Overview on models in homotopical algebra.

Resumen

Un funtor covariante [DELTA] [flecha diestra] A se dice un objecto modelo de A. Los objetos modelo producen en A un tema de estudio muy parecido a la topologia algebraica cuando A es la categoria de los espacios topologicos. En este trabajo se describen los escenarios en los cuales se desarrollan estos conceptos y las principales resultados desarrollados por el autor sobre objetos modelos.

Palabras clave: Categoria modelo, objetos simpliciales, cosimpliciales, homotopia, levantamiento.

Abstract

A covariant functor [DELTA] [right arrow] A is called a model object of A. Model objects produce in A a subject matter very much as algebraic topology when A is the category of topological spaces. Here we describe the settings on which such concepts are developed and describe the main features developed by the author about model objects.

Key words: Model category, Simplicial objects, Cosimplicial, Homotopy, Lifting.

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1. Model, closed model, and pre-model categories

By a model category (Daniel Quillen [QD67]) we mean a category A together with three distinguished classes of morphisms F (fibrations), C (cofibrations) and WE (weak equivalences) such that:

M.0. Axiom of admissibility: A is closed under finite projective and inductive limits.

M. 1. Lifting axiom: Given a solid arrow diagram in A, where i is a cofibrations and p is a fibration and where either i or p is a weak equivalence, then the dotted arrow exists.

M.2. Factorization axiom: Any morphism f in A can be factored f = pi where i is a cofibration and weak equivalence and p is a fibration. Also f = pi where i is a cofibration and p is a fibration and a weak equivalence.

M.3. F is closed under composition, base change and any isomorphism is a fibration. C is closed under composition, cobase change and any isomorphism is a cofibration.

M.4. The base extension of a morphism which is a fibration and weak equivalence is a weak equivalence. The cobase extension of a map which a cofibration and weak equivalence is a weak equivalence.

M.5. Triangular axiom: If in a commutative diagram

two of the morphisms are weak equivalences so is the third.

If (A, F, C, WE) is a model category, then there exists an associated category, called the homotopy category of (A,F,C,WE) denoted simply by [H.sub.[omicron]]A which is the localization of A with respect to WE. [H.sub.[omicron]]A is characterized by the existence of a functor r : A [right arrow] [H.sub.[omicron]]A and the following universal property of (r, [H.sub.[omicron]]A): for every f [member of] WE, r(f) is an isomorphism. If there exists another pair (F,B), F : A [right arrow] B such that for every f [member of] WE, F(f) is an isomorphism then, there exists a unique functor p : [H.sub.[omicron]]A [right arrow] B such that [rho]r = F.

Note that in general, if r(f) is an isomorphism, f need not be a weak equivalence. There is however, a special kind of model category in which r(f) is an isomorphism if, and only if f is a weak equivalence. They are called closed model categories. Before we define them, we give some notation: In a model category a morphism which is both a fibration and a weak equivalence is called a trivial fibration. TF denotes the class of such morphisms. A morphism which is both a cofibration and a weak equivalence is called a trivial cofibration, TC denotes the class of such maps. We call the classes F, C, TC, TF and WE the classes of structural maps of the model category.

From our point of view, the main feature of closed model categories is that, with the exception of WE, the classes of structural maps are characterized by lifting properties: PM2 in next definition.

On the other hand one can also have "almost" model categories which fail to be model because WE fails to behave well. They are "pre model categories".

By a pre-model category, (Roberto Ruiz, [RR76]) we mean a category A together with five classes of maps F, C, TF, TC and WE such that:

P.M.I. TF [subset or equal to] F and WE = TF [omicron] TC (i.e. a map is a weak equivalence if and only if it factors as a trivial cofibration followed by a trivial fibration).

P.M.2. F, C, TF, TC admit the following characterization by liftings.

i f is a fibration if and only if f has right lifting property with respect to TC.

ii f is a trivial fibration if and only if f has right lifting property with respect to C.

iii If f has left lifting property with respect to TF, then f is a cofibration.

iv If f has left lifting property with respect to F, then f is trivial cofibration.

P.M.3 Any map f admits two factorizations:

and

where h is a cofibrations and i is a trivial fibration and h' is a trivial cofibration and i' is a fibration.

Remarks:

i i and ii implies that iii and iv in P.M.2 become equivalences.

ii Any isomorphism belong to each one of the classes F, C, TF, TC (P.M.2) and hence to WE.

iii F, C, TF, TC are closed under composition and retracts.

iv TC [subset or equal to] C, and furthermore TF = F [intersection] WE and TC = C [intersection] WE.

v Let

be a Cartesian square in A. If f is a fibration (resp. trivial fibration) so is [bar.f]. Let

be a cocartesian square in A. If f is a cofibration (resp. a trivial cofibration) so is f.

vi F [intersection] C [intersection] WE = isomorphisms of A.

Thus a pre model category have good behavior in liftings (as good as in closed model categories) but bad behaved WE: it only goes up to containing all isomorphisms and factorization WE = TF [omicron] TC.

The "closure" of a model category is define as follows [RR77J: given a model category (A, F, C, WE) there exists a unique pre model category (A, [bar.F], [bar.C], [bar.TF], [bar.TC], [bar.WE]) such that F [subset or equal to] [bar.F], C [subset or equal to] [bar.C],TF [subset or equal to] [bar.TF], TC [subset or equal to] [bar.TC] and WE [subset or equal to] [bar.WE]. This pre model category measures the extent in which a model category is a closed model category. It is proved that when the original model category is closed, it coincides with its closure. The closure can also be viewed as "the theory of liftings" of the model category. The uniqueness of the closure is implied by the following fact, meaningful in itself. If 21 is a class of morphisms in A and we denote by  the class of all retracts of members of 21, then one has that [bar.F] = [F], [bar.C] = [C], [bar.TF] = [TF], [bar.TC] = [TC].

Hence a model category is closed if and only if the classes F, C, TF, TC (or equivalently F, C, WE) are closed under retracts. In pre model, model and closed model categories a "cylinder" object (resp. "path" object) of A is a diagram as the next first (resp. second) one.

