Ò X 9: where the functor is the

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229

Many of the statements in this book have their basis in properties of adjoint functors. Examples of more of more situation as such are induction of model structures by functors which allows us to give to the category of Simplicial Topological Spaces (?‰X 9:) a closed model structure induced by the one in ?‰S using the "adjoint pair"

?‰SÒd ?‰X 9:Òf ?‰S where the functor is the d discrete functor and is the f forgetful one. Similarly, the category of topological spaces (X 9:), has a closed model structure induced by the standard one in ?‰S by ?SÒX 9:Ò?S where the first one is the geometric realization and the second is the standard singular functor. Also we will see that ?‰S admits topologic-algebraic techniques, very similar to the standard ones, which are not in general model structures. Most of the proofs of the generalizations involved lie in the fact that for each functor ˜ ?À Ò?‰S one has an associated adjoint pair

?‰SÒR˜ ?‰SÒS˜ ?‰S

which can be regarded as generalizations of Sd and Ex, the subdivision and extension functors due to Kan. The background for this pair, which is natural in ˜, lies itself in the existence for each YÀ?ÒS of an adjoint pair

?‰SÒRY SÒSY ?‰S

the realization and singular functors associated to Y. This construction mimics the one for the standard geometric realization and singular functors except that Topology is not used. The fact that R , S˜ ˜ and their existence provides a systematic change in ?‰S and

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When working with model structures induced by functors of the kind YÀ?ÒX 9: as in usual cases, it becomes apparent that it may be better to consider "realizations" as functors which take us back into the original category from a specific enlargement of it. So, for example, if one considers ?‰S as an enlargement of , realizationsS are functors ?‰SÒS. In the case of X 9: realizations are functors ?‰X 9:ÒX 9:. Particular restrictions are also considered. For example

| |

ο

S

i

ο

Top

Top

R

the usual geometric realization is the restriction of the topologized realization to the sub category of simplicial sets. Here we consider specific techniques to build them up. They will also admit right adjoint which we will again call the associated singular functors. We will see that the constructions are natural in the categorical sense.

Realization and Singular Functors

Here we define the realization and singular functors associated to a specific cosimplicial object.

9.1 Definition:

Let YÀ?ÒS be a cosimplicial object of S.

i The realization functor RY À?‰SÒS is given by

R (X) X Y /

a Y œ nn µ

for a simplicial set, where X µ is the equivalence relation induced by identifying for each w [ n ]À Ò[ m ] elements of the form (W*(x ),y )m n

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b On morphisms, if f XÀ ÒK is a simplicial map, then R (f) [ x ,y ]Y À n n È[f (x ),y ] n n n which is a well defined

function.

ii As for the singular functor SY ÀSÒ?‰S it is

a given on objects by S (X)Y n œS(Y ,X) n and if s [n]À Ò[m] then w*(a)œaw*.

b For morphisms, S (f)Y is S(Y ,f) n at the n-th level

RY is a covariant functor, as it is easy to check. S is a covariantY functor as it is easy but tedious to prove, and the pair (R ,S )Y Y is an adjoint pair.

9.2 Definition:

When YÀ?ÒX 9: then the definitions of the singular and realization functors

SY À X 9:Ò?‰S and RY À?‰SÒX 9:

are the same as in the set theoretical case (9.1) except that in the realization the product, the sum and the quotient are topological ones and in the singular functor one considers continuous functions instead of simple functions

Again one has an adjoint pair (R ,S )Y Y .

9.3 Definition:

YÀ?ÒX 9: the topological realization and the topological singular functors

RY À?‰X 9:ÒX 9: and SY À X 9:Ò?‰X 9:

again are the same as in the set theoretical case the with the product, the sum and the quotient involved topological ones, the functions are continuous functions and the topology of the function space X 9: (Y ,X)n is the compact open topology.

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For YÀ?ÒX 9: the topological realization and the topological singular functors form an adjoint pair (R ,S ), YY Y if is level wise locally compact.

For these adjoint pairs one has the following properties:

9.5 Proposition:

In each of the adjoint pairs (R ,S )Y Y , the singular S and realizationY functors R , Y given above are natural. In other words, for each cosimplicial map a YÀ ÒZ there are natural transformations.

