:15) X α ! ! ❈❈❈❈❈❈❈❈❈ c ( α ) / / F ( α ) j $ $ ■■■■■■■■■ ˜ c ( j ) / / C ( j ) f ( j ) (cid:15) (cid:15) Cyl( A ) H / / Y j ′ / / M ( α , α )Then (4.12a) commutes and defines a map µ : N ( i ) → C ( j ). Using this map, (4.12b)commutes. Since ι (cid:3) f ( j ), (4.12b) has a lift ν .(4.12a) A i (cid:15) (cid:15) i / / h ●●●●●●●●● B ˆ h (cid:15) (cid:15) X c ( α ) / / F ( α ) ˜ c ( j ) (cid:15) (cid:15) Cyl( A ) λ / / C ( j ) (4.12b) N ( i ) µ / / (cid:15) (cid:15) C ( j ) f ( j ) (cid:15) (cid:15) N ( i ) χ % % ❑❑❑❑❑❑❑❑❑❑ (cid:15) (cid:15) Cyl( B ) ν B B X / / M ( α , α )The map ˆ h is as in (2.3).Expanding (4.12b), the diagrams in (4.13a) and (4.13b) commute.(4.13a) A i (cid:15) (cid:15) i / / Cyl( A ) λ $ $ ❍❍❍❍❍❍❍❍❍ (cid:15) (cid:15) N ( i ) µ / / (cid:15) (cid:15) C ( j ) B i / / Cyl( B ) ν : : ✈✈✈✈✈✈✈✈✈ (4.13b) B f (cid:15) (cid:15) i / / Cyl( B ) X % % ❑❑❑❑❑❑❑❑❑ ν / / C ( j ) f ( j ) (cid:15) (cid:15) Y j ′ / / M ( α , α )The diagram in (4.13a) shows B i −→ Cyl( B ) ν −→ C ( j ) and λ define a map N ( i ) → C ( j ).To verify this is a lift for (4.1), the restriction to Cyl( A ) commutes by (4.11) and therestriction to B commutes by (4.13b). (cid:3) The following result is an immediate consequence of Proposition 4.2 and Lemma 4.10.
Theorem 4.14 (Theorem 1.6. ii ) . Suppose i A (cid:3) f ( j ) , ι (cid:3) f ( j ) , and i ◊ ˜ f ( α ) . If i ◊ ⌈ α , α ⌉ and χ is trivial then i ◊ α . We now briefly turn to some special cases of Theorem 4.14.4.1. A is initial. If A is the initial object, the hypothesis of Definition 2.4 becomes( ; → B ) ◊ ˜ f ( α ).In Theorem 4.14 the hypothesis i ; (cid:3) f ( j ) is vacuous and ι (cid:3) f ( j ) becomes i B (cid:3) f ( j ).Then we have the following version of Theorem 4.14. Corollary 4.15.
Suppose i B (cid:3) f ( j ) and ( ; → B ) ◊ ˜ f ( α ) . If ( ; → B ) ◊ ⌈ α , α ⌉ and χ is trivialthen i ◊ α . Sections.
Suppose we have maps as in (1.5) with A i −→ B f −→ Y = A g −→ X α −→ Y and H the composite Cyl( A ) π −→ A i −→ B f −→ Y = Cyl( A ) π −→ A h −→ X α −→ Y . BSTRUCTIONS IN A MODEL CATEGORY 9
In this case the hypothesis i A (cid:3) f ( j ) of Lemma 4.10 can be removed sinceCyl( A ) π −→ A h −→ X c ( α ) −−−→ F ( α ) ˜ c ( j ) −−→ C ( j ).defines a lift λ in (4.11). Corollary 4.16.