C is the "actual" cylinder object of A, [i.sub.j] (j = 0,1) are trivial cofibrations (which are usually inclusions) and P is the actual "path object" of A and [C.sup.j] are trivial fibrations (which usually are evaluation functions). Then one has for f, g : A [right arrow] B that f is "left homotopic" to g if there exists a cylinder of A and h : C [right arrow] B such that h [omicron] [i.sub.0] = f and h [omicron] [i.sub.1] = g. The dual procedure provides "right homotopy". If the map [phi] [right arrow] A ([phi] initial object) belongs to C (A is a "co fibrant object") then left homotopy is an equivalence relation on Hom(A, B) and left homotopy implies right homotopy [QD67]. Since dual assertions hold, right and left homotopy coincide and are equivalence relations on Hom(A,B) when A is co fibrant and B is fibrant.

Elsewhere homotopy is done through homotopy systems: Let A be a category. A homotopy system (Kan, [KD55,58]) Z = (I, [J.sub.0], [J.sub.1],q) consists of the following:

i A "cylinder" covariant functor I : A [right arrow] A.

ii Three natural transformations [J.sub.0] : [1.sub.A] [right arrow] I, [J.sub.1] : [1.sub.A] [right arrow] I, g = I [right arrow] [1.sub.A] such that q[J.sub.0] = q[J.sub.1] = 1.

Homotopy is then given as follows: Let f,g:X[right arrow]Y be morphisms and A. We say that f is homotopic to g, denoted f [congruent to] g, if there exists a morphism [rho] : I(X) [right arrow] Y in A such that [ho] [omicron] [J.sub.0](X) = f and [rho] [omicron] [J.sub.1] (X) = g.

The homotopy relation as defined is not in general an equivalence relation. But it is reflexive and compatible with composition: if f,g : X [right arrow] Y, h : Y [right arrow] Z, k : K [right arrow] X then f [equivalent] g, implies that hf [equivalent] hg and fk [equivalent] gk.

If ~ denotes the equivalence relation generated by [equivalent], then one has the following definition: Let f : X [right arrow] Y a morphism in A. We say that f is a homotopy equivalence if [f] [member of] A(X, Y)/ ~ is an isomorphism in Mor A/~

Fibrations and cofibrations for a homotopy system are given in Kamps [KK69]. We will provide now examples of homotopy systems. The most common ones are given in categories with final object different from the initial one, if it exists. We will denote by * the final object and by 0 the initial one.

If A is a category, then say that A is pointed if * [congruent to] 0. Otherwise is unpointed. Let A be unpointed. For an object A of A the morphisms * [right arrow] A are called the points of A and A(*, A) is called the underlying set of A. In unpointed categories there are plenty of homotopy systems. In fact, Let A be an unpointed category closed for finite products. Let X be an object of A and (if that order) points of X. Then there exists a natural isomorphism [iota] : [1.sub.A] [right arrow] [1.sub.A] x *, and the bimap I : A [right arrow] A, A [??] A x X, f [??] f x [1.sub.X] is a covariant functor. Further, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are natural transformations on A [member of] ObjA and [[pi].sub.1] [omicron] [d.sub.i] (A) = [1.sub.A]. Thus (I, [d.sub.0], [d.sub.1],s) is a homotopy system on A.

Normal homotopies in the categories of topological spaces Top and simplicial sets [[DELTA].sup.[omicron]]S are of this kind, the first one induced by {I = [0,1],0,1} and the second by {[DELTA], [[epsilon].sup.0], [[epsilon].sup.1]} where [[epsilon].sup.i] :  [right arrow] , 0 [??] i induce simplicial points (still denoted by [[epsilon].sup.0], [[epsilon].sup.1]) on the second. We assume the reader familiar with them.

Let Z = (I, [J.sub.0], [J.sub.1], q) be a homotopy system in A. We say that a map F : E [right arrow] B is a fibration (or a Z fibration) if f has the right lifting property with respect to [J.sub.0] i.e. to the class of maps [J.sub.0](X) : X [right arrow] I(X), X [member of] ObjA. A map i : A [right arrow] X is called a (Z) cofibration if for any commutative diagram.

there exists a homotopy [bar.[phi]] : I(X) [right arrow] Y such that [bar.[phi]I(i) = g and [bar.[phi]][J.sub.0](X) = f.

One has the following properties for fibrations and cofibrations in a homotopy system [KK69]: Isomorphisms are fibrations and cofibrations. Projections are fibrations. Fibrations are closed under composition, base extension, and retracts. Cofibrations are closed under composition, co-base extension, and retracts.

2. Simplicial Systems, category, functor

We change a little Quillen's version of simplicial categories [QD67] to the notion of simplicial systems in a given category. The reason is that, as we will see, there may be more than one way in which a category is a simplicial category. For us then a "simplicial category" will be a pair formed by a category and a "simplicial system".

Let A be a category. By a simplicial system in A we mean a functor [[Hom.bar].sub.A] : A x A [right arrow] [[DELTA].sup.[omicron]]S such that the following conditions hold:

S.1. For any objects X, Y, Z of A there exists a "composition" (simplicial) map:

[[Hom.bar].sub.A] (X, Y) x [[Hom.bar].sub.A] (Y, Z) [right arrow] [[Hom.bar].sub.A] (X. Z)

level wise denoted by (f, g) [??] g [omicron] f, which is associative in the sense that, if f [member of] [[Hom.bar].sub.A] [(X,Y).sub.n]. g [member of] [[Hom.bar].sub.A] [(Y, Z).sub.n], and h [member of] [[Hom.bar].sub.A] [(Z, K).sub.n], then (h [omicron] f) [omicron] g = h [omicron] (f [omicron] g).