Ra ÀRY ÒRZ and Sa ÀSZ ÒSY

Proof:

R ,Sa a are defined by

R (X) [x ,y ]a n n œ[x ,a (y )]n n n

. (S (X))( )a α œαan from which naturality can be easily traced.

We are very much interested in the following property which establishes relations between retracts at the cosimplicial level and retracts at the set theoretical and topological level. Usually retracts among simplicial objects or simplicial maps are refered to as simplicial retracts. Similarly for the cosimplicial case Thus given . Y,Z cosimplicial sets (Res. cosimplicial topological spaces) we say that Z is a cosimplicial retract of if there are cosimplicial maps Y i ZÀ ÒY and r YÀ ÒZ such that riœ1Z

9.6 Proposition:

Given Y,Z cosimplicial sets (Res. cosimplicial topological spaces) if Z is a retract of , then for each in Y X ?‰S (Res. in X ?X 9:) R (X) is a

Z

retract of R XY( ) and for each simplicial function , f R (f)Z is a retract of R (f)Y . Furthermore, for each set (Res. topological space ) X X S (X)Y is a retract of S (Z)Y and for each function (Res. Continuous functionf f S (f)) Y is a retract of S (f)Z .

Proof:

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R (X)

Z

R (X)

Y

R (X)

i

R (X)

r

R (X)

Z

1

R (X)

z

S (X)

Z

S (X)

Y

S (X)

i

S (X)

r

S (X)

Z

1

S (X)z

9.7 Proposition:

With the same hypothesis of proposition 9.6 R (S (X))Z Z is a retract of R (S (X))Y Y . Similarly R (S (f))Z Z is a retract of R (S (f))Y Y . The same statements hold for the compositions S R Z Z and S R .Y Y

Proof:

The proof is given by the following sequence of diagrams, the first and third being given by proposition 9.6. The second and fourth diagrams are induced by the first and third respectively using fœR (X)i and R (X)r for the first case, and fœS (X)r and S (X)i for the

second. The retracts are given in the second and fourth diagrams by the double arrows.

Fig. 1

S

Z

(K)

S

Y

(K)

S

Z

(K)

S

r

(K)

S

i

(K)

S

Z

( f )

S

Y

( f )

S

Z

( f )

S

Z

(L)

S

Y

(L)

S

Z

(L)

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Fig. 2

S R (X)Z Z

S R (X)r Z

S R (X)Z Y S R (X)Z Z S R (X)i Z

S R (X)Z i S R (X)Y i S R (X)Z i

S R (X)Z Y

S R (X)r Y

S R (X)Y Y S R (X)Z Y S R (X)i Y

S R (X)Z r S R (X)Y r S R (X)Z r

S R (X)Z Z

S R (X)r Z

S R (X)Y Z S R (X)Z Z S R (X)i Z

Fig. 3

R

Z

(K)

R

Y

(K)

R

Z

(K)

R

r

(K)

R

i

(K)

R

Z

( f )

R

Y

( f )

R

Z

( f )

R

Z

(L)

R

Y

(L)

R

Z

(L)

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Fig. 4

R S (X)Z Z

R S (X)i Z

R S (X)Y Z R S (X)Z Z R S (X)r Z

R S (X)Z r R S (X)Y r R S (X)Z r

R S (X)Z Y

R S (X)i Z

R S (X)Y Y R S (X)Z Y R S (X)r Z

R S (X)Z i R S (X)r i R S (X)Z i

R S (X)Z Z

R S (X)i Z

R S (X)Y Z R S (X)Z Z R S (X)r Z

That for each , f R (S (f))Z Z is a retract of R S (f)Y Y follows from Fig. 3 changing by . For the case X f S R (f)Z Z being a retract of S R (f)Y Y one uses Fig. 2 changing X by . Note that in Figures 2 and 4 thef columns and rows are the identity morphisms.

