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# 8 Homogeneous Cofunctors

p ; Vki

\Q

is a homotopy equivalence. Hence all arrows in () are homotopy equivalences.

### 8Homogeneous Cofunctors

8.1 Denition A cofunctorE inF ishomogeneous of degree k, where k0, if it is polynomial of degreekand ifTk1E(V) is contractible for eachV 2 O. Remark The cofunctor given by E(V) =for all V is homogeneous of degree kfor any k0. Conversely, if E is homogeneous of degree kand homogeneous of degree ‘, where ‘ < k, then clearly E(V)’Tk1E(V).

8.2 Example Let F in F be arbitrary, and select a point in F(M), if one exists. Then TkF(V) is pointed for all V and k. Therefore a new cofunctor LkF can be dened by

LkF(V) := hober [TkF(V)−!Tk1F(V)]: It follows from 6.1 that E is homogeneous of degree k.

8.3 Example Starting with a bration p:Z ! Mk

, dene F as in 7.1 and dene G as in 7.2. Select a point in G(M). Then

E(V) := hober [F(V)−! G(V)]

is dened. It follows from 7.6 that E is homogeneous of degree k.

Example 8.3 deserves to be studied more. Ultimately E has been constructed in terms of the bration p, and a partial section of p dened near the fat diagonal NkM. Is it possible to recover p from E? In particular, for S 2 Mk

, can we describe the ber p1(S) in terms of E?

Note thatS is a subset ofM withkelements. LetV be a tubular neighborhood of S M, so that V is dieomorphic to a disjoint union of k copies of Rm. Then S belongs to Vk

Mk

and therefore we have maps E(V)−!F(V) = Γ(p ; Vk

)−−−−−−−!evaluation p1(S):

8.4 Proposition The composite map E(V)!p1(S) is a homotopy equiva-lence.

Hence we can indeed describe p1(S) in terms of E, up to homotopy equiva-lence: namely, as E(V) for a tubular neighborhood V os S in M.

Proof of 8.4 Much as in the proof of 7.6 we choose a sequence of open subsets V =V0V1V2V3: : :

such that the inclusions Vi+1 ! Vi are isotopy equivalences, such that the closure of Vi+1 in Vi is compact, and such that \iVi=S. We note that

F(V) =Y

j

Γ(p ;Uj) where the Uj are the connected components of Vk

. Among these components we single outU0, the component containing S. It is the only component whose closure in spkV does not meet NkV. For the remaining components we can use an idea as in the proof of 7.6, and nd

G(V) Y

j6=0

Γ(p ;Uj): Therefore F(V)’E(V)G(V) and the composition

E(V)!F(V)!Γ(p ;U0)!p1(S) is a homotopy equivalence.

Digression Knowing all the bers of a bration is not the same as knowing the bration. However, in the present case we can also \describe" the entire bration p in 8.3 in terms of the cofunctor E. Recall from the proof of 3.5 the poset I(k). Its elements are the open subsets of M which are dieomorphic to a disjoint union of k copies of Rm, and for V; W 2 I(k) we decree V W if and only if V W and the inclusion is an isotopy equivalence. We saw that

jI(k)j ’ M

k

:

Since I(k) O, we can restrict E to I(k). The restricted cofunctor takes all morphisms to homotopy equivalences, so that the projection

hocolim

I(k) E −! jI(k)j

is a quasibration. The associated bration is the one we are looking for. This motivates the following classication theorem for homogeneous cofunctors.

8.5 Theorem Up to equivalence, all objects in F which are homogeneous of degree k are of the type discussed in 8.3.

Outline of proof Of course, the digression just above already gives the idea of the proof, but we have to proceed a little more cautiously. The plan is: Given E, homogeneous of degree but not necessarily dened in terms of some bration, construct the appropriate F, polynomial of degree k, and a morphism E ! F. Then show that F is equivalent to a cofunctor of type V 7! Γ(p ; Vk

).

as in 7.1. This step requires a lemma, 8.6 below. Finally identify E with the homotopy ber of the canonical morphism from F to Tk1F.