S.2. There exists a natural isomorphism

[lambda]:A(,) [right arrow] [[[[Hom.bar].sub.A] (, )].sub.0]

denoted by A(X,Y) [right arrow] [([[Hom.bar].sub.A] (X,Y)).sub.0;] u [??] [??] such that if u [member of] A(X,Y), f G [[Hom.bar].sub.A](Y, Z)" and g [member of] [[Hom.bar].sub.A] [(W,X).sub.n] then

[s.sup.n.sub.0] (u) = [[Hom.bar].sub.A] [(u,Z).sub.n](f)

and

[s.sup.nsub.0](u) [omicron] g = [[Hom.bar].sub.A] [(W,u).sub.n](g)

where (abusing notation) [s.sup.n.sub.0] denotes the composite of the (in general different) functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By a simplicial category we mean a pair (.A, [[Hom.bar].sub.A]) where A is a category and [[Hom.bar].sub.A] is a simplicial system on A. As for functors among them:

Let (A, [[Hom.bar].sub.A]) and (B, [[Hom.bar].sub.B]) be two simplicial categories. By a simplicial functor

F : (A. [[Hom.bar].sub.A]) [right arrow] (B, [[Hom.bar].sub.B])

we mean a functor F : A [right arrow] B together with maps

[[Hom.bar].sub.A] (X, Y) [right arrow] [[Hom.bar].sub.A] (F(X), F (Y)); f [??] F(f)

such that F(f [omicron] g) = F(f) [omicron] F (g) and F([??]) =F([??]). We say that F is strictly simplicial if the maps

[[Hom.bar].sub.A] (X,Y) [right arrow] [[Hom.bar].sub.B] (F(X), F(Y))

define a natural transformation (denoted again by F).

F : [[Hom.bar].sub.A] [right arrow] [[Hom.bar].sub.B] (F x F)

Since the simplicial functors that we will use are always strictly simplicial we will talk simply of "simplicial functors" and refer to the "strict" part only when specially necessary.

Simplicial categories have cylinders and path objects: Let (A, [[Hom.bar].sub.A]) be a simplicial category. Let X be an object of A and K a simplicial set.

i By a cylinder object associated to (X, K) we mean a pair (X [cross product] K, [alpha]) where X [cross product] K is an object of A and

[alpha] : K [right arrow] [[Hom.bar].sub.A] (X, X [cross product] K)

is a simplicial map such that for each Y in A the simplicial map

[phi] : [[Hom.bar].sub.A] (X x K,Y) [right arrow] [([[Hom.bar].sub.A] (X.Y)).sup.K]

next defined is an isomorphism: <pn has domain [[Hom.bar].sub.A] [(X [cross product] K, Y).sub.n], codomain [[DELTA].sup.[omicron]] S (K x [DELTA][n], [[Hom.bar].sub.A] (X,Y)) and [rho] [??] (o) [omicron]([alpha]x 1)[omicron](1 x [[rho].bar]). More explictly the image of [rho] is the composition of the maps: 1 x [[rho].bar] with domain K x [DELTA][n], and codomain K x [[Hom.bar].sub.A] (X x K,Y); [alpha] x 1 has domain K x [[Hom.bar].sub.A] (X x K,Y), and codomain [[Hom.bar].sub.A] (X,X [cross product] K) x [[Hom.bar].sub.A] [cross product] K,Y); o with domain [[Hom.bar].sub.A] (X.X [cross product] K) x [[Hom.bar].sub.A] (X [cross product] K,Y), and codomain [[Hom.bar].sub.A] (X,Y) where

[[rho].bar] : [DELTA][n] [[Hom.bar].sub.A] (X [cross product] K, Y)

is the simplicial map associated to [rho], namely the unique simplicial map such that [[rho].bar]([l.sub.[n]]) = [rho].

ii By a path object associated to (X,K) we mean a pair ([X.sup.K], [beta]) where [X.sup.K] is an object of A and [beta] : K [right arrow] [[Hom.bar].sub.A](XK. X) is a simplicial function such that the induced map.

[psi] : [[Hom.bar].sub.A](Y, [X.sup.K]) [right arrow] [[Hom.bar].sub.A][(Y, X).sup.K]

described below is an isomorphism: [[psi].sub.n] with domain [[Hom.bar].sub.A] [(Y, [X.sup.F]).sub.n], and codomain

(K x [DELTA][n], [[Hom.bar].sub.A](Y.X)),

where

[rho] [??] (o) [omicron] ([Pr.sub.2], [Pr.sub.1]) [omicron] (1 x [[rho].bar])

i.e. the image of [rho] is the composition of the maps: 1 x [rho] with domain K x [DELTA] [n], and codomain K x [[Hom.bar].sub.A](Y, [X.sup.K]); ([Pr.sub.1], [Pr.sub.2]) with domain K x [[Hom.bar].sub.A](Y, [X.sup.K]), and codomain [[Hom.bar].sub.A](Y, [X.sup.K]) x [[Hom.bar].sub.A]([X.sup.K],X): o with domain [[Hom.bar].sub.A](Y, [X.sup.K]) x [[Hom.bar].sub.A] ([X.sup.K], X). and domain [[Hom.bar].sub.A](Y. X)

Let (A, [[Hom.bar].sub.A]) be simplicial category. Let X be an object of A and K, L simplicial sets. There are canonical isomorphisms X [cross product] (K x L) [congruent to] (X [cross product] K) [cross product] L and [([X.sup.K]).sup.L] [congruent to] [X.sup.K[cross product]L] when all of the objects involved are defined.

Recall that the homotopy relation on simplicial sets in general is not an equivalence relation. We denote by [equivalent] the homotopy relation in [[DELTA].sup.[omicron]]S and by ~ the equivalence relation induced by [equivalent]. Recall further that the functional simplicial set associated to spaces X and Y (where the idea of simplicial categories was taken from) is the simplicial set Hom(X, Y) whose n-th level is given as follows: [[DELTA].bar] denotes the cosimplicial space with [[[DELTA].bar].sup.n] = {([x.sub.0],..., [x.sub.n]) [member of] [R.sup.n+1] | [x.sub.i] [greater than or equal to] 0, [SIGMA] [x.sub.i] = 1} the co faces [[delta].sup.i]: [[[DELTA].bar].sup.n] [right arrow] [[[DELTA].bar].sup.n+1] is the function that adds 0 in the i-th coordinate and [[rho].sup.i] : [[[DELTA].bar].sup.n] [right arrow] [[[DELTA].bar].sup.n-1], [x.sub.i] + [x.sub.i+1] in the i-th coordinate. [[DELTA].bar] will be called "the standard model" in Top. Now, back to Hom(X,Y) we take