We give now some examples of realization and singular functors: i When Y is a cosimplicial point, in other words, Yn is a set

with only one element for each , n then R (X) is isomorphic inY S to 10(X) where 10(X) is the set of path components of .X In fact, the functors, RY and 10 À?‰SÒS are naturally isomorphic. That is so, simply because the singular functor SY ÀSÒ?‰S which associates to each set the simplicialX set X each one of whose levels is X and defined on morphisms in the obvious way. It is known that these last functor is right adjoint to 10. Hence the result follows from properties of adjointness. More generally

ii If Yµ is the degenerated cosimplicial set associated to a set Y then

R (X) (X) Y

Y

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As we mentioned before, all of the properties which we have developed for realization and singular functors also hold for R˜ and S˜ where ˜ ?À Ä?‰S is a covariant functor i.e. a model of ?‰S. In that case In fact S˜ À?‰SÄ?‰S is given by S (X)˜ n œ?‰S(˜ ,X)

n

and if w [n]À Ä[m]−?, then (w)˜ À˜n Ęm and S (w)˜ À?‰S(˜n,X)Ä?‰S(˜n,X)

maps ˜n Äα X into ˜ Ò ˜n˜(w) mÄα X. On the other hand R˜ À?‰SÄ?‰S is given by R (T)˜ n œR (T)˜n , for both TœX and Tœf.

9.8 Proposition:

i S˜ À?‰SÄ?‰S and R˜ À?‰SÄ?‰S are covariant functors. ii (R ,S )˜ ˜ is an adjoint pair.

Proof:

In fact, for each simplicial set we takeX

(<X)n Xn ? S (˜n,R (X)) S R (X)

n

À Ò ‰ œ

˜ Š ˜ˆ ˜ ‰‹

As the function which sends xnXn into the simplicial map (<X) (x )n n , whose p-th level is y n [x ,y ]n . Each one of the claims that

p È n p

(<X n) (X )n is a simplicial map, and <X itself is a simplicial map, can

be verified easily. Furthermore if f XÀ ÒK is a simplicial map, then the following diagram commutes

X

K

f

S R (X)

Y Y

y

X

y

K

S R (K)

Y Y

S R ( f )

using the non simplicial case.Hence <À1?‰S ÒS R is a natural

Y Y

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(:X)p À[r,y ]pn Èr (y )p pn wich is natural in X. That ends the proof that

(R ,S ) ˜ ˜ is an adjoint pair.

This shows three things:

i In general realizations do not distribute over products. In fact, for any non pointed set Y R, Y does not distribute over products since R (XYZ)zXZY while

R (X)YR (Z)Y zXYZY.

ii Using adjointness of RY and SY, if one considers the inclusion SÒ?‰S via a set (i.e. the functor which associates to aY set the degenerated simplicial set X S(Y,X)) which is SY, then it is right adjoint to the functor 10( )YÀ?‰SÒS. Note

that SY is the composite S S(Y,__)Ò SÒi ?‰S

iii Third, realizations do not distribute over Y. In other words, in general RY L (X)ϵ/ R (X)YR (X). L In fact,

RY K (X)œRY K (X)œ10(X)KY but

. R (X)YR (X)K œ10(X)YK

More generally, one has the following fact which we will use heavily and is responsible for many of the properties of the simplicial and homotopy structures and their relations.

9.9 Proposition:

Let be a cosimplicial set (res. a cosimplicial topological space) andY let be a set (res. a topological space) then there exists a naturalZ isomorphism a RÀ Y Z ( )ÒR ( ) Y Z which is also natural (of course) in and . Y Z

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ˆSY X(K)S(Yn X,K) (S(Y ,K))n X (S (K) )X n

‚ 8œ ‚ œµ œ Y

This functor (S ( ))Y X is therefore, according to the general theory, right adjoint to R ( )YX. Consequently, 10( )‚X as given before is left adjoint to ( )X where the underling again means the degenerated functor.