Suppose (4.17) commutes, ι (cid:3) α , ι (cid:3) f ( j ) , and i ◊ ˜ f ( α ) . (4.17) A / / i (cid:15) (cid:15) X α (cid:15) (cid:15) B / / YIf i ◊ ⌈ α , α ⌉ and χ is trivial then there is a (strict) lift in (4.17) .Proof. Using Theorem 4.14 we have maps K and g so that (1.5) commutes. Since ι (cid:3) α and the diagram in (4.18a) commutes, (4.18a) has a lift J . Then B i −→ Cyl( B ) J −→ X is anextension of g lifting f since (4.18b) commutes.(4.18a) N ( i ) H ∐ g / / ι (cid:15) (cid:15) X α (cid:15) (cid:15) Cyl( B ) K / / Y (4.18b) A i ❋❋❋❋❋❋❋❋❋ i (cid:15) (cid:15) h / / X α (cid:15) (cid:15) Cyl( A ) Cyl( i ) (cid:15) (cid:15) H ; ; ①①①①①①①①① Cyl( B ) K ❋❋❋❋❋❋❋❋❋ J E E ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ B i ; ; ①①①①①①①①① f / / Y (cid:3)
5. M
ODEL CATEGORIES
The last step in the proof of Theorem 1.6 is to compare the hypotheses in that state-ment to those in Proposition 2.7 and Theorem 4.14. We make that comparison here andcomplete the proof of Proposition 3.5.
Lemma 5.1.
If i (cid:3) α and i ◊ id Y then i ◊ α .Proof. We start from the diagram in (1.5). Since i ◊ id Y , there are dotted maps complet-ing (5.2a).(5.2a) A i (cid:15) (cid:15) i / / Cyl( A ) (cid:15) (cid:15) H { { ①①①①①①①①① A i (cid:15) (cid:15) i o o β ◦ h (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ Z Z id Z o o B f ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ i / / Cyl( B ) K c c B i o o ℓ _ _ (5.2b) A h / / i (cid:15) (cid:15) X β (cid:15) (cid:15) B ℓ / / θ > > ⑦⑦⑦⑦⑦⑦⑦⑦ Z Since i (cid:3) α , there is a lift in (5.2b). Then K and θ are maps completing (1.5). (cid:3) Recall that if α is an acyclic fibration and i is a cofibration then i (cid:3) α . Lemma 5.3.
Let i : A → B and α : X → Y .i . If Y is fibrant and ι is an acyclic cofibration then i ◊ id Y .ii . If i is a cofibration, Y is fibrant and ι is an acyclic cofibration, then i ◊ ˜ f ( α ) .iii . If A is cofibrant then i A (cid:3) f ( j ) and i A (cid:3) f ( j ) . BSTRUCTIONS IN A MODEL CATEGORY 10
Proof.
Proof of Item i : Since N ( i ) → Cyl( B ) is an acyclic cofibration and Y is fibrant,(5.4a), with maps as in in (5.4b), has a lift. The lift defines the dotted maps in (5.4b).(5.4a) N ( i ) K ∐ φ / / (cid:15) (cid:15) Y (cid:15) (cid:15) Cyl( B ) / / H < < ∗ (5.4b) A / / (cid:15) (cid:15) Cyl( A ) (cid:15) (cid:15) K { { ①①①①①①①①① A (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ o o (cid:15) (cid:15) Y YB / / φ ? ? ⑧⑧⑧⑧⑧⑧⑧⑧ Cyl( B ) c c B o o _ _ Proof of Item ii : Since Y is fibrant and N ( i ) → Cyl( B ) is an acyclic cofibration, i ◊ id Y . Since i is a cofibration and ˜ f ( α ) is an acyclic fibration, i (cid:3) ˜ f ( α ) and Lemma 5.1implies i ◊ ˜ f ( α ). Proof of Item iii : If A is cofibrant then i is an acyclic cofibration and i (cid:3) f ( α ). (cid:3) Proof of Theorem 1.6. If i is a cofibration, Y is fibrant and ι is an acylic cofibration then i ◊ ˜ f ( α ) and the hypotheses of Definition 2.4 and Proposition 2.7 are satisfied.If A is cofibrant, then i (cid:3) f ( j ). If ι is an acyclic fibration then ι (cid:3) f ( j ). If i is acofibration, Y is fibrant and ι is an acyclic cofibration, then i ◊ ˜ f ( α ). Then the hypothesesof Theorem 4.14 are satisfied. (cid:3) We now complete the proof of Proposition 3.5.