[Hom.bar][(X, Y).sub.n] = Top ([[[DELTA].bar].sup.n] x X, Y)

faces and degeneracies induced by those of [[DELTA].bar]. It is clear that

[Hom.bar][(X, Y).sub.0] [congruent tl] Top{X, Y)

and is well known that two maps f,g : X [right arrow] Y are homotopic in Top if as members of [Hom.bar][(X, Y).sub.0] they are homotopic. The generalization of this situation to simplicial categories is as follows: Let (4, [[Hom.bar].sub.A]) be a simplicial category, f, g : X [right arrow] Y be morphisms in A. We say that f is strictly homotopic to g, and denote it f [equivalent] g, if their images [bar.f], [bar.g] [member of] [[Hom.bar].sub.A][(X,Y).sub.0] are homotopic i.e. if [bar.f] [equivalent] [bar.g]. We say that f is homotopic to g, denoted f ~ g if [bar.f] ~ [bar.g] in [[Hom.bar].sub.A](X, Y). We denote [[pi].sub.0]([[Hom.bar].sub.A](X, Y)) by [[pi].sub.0](X, Y). The category [[pi].sub.0]A is defined as having as objects those of A and for each pair X, Y of ObjA = (Obj[[rho].sub.0]A).

[[pi].sub.0]A(X,Y) = [[pi].sub.0][[[Hom.bar].sub.A](X,Y)]

with composition induced by the one in A.

The existence of path and cylinder objects provides a nice representation of [[Hom.bar].sub.A](X, Y). In fact if (A, [[Hom.bar].sub.A]) is a simplicial category and B is a subcategory of [[DELTA].sup.[omicron]]S, then we say that A admits a cylinder through B if there exists a functor A x B [right arrow] A such that for each X [member of] A and K [member of] B the image of (X,K) is a cylinder object of X in A, say

We say that A admits paths through B if there exists a functor A x [B.sup.[omicron]] [right arrow] A such that for each X [member of] A and K [member of] B the image of (X, F) is a path object of X in A say

Now we consider [DELTA] : [DELTA] [right arrow] [[DELTA].sup.[omicron]]S, the standard models of [[DELTA].sup.[omicron]]5. On it [[DELTA].sub.n] = [DELTA][n] is the simplicial set with [DELTA][[n].sub.m] the set of increasing functions [n] [right arrow] [m] where [p] = {0, 1,...,p}. [d.sub.i] : [DELTA][[n].sub.m] [right arrow] [DELTA][[n].sub.m-1] maps [alpha] : [m] [right arrow] [n] in [alpha] [omicron] [[delta].sub.i] where [[delta].sub.i] : [m - 1] [right arrow] [m] is the 1 - 1 and increasing function which misses i in [m]. Further [[rho].sub.j] : [DELTA][[n].sub.m] [right arrow] [DELTA][[n].sub.m+1] maps a in [alpha] [omicron] [[rho].sub.j] where [[rho].sub.j] : [m+ 1] [right arrow] [m] is the onto increasing function which repeats j in [m]. [DELTA][n] is thus a simplicial set. The [DELTA] [n]'s form a cosimplicial object of [[DELTA].sup.[omicron]]S whose n-th level is [DELTA][n], [d.sup.i.sub.p] : [DELTA][[n].sub.p] [right arrow] [DELTA][[n + 1].sub.p] sends [alpha] in [[delta].sub.i] [omicron] [alpha] and [s.sup.j.sub.p] : [DELTA][[n].sub.p] [right arrow] [DELTA][[n - 1].sub.p] sends [alpha] in [[rho].sub.j] [omicro] [alph a]. One thus have that [DELTA][n], [d.sup.i] : [DELTA][n] [right arrow] [DELTA][n + 1] and [s.sup.j] : [DELTA][n] [right arrow] [DELTA][n - 1] conform a cosimplicial object of [[DELTA].sup.[omicron]]S, "the model of the simplicial [DELTA][n]'s".

We denoted by [DELTA] again the image [DELTA] ([DELTA]) which is a subcategory of [[DELTA].sup.[omicron]]S.

Suppose that A admits cylinders through [DELTA]. Then for each X [member of] A there exist a composition functor [DELTA] [right arrow] A x [DELTA] [right arrow] A with [n] [??] X [cross product] [DELTA][n] and (w : [n] [right arrow] [m]) [??] ([1.sub.X] [cross product] [w.sup.X] : X [cross product] [DELTA][n] [right arrow] X [cross product] [DELTA][m])

This composition is a cosimplicial object of A. Similarly if A admits paths through [DELTA], then for each X [member of] A one has a simplicial object of A, [DELTA] [right arrow] A x [DELTA] [right arrow] A with [n][??](X, [DELTA][n])t[right arrow][X.sup.[DELTA][n]] and (w : [n] [right arrow] [m]) [??] ([1.sub.X], [w.sup.*]) [??] ([X.sup.[w.sup.*]]: [X.sup.[DELTA][m]] [right arrow] [X.sup.[DELTA][n]]).

One uses these two object to prove the following.

Suppose A admits cylinders through [DELTA]. Then for each pair X, Y of objects of A, [[Hom.bar].sub.A](X, Y) is (up to isomorphism) the simplicial set whose n-th level is [[Hom.bar].sub.A][(X,Y).sub.n] = A(X [cross product] [DELTA][n],Y). Suppose A admits paths through [DELTA]. Then for each X, Y objects of A, [[Hom.bar].sub.A](X. Y) is (up to isomorphism) the simplicial set whose n-th level is

[[Hom.bar].sub.A][(X,Y).sub.n] =A(X, [Y.sup.[DELTA][n]])

Therefore in case A admits cylinders through A, homotopy in A is a left homotopy. For f,g : X [right arrow] Y morphisms of A, f ~ g if and only if there exists a morphism H : X [cross product] [DELTA]  [right arrow] Y such that H [omicron] [d.sup.1] = f and H [omicron] [d.sup.0] = g.

Similarly when A admits paths through [DELTA], homotopy in A is a (right) homotopy: f ~ g if and only if there exists a morphism T : X [right arrow] [Y.sup.[DELTA]] in A such that [d.sub.1] [omicron] T = f and [d.sub.0] [omicron] Tg.

Of course when A admits path and cylinders through [DELTA], homotopy in A is given by the (then) equivalent ways above.