In the case of simplicial sets and topological spaces, one has the following well known relation which is responsible (together with the fact that the geometric realization distributes over products) for the transmission of homotopy from simplicial sets into homotopy of topological spaces for each À n in *, | [n]|? œµ?n where ?[n], denotes the standard simplicial n-simplex and ?n the standard (Milnor's) topological n-simplex. This fact will be called "singularity". More formally

9.10 Definition (Singular Pairs):

Let ˜ ?À Ò?‰S be a cosimplicial object of ?‰S and YÀ?ÒS a cosimplicial set. We say that the pair ( ,Y)˜ is a singular pair if the following diagram commutes up to natural isomorphism:

RY

S

Y

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R

Y

Top

Y

commutes up to natural homeomorphism.

Of course this can be rephrased by saying that the cosimplicial maps RY‰˜ and are isomorphic. Depending on the categories in use,Y one may have that singularity always holds when the cosimplicial simplicial set is the standard one of the simplicial ?[n]'s. More formally

9.11 Proposition:

Let ??À?Ò?‰S denote the cosimplicial object of ?‰S of the simplicial ?[n]'s. Then for any cosimplicial set the pair Y (??,Y) is singular i.e. there exist a natural isomorphism

RY?? ÒY (with inverse 3ÀYÒRY??)

That permits us to identify for each , n R (Y ??[m]) with Yn, and for each w [n]À Ò[m] in , ? R (Y ??(w)) with Y(w).

Proof:

We take -n ÀR ( [n])? Yn as the function defined by

Y Ò

-n p p ? ?

p

([r,y ])œr*(y ) where belongs to r [n] œ ([p],[n]). One can see that -n is well defined, 1-1 and onto and furthermore {-n}, n*, form a cosimplicial map.

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9.12 Proposition:

Let be a cosimplicial topological space. Then YRY?? ÒY with the same underlying function as in proposition 9.11 is a natural continuous set theoretical isomorphism.

Proof:

Since is the same map as in proposition 9.11 naturality follows, a -well as the set theoretical isomorphic situation. As for continuity it is enough to notice that if A is contained in Yn then

p ((-1 n -1) (A)) [n] Yr {a} (a*) (A)-1 r

- ∩? ‚ œ ∪ ‚

where a runs into ?[n]r.

Note that given a YÀ n ÒX there are two functions associated to aÀFirst since belongs to a S (X) Y n then there exist a unique simplicial map AÀ?[n]ÒS (X)Y such that a(1 )n œa, where 1n denotes the identity function of [n]. On the other hand since R ( [n])Y ? has been identified with Yn then by adjointness there exist (a) ÀYn ÒX. What we question now is the connection between and a (a) in general, since in the case of singularity they are identified. In the proposition below we show that relation and why in case of singularity they are in fact identifiable.

9.13 Proposition:

Let be a cosimplicial topological space and let Y a YÀ n ÒX be a continuous function.

Then:

i. A is given at the p-th level by A(r)œar* where r [p]À Ò[n] and r*œY(r).

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X

l

n

( a )

a

R ( [n])

Y

We concentrate now in realization and singular functors associated to cosimplicial simplicial sets ˜ ? Ò ?À ‰S. They induce functors which behave very much like realizations and singular functors except that the domain and range of them are both ?‰S. They correspond to the idea of Subdivision and Ext functors defined by Kan but since we are interested in their properties similar to the ones given in Part 1, we still denote them by S˜ and R˜.

Recal that if ˜ ?À Ò?‰S be a cosimplicial simplicial set then S˜ À?‰SÒ?‰S maps XÈS (X), ˜ the simplicial set whose n-th level is S (X)Y n œ?‰S(˜n,X) and if w [n]À Ò[m] then (S (X))(w)˜ maps rÈrw* for rS (X)˜ m. For a simplicial function f XÀ ÒZ, S (f)˜ is the simplicial function whose n-th level S (f)˜ n ÀS (X)˜ n ÒS (Z)˜ n maps aÈfa for each in a S (X)Y n.

One can see that this looks very much like the standard singular functor from X 9: into ?‰S, except that the one used here is a "simplicial version" of it. The definition of the realization functor R˜ À?‰SÒ?‰S is a little more complicated and we review it now.

We begin by noticing that if ˜ ?À Ò?‰S is a cosimplicial simplicial set then for each in p *, ˜ defines a cosimplicial set ˜p À?Ò?‰S given by ˜p([n])œ˜np and if w [n]À Ò[m] then

˜p(w)œ(Y(w))p.