8.6 Lemma [3, 3.12] Suppose that Y is a functor from a small category A to the category of Spaces. If Y takes all morphisms in A to homotopy equiva-lences, then the canonical projectionhocolimAY ! jAjis a quasibration. The section Space of the associated bration is homotopy equivalent to holimAX. Sketch proof of 8.6 The quasibration statement is obvious. We denote the total Space of the associated bration by T, so that hocolimAY T by a homotopy equivalence. For the statement about the section Space, recall that holimY can be dened as the Space of natural transformations ~A!Y, where A is the constant functor a7! on A, and ~A is aCW{functorweakly equivalent to it (some explanations below). The standard choice is

~A(a) :=jA#aj:

CW{functor refers to a functor with a CW{decomposition where the cells are of the form Rimor(b;|) for some b2 A and some i. Weakly equivalent to A means here that there is an augmentation ~A(a) ! A(a), natural in a, which is a homotopy equivalence for each a. In other words, ~A(a) is always contractible.| Suppose now that X is any CW{functor from A to spaces.

There are obvious embeddings

nat(X; Y)−! mapjAj(hocolimX ; hocolimY)−! mapjAj(hocolimX ; T) where mapjAj is for Spaces of maps over jAj. One shows by induction over the skeletons of X that the composite embedding is a homotopy equivalence. In particular, this holds for X= ~A.

Proof of 8.5 Suppose that E in F is homogeneous of degree k. Dene a cofunctor F0 from O to Spaces by

F0(V) := holim

U2I(k)(V)

E(U):

Here I(k)(V) I(k) is the full sub{poset consisting of all U 2 I(k) which are contained in V. For the meaning of I(k), see the digression preceding 8.5. By 8.6, the cofunctor F0 is equivalent to another cofunctor F1 given by a formula of type

F1(V) = Γ(qV)

where qV is a certain bration on jI(k)(V)j. The bration qV is natural in W, in the sense that a morphism V W in O induces a map from the total Space of qV to that of qW, covering the inclusion

jI(k)(V)j,! jI(k)(W)j:

By inspection, this map of total Spaces maps each ber of qV to the corre-sponding ber of qW by a homotopy equivalence. Hence F1 is equivalent to the cofunctor F2 given by

F2(V) := Γ

qM;jI(k)(V)j :

Finally we know from 3.5 (and proof) that jI(k)(V)j ’ Vk

, and this can be understood as a chain of natural homotopy equivalences (natural in V 2 O).

It follows easily that F2 is equivalent to a cofunctor F3 given by a formula of type

F3(V) := Γ

p ; Vk where p is a bration on Mk

. This is exactly the kind of cofunctor introduced in section 7, so we now write F := F3. From the denition, F belongs to F. Replacing E by an equivalent cofunctor if necessary, we can assume that E maps directly to F instead ofF0. If S2 Mk

and V is a tubular neighborhood of SM, then the composition

E(V)−!F(V) = Γ

p ; Vk eval.

−−−−!p1(S)

is a homotopy equivalence, by construction and inspection. This is of course reminiscent of 8.4. Now form the commutative square

()

E −−−−! F

??

yk1 ??yk1 Tk1E −−−−! Tk1F

and recall that Tk1E(V) is contractible for all V 2 O. Given our analysis of Tk1F in section 7, we can complete the proof of 8.5 by showing that () is

homotopy Cartesian. By 2.5 and 5.1, it suces to check that

()

E(V) −−−−! F(V)

??

yk1 ??yk1 Tk1E(V) −−−−! Tk1F(V)

is homotopy Cartesian for all V 2 Ok. If it happens that V 2 Or Ok for some r < k, then we have E(V) by homogeneity and F(V)!Tk1F(V) is a homotopy equivalence, by section 5 and section 6. If not, then V has k connected components and is a tubular neighborhood of some S M, where S2 Mk

. Using 8.4 now (and 7.6), and our observation above which seemed so reminiscent of 8.4, we nd that () is again homotopy Cartesian.

Outline

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