We present now in some detail the most typical example of a "model object" at work which at long last produces a model category. Then we introduce its generalization to "model object" (or simply "model") and show that they produce pre model categories. In order to deal with the difference (from model to pre model categories) we work the missing parts (homotopy) by means of the homotopy system of its simplicial system.

3 Kan Fibrations and Trivial Fibrations

Let X be a simplicial set. Let n [member of] N*. By a (simple) n-box we mean a subset {[x.sub.0],..., [[??].sub.k],..., [x.sub.n]} of [X.sub.n], (where [[??].sub.k] means that there is no element indexed on k), such that for each pair i,j = 0,1,2,..., n + 1, if i < j then [d.sub.i][x.sub.j] = [d.sub.j-1][x.sub.i]. By a trivial n-box of X we mean a subset {[x.sub.0],..., [x.sub.n+1]} of X such that for each pair i,j = 0,1,..., n + 1 with i < j then [d.sub.i][x.sub.i] = [d.sub.j+1][x.sub.i].

The equalities [d.sub.i][x.sub.j] = [d.sub.j-1][x.sub.i] for simple and trivial boxes will be refereed to as the compatibility relations of the [x.sub.i]'s. They are fulfilled emptily for n = 0. So 0--boxes of X are subsets of [X.sub.0] with one element, and trivial 0--boxes, pair ([x.sub.0], [x.sub.1]), [x.sub.0] not necessarily different from [x.sub.1].

Let S = {[x.sub.0],..., [[??].sub.k],..., [x.sub.n]} (respectively, S' = = {[x.sub.0],..., [x.sub.n+1]}). We say that x is a filler of S (resp. of S") if for each i [not equal to] k, [d.sub.i](x) = [x.sub.i] (resp. for each i, [d.sub.i] (x) = [x.sub.i]).

If f : X [right arrow] Y is a simplicial function and S = {[x.sub.0],..., [[??].sub.k],..., [x.sub.n]} is a box in X (resp. S' = {[x.sub.0],..., [x.sub.n+1]} is a trivial n--box in X) then f(S) = {f([x.sub.o]),... f([x.sub.n+1])} is an n-box in Y, (resp. f(S') is a trivial n--box in V).

Let f : X [right arrow] y be a simplicial map. f is said to be a Kan fibration if given any n--box S such that the image box f(S) admits a filler y [member of] [Y.sub.n+1], then there exists a filler x [member of] [x.sub.n+1] of S, such that [f.sub.n+1] (x) = y. f is said to be a trivial fibration if [f.sub.0] is onto and if given a trivial n-box S in X such that f(S) admits a filler y [member of] [Y.sub.n+1], then there exists a filler x [member of] [x.sub.n+1] of S such that f(x) = y.

A simplicial set X is called a Kan complex (also is said to be (Kan) fibrant, or to have Kan extension condition) if the simplicial function X [right arrow] * (where * is any simplicial point) is a Kan fibration.

Let X be a simplicial set and x [member of] [X.sub.n]. Recall that there exists a unique simplicial map x: [DELTA][n] [right arrow] X such that, x ([1.sub.[n]]) = x. It is clear that if f : X [right arrow] Y is a simplicial map, then [f.sub.[omicron]] [??] = [??].One also have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We call a set {[a.sub.0],..., [[??].sub.k],..., [a.sub.j]} = S with [a.sub.j] : [DELTA][n] [right arrow] X a functional n--box in X if [a.sub.j] [omicron] [d.sup.i-1] [omicron] [a.sub.i], for i < j i,j [not equal to] k. When [a.sub.k] is not omitted we call the set a trivial functional n--box in X. If X [??] Y, then we denote by f(S) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the functional n--box image of S.

We call [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a filler of S if a [omicron] [d.sub.i] = [a.sub.j] for i [not equal to] k. Similarly for a trivial functional n--box.

Fibrations and trivial fibrations can be given by means of functional boxes: f is a Kan fibration if and only if for each functional n-box S whose image f(S) admits a filler 6, there exists a filler a of S such that f [omicron] a = b. f is a trivial fibration if and only if [f.sub.0] is onto and for each functional trivial n--box S whose image f(S) admits a filler b there exists a filer a of 5 such that f [omicron] a = b. One can produce trivial boxes from standard ones. A simple but tedious proof can be supplied for the following: if S = {[x.sub.0],..., [[??].sub.k],..., [x.sub.n+1]} is an n--box (n > 1) in a simplicial set X, then the set {[[x.bar].sub.0],..., [[x.bar].sub.n]} [subset or equal to] [X.sub.n-1] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a trivial n--1 box in X, which we denote by [d.sub.k] (S).

If S = {[x.sub.0],..., [[??].sub.k],..., [x.sub.n+1]} is a trivial n--box, such that [d.sub.k] (S) admits a filler x, then

{[x.sub.k+1],..., [x.sub.k+1],..., [x.sub.n+1]}

is a trivial n--box.

It can be seen that any trivial fibration is a Kan fibration. Then one have a pre model category as follows: F is the class of Kan fibrations, TF is the class of trivial fibrations, C (resp. TC) is the class of maps with left lifting property for TF (resp. F) and WE (weak equivalences) is TF [omicron] TC. Quillen [QD67] proves that in fact F, C, WE is a closed model structure in [DELTA] [omicron]S, that is ([DELTA] [omicron] S, F, C, WE) is a closed model category.

The use of maps of the kind [DELTA][n] [right arrow] X on the theorems allows us to generalize the concepts of fibrations, trivial and Kan complexes changing maps [DELTA][n] [right arrow] X to maps [Y.sup.n] [right arrow] X where Y : [DELTA] [right arrow] [[DELTA].sup.[omicron]] S is a covariant functor. The version by boxes of the kind {[x.sub.0],..., [[??].sub.k],..., [x.sub.n+1]} and {[x.sub.0],..., [x.sub.n+1]} will be useful in order to relate the generalization through Y with the standard theory.

Now we provide the generalization of the concept of fibrations to get fibrations and trivial fibrations associated to a functor Y : [DELTA] [right arrow] A.