One the other hand, if r [p]À Ò[q], then there exists a cosimplicial map that we will denote by

Y( )(r) Y Y

q p

À Ò

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Since in this case the precise meaning of the maps involved is particularly important, we will use functorial rather than simplicial notation. So, for example we will use ˜(w)p instead of w*p.

The definition of R˜ À?‰SÒ?‰S is the following: Let X be a simplicial set. R (X)˜ is the simplicial set whose p-th level is R (X)Yp . If r [p]À Ò[q], one uses the natural transformation

Y( )(r)

q p

À˜ Ò˜

to define R (X)(r)˜ as the function R (X)(r)˜ œR ( ) (X) ˜ (r) which, as we recall, is given by

R (X)˜q ÒR (X); [X ,Y ]˜p n qn È[X,Y (r)(y )]n qn

From the properties of realizations ?‰SÒS, it is possible to prove that R (X)˜ so defined is in fact a simplicial set. As for the action of R˜ on simplicial functions, if f XÀ ÒK is a simplicial map, then R (f)˜ is defined by

R (f)˜ p œR (f)˜p

Again, one can prove that the family of functions R (f)˜p define a simplicial map R (f) R (X)˜ À ˜ ÒR (K)˜ and that R˜ is a covariant functor.

As we mentioned before all of the properties which we have developed for realization and singular functors also hold for R˜ and S˜.

9.14 Proposition:

The pair (R ,S )˜ ˜ is an adjoint pair. Proof:

For each simplicial set we takeX

(<X n) ÀXn Ò?‰S(˜n,R (X))˜

is the function which sends xnXn into the simplicial map (<X n) (x )n À˜n ÒR (X)˜

whose p-th level is

((<X n) (x ))n p À˜np ÒR (X); y˜p pn È[x ,y ]n pn

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X

K

f

S R (X)

y

X

y

K

S R (K)

S R ( f )

Hence <À1?‰SÒS R˜ ˜ is a natural transformation.

On the other hand for each simplicial set X we take :X ÀR S (X)˜ ˜ ÒX to be the simplicial map whose p-th level is

(:X)p R (S (X))p X ; [r,Y ]p r (Y )p

n n

p p

À ˜ ˜ Ò È

which can be shown to be well defined. One can see easily that :X is a natural transformation :ÀR SY Y Ò1?‰S.

Finally, one completes the proof of adjointness from the commutativity of the following two diagrams for a R (X)À ˜ ÒK and b XÀ ÒS (K)˜ À

K

a

R S R (X)

j

K

R S (K)

R S ( a )

R (

y

X

)

R (X)

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X

b

S (R (X))

ψ

X

S R S (K)

S R ( b )

S (

ϕ

K

)

S (X)

That ends the proof that (R ,S )˜ ˜ is an adjoint pair.

As for naturality of R˜ and S˜ with respect to they are given in the˜ following proposition.

9.15 Proposition:

Let - ˜À Ò™ be a cosimplicial simplicial map. Then: i. The correspondence

S (X) S (X)- À ™ ÒS˜(X)

given at the n-th level by

S (X)- n À?‰S(Z ,X)n Ò?‰S(Y ,X); an Èa-n

defines a simplicial map, for each simplicial set X , and S- ÀSÒS˜ is a natural transformation.

ii. The correspondence R (X) R (X)- À ˜ ÒR (X)™ given at the p-th level by

(R (X))- p œR ( )(X) R (X)p À ˜p ÒR (X)p

is a simplicial map and RR˜ ÒR™ is a natural transformation.

As far as retracts are concerned, we have seen that their properties are consequences of naturality and the diagrams of the set theoretical and topological cases can be used as well for the proof of the following proposition.