Lemma 5.5.
Suppose the model category is right proper. If i : A → B is a cofibration, Ais cofibrant, the induced map ι : N ( i ) → Cyl( B ) is a cofibration, X and Y are fibrant, and θ : X → Y is a weak equivalence then i ◊ θ .Proof. Let Cocyl( Y ) be a good cocylinder for Y and Y i −→ Cocyl( Y ) ev × ev −−−−−→ Y × Y be thediagonal. We have the two pullback diagrams in (5.6a) and (5.6b).(5.6a) X × θ ,ev Cocyl( Y ) ξ / / (cid:15) (cid:15) Cocyl( Y ) ev (cid:15) (cid:15) X θ / / Y (5.6b) X × θ ,ev Cocyl( Y ) / / ζ (cid:15) (cid:15) Cocyl( Y ) ev × ev (cid:15) (cid:15) X × Y θ × id / / Y × Y The composites in (5.7a) and (5.7b) are the same map.(5.7a) X × θ ,ev Cocyl( Y ) ξ −→ Cocyl( Y ) ev −−→ Y (5.7b) X × θ ,ev Cocyl( Y ) ζ −→ X × Y proj −−→ Y We first show that if X and Y are fibrant and θ is a weak equivalence then (5.7a) is anacyclic fibration.If Y is fibrant, the maps ev , ev : Cocyl( Y ) → Y are acyclic fibrations. The modelcategory is right proper and θ is an weak equivalence, so the top horizontal map in(5.6a) is a weak equivalence. Composing with ev implies (5.7a) is a weak equivalence.The first map in (5.7b) is the left vertical map in (5.6b). It is the pullback of a fibrationand so is a fibration. The second map in (5.7b) is the projection. It is a fibration since X is fibrant. Therefore (5.7b) is a fibration.If A is cofibrant and we have the maps in (1.5), then (5.8a) commutes and has a liftˆ J : Cyl( A ) → Cocyl( Y ). If we take the maps f , h and H in (5.8b) as in (1.5) and let BSTRUCTIONS IN A MODEL CATEGORY 11 J = ˆ J ◦ i , then (5.8b) commutes. Let g × L be a lift in (5.8b).(5.8a) A i (cid:15) (cid:15) f ◦ i / / Y i / / Cocyl( Y ) ev × ev (cid:15) (cid:15) A i / / Cyl( A ) ( f ◦ i ◦ π , H ) / / Y × Y (5.8b) A h × J / / i (cid:15) (cid:15) X × θ ,ev Cocyl( Y ) (5.7a) (cid:15) (cid:15) B f / / Y The diagram in (5.9a) defines a map N ( i ) → Cocyl( Y ) so that (5.9b) commutes. If ι is anacyclic cofibration, the diagram in (5.9b) where the top map is induced by (5.9a) and theleft vertical map is the inclusion commutes and has a lift M . Then K = ev ◦ M : Cyl( B ) → Y and g : B → X make (1.5) commute.(5.9a) A ∐ A i ∐ i / / i ∐ i (cid:15) (cid:15) Cyl( A ) (cid:15) (cid:15) ˆ J (cid:24) (cid:24) B ∐ B / / ( i ◦ f ) ∐ L , , N ( i ) % % Cocyl( Y ) (5.9b) N ( i ) (cid:15) (cid:15) / / Cocyl( Y ) ev (cid:15) (cid:15) Cyl( B ) π / / B f / / Y (cid:3) Remark . The diagram in (5.8b) is the right homotopy version of (1.5) and the resultsof this paper could be reworked to prioritize (5.8b) over (1.5). In particular, many of thehypotheses of Lemma 5.5 only serve the transitions between right and left homotopies.R
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