We reestablish in detail the general definitions so as to point out how the standard theory is in fact a particular case of the theory induced by Y. Before we complete the pre model category associated to Y we give the counterpart (for Y) of the theorems in the previous paragraph of fibrations and trivial fibrations (when [[DELTA].bar] was used).

Next we show how these two concepts, Y fibrations and Y trivial fibrations, can be characterized (as well as some of their properties) by the use of the singular functor [S.sub.Y]: A [right arrow] [DELTA] [omicron]S associated to Y. Finally we complete the pre model category associated to Y.

4. Y fibrations and Y Trivial Fibrations

The generalizations of the concepts of boxes, trivial boxes and fillers can be given as follows.

Definition:

i Let X be an object of A. By a simple [Y.sup.n] box (or a [Y.sup.n] box) in X we mean a family of maps {[a.sub.0],..., [[??].sub.k],..., [a.sub.n+1]}, 0 [menor que o igual a] k [menor que o igual a] n + 1, where [a.sub.i]: [Y.sup.n] [right arrow] X is such that if i < j, i, j [not equal to] k, then [a.sub.i] [omicron] [d.sup.i] = [a.sub.i] [omicron] [d.sup.j-1], [d.sup.i] [d.sup.j-1]: [y.sup.n-1] [right arrow] [Y.sup.n].

ii By a trivial [Y.sup.n] box for n [mayor que o igual a] 1 we mean a family {[a.sub.o],..., [a.sub.n+1]} of maps [a.sub.i] : [Y.sup.n] [right arrow] X such that [a.sub.j] [omicron] [d.sup.i] = [a.sub.i] o [d.sup.j-1] if i < j, then [a.sub.j] [omicron] [d.sup.i] = [a.sub.i] [omicron] [d.sup.j-1]. A trivial [y.sup.0] box is a family {[a.sub.0], [a.sub.1]}, [a.sub.i] : [y.sup.0] [right arrow] X.

iii Let S {[a.sub.0],..., [[??].sub.k],..., [a.sub.n+1]} be a [Y.sup.n] box (resp. S' = ([a.sub.'0],... [a.sub.'n+1]) be a trivial [Y.sup.n] box). By a filer of S (resp. S') we mean a map a : [Y.sup.n+1] [right arrow] X such that for each i [not equal to] k (resp. for each i) the following diagram commutes

If f : X [right arrow] Y is a simplicial map and if S = {[x.sub.0],..., [[??].sub.k],..., [x.sub.n+1]} is a [Y.sup.n] box in X then it is clear that {f [a.sub.0],..., f [[??].sub.k],..., f [a.sub.n+1]} is a [Y.sup.n] box in Y. Similarly the [Y.sup.n] box (resp. trivial [Y.sup.n] box) of the f [a.sub.i]'s will be called the image of S by f and will be denoted in both the simple and trivial case by f(S).

We require Y to have the following property : Let [K.sub.n+1] be any set formed with at least n - 1 of the maps [d.sup.i] : [y.sup.n] [right arrow] [y.sup.n+1] in A Then for any [K.sup.n+1] there exists an object B on A and a map j : B [right arrow] [Y.sup.n+1] such that each equation j [omicron] X = [d.sup.i] has solution (denoted [d.sub.i]|) and further if B' and j' : B' [right arrow] [Y.sup.n+1] admit solutions to j' [omicron] X = [d.sup.i] then there exists a unique H : B [right arrow] B' such that j' [omicron] H = j.

Thus j is unique up to isomorphism. When in [K.sub.n+1] the map [d.sup.k] is missing we denote j by i : Y[n + 1, k] [right arrow] [Y.sup.n+1] and when [K.sub.n+1] = {[d.sub.0], [d.sub.1],..., [d.sup.n]}, j is denoted by i : [delta][Y.sup.n+1] [right arrow] [Y.sup.n+1].

When [d.sup.i] : [Y.sup.n] [right arrow] [Y.sup.n+1] have underlying functions (and [Y.sup.n] underlying set) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Of course in general there exist a unique

[i.sub.k] : Y[m + 1, k] [right arrow] [delta][Y.sup.n+1]

such that (abusing the notation) i o [i.sub.k] = i.

i Let f : X [right arrow] K be a map in A. We say that f is a Y fibration if given any [Y.sup.n] box S such that f(S) admits a filler b, then there exists a filler a of S such that fa = b.

ii f is said to be a y trivial fibration if for any trivial [Y.sup.n] box whose image f(S) admits a filler b, there exists a filler a of S such that f a = b. Furthermore f has "the [y.sup.0] lifting property" i.e. for any b : [Y.sup.0] K there exists a : [y.sup.0] [right arrow] X such that f a = b.

iii X is y Kan complex (resp. F trivial complex) if X [right arrow] * is a y fibration (resp. Y trivial fibration).

Consider a family {[a.sub.o],..., [[??].sub.k],..., [a.sub.n+1]} = S, [a.sub.i] : [Y.sup.n] [right arrow] X (n [mayor que o igual a] 1). Then S is a [Y.sup.n] box (resp. a trivial [Y.sup.n] box) iff there exists a map (filler)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

such that for each i [not equal to] k then a [omicron] [d.sup.i]|= [a.sub.i]. The family {[a.sub.0],..., [a.sub.n+1]} is a trivial [Y.sup.n] box if there exits a map a : [delta][Y.sup.n] [right arrow] X such that a [omicron] [d.sup.i] = [a.sub.i] for each i.

The effect on Y fibrations and Y trivial fibrations is the following:

Let f : X [right arrow] K be a map on A. Then f is a Y fibrations (resp. Y trivial fibration) iff f has right lifting property with respect to inclusions of the kind [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that X is a Y Kan complex iff every [Y.sup.n] box admits a filer. Furthermore X is an F trivial complex iff every trivial [Y.sup.n] box admits a filler.

Recall that given Y : [DELTA] [right arrow] A, there exists associated to it the singular functor Sy : A [right arrow] [[DELTA].sup.[omicron]]S, given by [([S.sub.Y](X)).sub.n] = A([Y.sup.n],X) which gets faces and degeneracies from the ones of Y by composition, and if F : X [right arrow] K then [S.sub.Y](f) is the simplicial functions whose n-th level is given by [([S.sub.Y](f)).sub.n](a) = fa. Note that Sy(X) is a simplicial set and that the elements of [([S.sub.Y](X)).sub.n] are the kind of maps [Y.sup.n] [right arrow] X we have used to define [Y.sup.n] boxes and trivial [Y.sup.n] boxes.