9.16 Proposition:

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i. For each simplicial set , X R (X)˜ is a retract of R (X)™ and S (X)˜ is a retract of S (X)™ .

ii. For each simplicial map , f R (f)˜ is a retract of R (f)™ and S (f)˜ is a retract of S (f)™ .

iii. For each simplicial set , X S R (X)˜ ˜ is a retract of S R (X)™ ™ and R S (X)˜ ˜ is a retract of R S (X)™ ™ .

iv. For each simplicial map , f S R (f)˜ ˜ is a retract of S R (f)™ ™ and R S (f)˜ ˜ is a retract of R S (f)™ ™ .

Note that, again, if is a cosimplicial simplicial point of , meaning a˜ ™ cosimplicial simplicial subset of each of whose levels is a simplicial™ point, then is a retract of . In that case ˜ ™ R (X)˜ is a degenerated simplicial set whose p-th level is 10(X) and S (X)˜ is a degenerated simplicial set whose p-th level is the set of simplicial points of . IfX we call SP(X) the set of simplicial points of , then X S (X)˜ œSP(X). Hence the pair ( 10( ) , SP( ) ) is an adjoint pair. Furthermore 10( )X is a retract of R (X)™ and SP(X) is a retract of S (X)™ . Similarly 10(f) is a retract of R (f)™ and SP( )0 (where SP( ) SP(X)0 À ÒSP(X) is the function which sends a simplicial point Bn, n* into the simplicial point f (x ), nn n* ) is a retract of S (f)Z . Hence if it is known that R (f)™ is for example a cofibration so is 10(f) and similarly for weak equivalences. The same thing can be said about SP(f) and S (f)™ . The case of fibrations is of no interest here since 10(f) and SP(f), being degenerated simplicial maps, are Kan fibrations independently of R (f)™ (res. S (f)Z ) being so.

9.17 Remark (On cosimplicial points):

It is important to notice that contrary to what happens to simplicial sets which always admit simplicial points, which are therefore retracts of the original set, cosimplicial set do not necessarily admit cosimplicial points. For example the underlying cosimplicial set of the cosimplicial topological space of Milnor's topological n-simplexes does not admit a cosimplicial point. Similarly cosimplicial simplicial sets do not necessarily admit cosimplicial simplicial points, which is clear from the example above since one can take ˜ to be the cosimplicial simplicial set whose (n,p) level is Yn .

p n

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We consider now the concept of singular pairs, at the cosimplicial simplicial level:

9.18 Definition:

Let ˜ ?À Ò?'S and ™ ?À Ò?‰S be cosimplicial simplicial sets. We say that the pair ( , )˜ ™ is a singular pair if the following diagram commutes up to natural isomorphism:

R

We will also say that is singular for .˜ ™

We want to show that, here again, the cosimplicial simplicial set of the standard models is singular for any . In order to do so we need™ a small technical device.

9.19 Proposition:

Let Y, Z be cosimplicial sets and let (ÀYÒZ be a cosimplicial function. Let -Y ÀRY?? Ò Y and -Z ÀR?? ÒZ be the natural isomorphisms associated to the singular pairs (??,Y) and (??,Z). Finally let R(??ÀRY?? ÒRZ?? be the natural transformation give by R(??([n])œR ( [n])( ? . Then the following diagram commutesÀ

R

Y

Z

Y

R

Z

R

η

λ

Y

λ

Z

η

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The desired commutativity of the diagram

R

Y

( [n])

Z

n

Y

n

R

Z

( [n])

R

η

( [n])

λ

Y

λ

Z

η

n

n n

follows immediately from the fact that since (ÀYÒZ is a cosimplicial function, then for each r [p]À Ò[n] the following diagram commutes:

Z

p

Y

p

η

p

Y( r )

Z( r )

Z

n

Y

n

η

n

We give now the implication of this result:

9.20 Proposition:

Let ˜ ?À Ò?‰S be an arbitrary cosimplicial simplicial set. For each p* let -Yp ÀR˜?? Ò˜p be the natural isomorphism associated to the singular pair (??,˜p). Then, the family -Yp, p* defines a natural isomorphism

-˜ ÀR˜?? Ò˜

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a simplicial map aÀ?[n] ÒS (X)˜ since aS (X)˜ n. Then aœa* since a*(1 )n œa by the above identification. Therefore the simplicial function associated to by adjointness is again i.e.a a aœa.