Similarly fillers on Yn boxes, trivial of otherwise, are fillers in the corresponding simplicial set [S.sub.Y](X). It is rather simple to verify also, that f : X [right arrow] K has the [Y.sup.0] lifting property if and only if [([S.sub.Y] (f)).sub.0] is onto. One gets then the following results.

Let f : X [right arrow] K be a A map: f is a Y fibration if and only if Sy(f) is a Kan fibration. f is a Y trivial fibration if and only if Sy(f) is a trivial fibration. Finally, any Y trivial fibration is a Y fibration.

5. Completing the pre model category associated to Y

Now we complete the Y structure on A. To Kan fibration and trivial fibration one adds the following definition:

i A map f : X [right arrow] K is called a Y trivial cofibration if it has left lifting property with respect to the class of Y fibrations.

ii f is called a Y cofibration if it has left lifting property with respect to the class of Y trivial fibration.

iii f is called a Y weak equivalence if it factors as j [omicron] h where h is a Y trivial cofibration and j is a Y trivial fibration.

From these classes of maps we are primarily concerned with fibrations, cofibrations and weak equivalences.

In what follows we will give conditions under which this five classes of maps form a pre model structure in A. Thus far we have:

i The following classes of maps are closed under composition and contain all of the isomorphism of A: Y-fibrations, Y trivial fibrations, Y cofibrations, Y trivial cofibrations.

ii The base extension of a map which is a y fibration (resp. y trivial fibration) is again a Y fibration (resp. a Y trivial fibration) The co base extension of a map which is a y cofibration (resp. a Y trivial cofibration) is again a Y cofibration (resp. a Y trivial cofibration).

iii The classes of Y fibration, Y trivial fibration, Y cofibration, and Y trivial cofibrations are closed under retracts.

iv Suppose a map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] factors as k o h with h a. Y trivial cofibration and k a Y fibration (resp. with h a, Y cofibration and k a Y trivial fibration). If f has right lifting property for Y trivial cofibrations, then f is a Y fibration. (resp. if f has right lifting property for Y cofibrations, then f is a Y trivial fibrations).

If the factorization axiom holds for the Y structure then: f is a Y fibration if and only if it has right lifting property for Y trivial cofibrations. Also f is a Y trivial fibration if and only if it has right lifting property with respect to Y cofibration.

As we have mentioned in a model category the prefix "trivial" has special meaning which for the Y structure is still valid. In fact f is a Y trivial fibration if and only if f is a Y fibration and a Y weak equivalence. Also f is a y trivial cofibration if and only if is a y cofibration and y weak equivalence.

Thus so far the following axioms for model and closed model hold for the classes of structural maps associated to Y; for model categories: M.0, M. 1, M.3, MA. For closed model categories: C.M.1, C.MA and partially C.M.3: Y - F, Y - TF, Y - C are closed under retracts.

Now we want to give conditions on Y so that the factorization axiom hold. From the remark of proposition 2.24 it will follow as well that the classes of structural maps associated to Y form a pre model category.

6. Smallness and the factorization axiom

The concept of smallness that we use here is actually the sequential one used by Quillen [QD67] as well as the procedure to build up factorization of maps whenever there exists a family {[A.sub.i] [right arrow] [B.sub.i]} with the [A.sub.i] small. A more refined version can be found in RR94.

Let A be an object of A. We say that A is small if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for any family {[Z.sup.m]}, m [member of] N. More precisely, A is small if for any sequence [Z.sup.0] [member of] [Z.sup.1] [member of] [Z.sup.2] [member of] ... of maps and for any [alpha] : A [right arrow] lirn [Z.sup.n], there exits m [member of] N and a map [alpha]: A [right arrow] [Z.sup.n] such that j [omicron] [[alpha].bar] = [alpha].

Let Y : [DELTA] [right arrow] A be a covariant functor. We say that Y is small if for each n [member of] N*, [Y.sup.n] is small.

The cosimplicial character of Y implies smallness of some subobjects of the [Y.sup.n]'s whenever Y is small. In fact let I [subset or equal to] [n] = {0,1,..., n}. Then Y[n, I] is small. In particular Y[n, k] and [DELTA][Y.sup.n] are small.

Notice also that if A is small and B is retract of A in the usual sense i.e. there are maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that [eta][rho] = [1.sub.B], then B is also small.

Here are some examples of small Y's.

i [DELTA] is obviously small i.e. for each n [member of] M, [DELTA][n] is small. Hence so are [DELTA][n,k] and [delta][DELTA][n].

ii If Y is small then for any n [member of] TV, the object obtained by dropping the first n levels and last n cofaces and co-degenacies, R[C.sup.n](Y), is small. Hence in particular R[C.sup.(n)]([DELTA]) is small for any n [member of] N.

iii If Y is small then the functor [R.sub.Y] : [DELTA] [omicron]S [right arrow] [DELTA] [omicron]S, left adjoint of [S.sub.Y] : [DELTA] [omicron]S [right arrow] [[DELTA].sup.[omicron]]S, send small objects into small object. That follows since SY commutes with sequential direct limits. Even more: Y is small if and if SY commutes with sequential direct limits.

We establish now the general situation implied by smallness about factorization of arrows.

Proposition: Let A = {[A.sub.i] [right arrow] [B.sub.i] | i [member of] 1} be a family of maps in A. If for each i [member of] I, [A.sub.i] is small, then any map f : X [right arrow] Y factors as f = k [omicron] h where h and k have the following properties.

i h has the left lifting property for the class of maps with right lifting property for A.

ii k has right lifting property for A.

Thus if Y : [DELTA] [right arrow] A is small then any map F : X [right arrow] K factors in two ways: F = k [omicron] h with h [right arrow] Y - C and k [member of] Y - TF and also as f = k' [omicron] h' with h' [member of] Y - TC and k' [member of] Y - F. That is to say, if Y is small, then factorizations axiom holds for the Y--structure. Also f is a Y fibration if and only if f has right lifting property for Y - TC and f is a Y trivial fibration if and only if f has right lifting property for Y - C.