This apparently unimportant fact will be responsible (whenever R˜ commutes with finite products) for the identification of homotopy groups 1n(S (X))˜ with the groups defined using Y homotopy i.e. homotopy where ?[1] has been replaced by Y1 and where the "n-th" Y homotopy group of X" is defined using diagrams of the kind

n

×

1

n

X

n

(See the paragraph of -homotopy of the chapter "The Change of˜ models in?‰S")

9.21 Remarks (On the model ??):

i. We have used repeatedly the following property of the standard models:

Let be a simplicial set and let X xnXn. Then there exists a unique simplicial function xn À?[n]ÒX such that (x ) (1 )n n n œxn ,where 1n stands for the identity function

[n] Ò[n].

Of course it simply means that there exists a natural isomorphism

S??Ò1?‰S

When so interpreted, one has the immediate consequence that there exists also a natural isomorphism

R?? Ò1?‰S

since R?? is right adjoint to S .??

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the standard one. In other words when the functor of the˜ generalization is ??, the theory of -lifting, the -homotopy˜ ˜ theory and the -simplicial structure of ˜ ?‰S are the standard ones.

ii. If ( ,Y)˜ is a singular pair, ˜ ?À Ò?‰S and YÀ?ÒS (res. YÀ?ÒX 9:), then in addition to the identification R (Y ˜n)œYn (and R ( (w))Y ˜ œY(w) ) one also has the following useful one

R ( Y )Y ` n œ `Yn

where ` œ ∪ and ` œ . This

œ

Y d (Y ) Y n d (Y )

i 0

n i n 1 n - i n 1

becomes especially important in the study of ˜-homotopy groups since in that case the -homotopy is relative to ˜ `˜n, (`Y for Yn ) and the above fact implies that RY preserves relative homotopies.

9.22 Example (Subdivision and Ex):

In the paper "On C.C. complexes" Kan defines the functors Sd (Subdivision) and Ex. This is done by first defining a "subdivision" of the standard models. That is to say Kan defines a functor

??'À?Ò?‰S

where ??'([n]) or as he denotes it, ?'[n] is (up to natural isomorphism) the nerve of the following category: The objects are injections [p]Ò[n] and morphisms from [p]Ò[n] to [q] Ò[n] are injections [p] Ò[q] such that the followings triangle commutes

[ p ]

[ q ]

[ n ]

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becomes a particular case of -homotopy, for ˜ ˜ ?À Ò?‰S. The idea of change of models in ?‰S that we have introduced shows that much more than homotopy can be developed using ??'. In fact new fibrations (which replace Kan fibrations), cofibrations and the like, which work amazingly well as far as liftings are concerned, are available and the homotopy itself is a "simplicial" homotopy (See the ˜-simplicial system in ?‰S The theory that we develop here,).

however, is given in a general fashion and leaves open the discussion of the applications of it when the functor is Kan's˜ ??'.

The following is an extension of a previous proposition whose proof, being simple, is omitted.

9.23 Proposition:

Let be a cosimplicial simplicial set. Let be a simplicial set and ˜ X X the degenerated cosimplicial simplicial set associated to . ThereX exists a natural isomorphism R˜X ÒR ( )˜ ‚X, an extension of R˜pXp ÒR ( )˜pX . p

Notice that

(S˜X)(K)n œ?‰S((˜‚X) ,K)n œ?‰S(˜nX,K)

which is the -version of the function complex ˜ KX. This shows that the function functor ( )X is right adjoint to ( )X since

R??X(K)œR (K)??XœµKX

and this well known fact for the standard models extends to any cosimplicial simplicial set ˜

9.24 Corollary:

any cosimplicial simplicial set ˜

Figure

Fig.  1SZ(K)SY(K) S Z (K)Sr(K)Si(K)SZ( f )SY( f )SZ ( f )SZ(L)SY(L)SZ(L)Sr(L)Si(L)
Fig. 1SZ(K)SY(K) S Z (K)Sr(K)Si(K)SZ( f )SY( f )SZ ( f )SZ(L)SY(L)SZ(L)Sr(L)Si(L) p.5

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