Hence we also have the following: when Y is small, A together with the classes (or structural maps) Y - F, Y - TF, Y - C, Y - TC and Y - WE form a pre model category.

7. The Homotopy System Associated to Y

Although in Kan [KD55,561] the development of homotopy groups associated to homotopy system is done through cubical complexes, we have found easier to use simplicial sets and their standard homotopy groups in order to associate homotopy groups to a homotopy. In fact, as we will see, Y homotopy as we define it will become, for a suitable Y, a "simplicial homotopy", namely the homotopy associated to the Y simplicial system (a concept studied in next paragraph) in which the machinery of standard homotopy of simplicial sets is available. Here, however, we develop Y homotopy independently of the Y simplicial system.

In this and the next paragraph we assume that Y is a "pointed" model of A, that is to say [Y.sup.0] is the final object of A.

Note that the following defines a homotopy system in A :

i [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

ii [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

iii [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We call it the Y homotopy system in A and the corresponding homotopy is the Y homotopy. If f, g : X [right arrow] K [member of] A, and / is homotopic to g through this homotopy we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] or when the use of [Y.sup.1] is to be emphasized we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if here exits H : X x [Y.sup.1] [right arrow] K such that H [omicron] [d.sup.0] = F and H [omicron] [d.sup.1] = g.

As for the effect of SY : A [right arrow] [[DELTA].sup.[omicron]] S itself on Y homotopy, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [S.sub.Y](f) ~ [S.sub.Y](g), where ~ denotes the standard homotopy in [[DELTA].sup.[omicron]] S. However we will present a structure which permits the use of simplicial homotopy in full power in favor and the one in A induced and Y, including homotopy groups, exact homotopy sequences, etc, as done by Quillen [QD67].

8. The Y Simplicial Structure of A

The Y simplicial structure is, roughly speaking, the generalization of functional complexes in [[DELTA].sup.[omicron]]S, when in [([X.sup.K]).sub.n] = [[DELTA].sup.[omicron]]S(K x [DELTA][n], X) the models [DELTA][n] are substituted by [Y.sup.n], where Y : [DELAT] [right arrow] [[DELTA].sup.[omicron]]S.

The formal definition of simplicial system in categories, and the concept of simplicial category was given in the introduction. Here study first the machinery available in the simplicial system induced by special functors Y : [DELTA] [right arrow] A, then the relations with the Y structure and Y homotopy. In this paragraph we consider a pointed Y which realizes products on standard models. Therefore its realization RY : [[DELTA].sup.[omicron]]S [right arrow] A commutes with finite products. On the other hand the following assignments define a functor HomY : A x A [right arrow] [DELAT]oS:

i For X,K [member of] A, [Hom.sub.Y] (X, K) = [S.sub.Y] x [bar.x] (K).

ii For f : [X.sup.1] [right arrow] X and g : K [right arrow] [K.sup.1], [Hom.sub.Y](f,g) is the map given level wise by

[Hom.sub.Y][(f,g).sub.n] : [Hom.sub.Y][(X,K).sub.n] [right arrow] [Hom.sub.Y][([X.sup.1], [K.sup.1]).sub.n] which maps [alpha][??] g [omicron] [alpha] [omicron] (1 [Y.sup.n] x f).

Notice that Homy(X, K) is a generalization of [K.sup.X]. In fact [Ham.sub.Y][(X, K).sub.n] = [S.sub.Y] x [bar.X][(K).sub.n] = [[DELTA].sup.[omicron]]S([Y.sup.n] x X,K).

We show next that Homy admits an associated simplicial composition, as required in a simplicial systems.

i The following defines a simplicial functor for any simplicial sets X, K, Z:

[Hom.sub.Y][(X, K).sub.n] x [Hom.sub.y][(X, Z).sub.n] [right arrow] [Ham.sub.Y][(X, ZK).sub.n] ([alpha], [beta]) [??] [beta] [omicron] [alpha] = [beta] [omicron] ([alpha] x 1[y.sub.n]) [omicron] (1 x [DELTA]([Y.sup.n])) Or more graphically [beta] o [alpha] is the following composition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

ii The composition of part i is associative in the sense that for X, K, Z, T and f [member of] [Ham.sub.Y][(X, K).sub.n], g [member of] [Hom.sub.Y][(Z, K).sub.n] and h [member of] [Hom.sub.Y][(Z,T).sub.n] then (h [omicron] f) [omicron] g = h [omicron] (f [omicron] g).

To complete the simplicial system we are required to show that Homy (X, K) is a simplicial set built up on [[DELTA].sup.[omicron]]S(X, K) or to say better whose 0-level is [[DELTA].sup.[omicron]]S(X, K) which behave appropriately with the simplicial composition.

i The functions [lambda](X, K) given by A(X, K) [right arrow] [Hom.sub.Y][MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] define a natural isomorphism.

ii Let f [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [Hom.sub.Y][(K, Z).sub.n], u [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[DELTA].sup.[omicron]]S(X,K) and let [s.sup.n.sub.0] denote the composition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If g [member of] [Hom.sub.Y][(W, X).sub.n] and u [member of] [[DELTA].sup.[omicron]](X, K) then

[Hom.sub.Y][(W,u).sub.n] (g) = [s.sup.n.sub.0] (u) [omicron] g

Part i is clear. For part ii notice that [s.sup.n.sub.0]([??]) = [u.sub.0]([([S.sup.0]).sup.n] x 1 x). Furthermore if f : [Y.sup.n] x K [right arrow] Z one has a composition

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

On the other hand recall that

[Hom.sub.Y](u, Z)n : A([Y.sup.n] x K,Z) [right arrow] A([Y.sup.n] x X, Z)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We want to prove that the composition given above and the image of f by the last function coincide. But (level wise) the image of ([Y.sup.n],X) by the composition above is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Part ii is proved similarly.

Remark:

i Homy with the simplicial composition given and the natural isomorphism

[[DELTA].sup.[omicron]]S(X, K) [right arrow] Homy(X, [K.sub.0])

form a simplicial system which we refer to as the simplicial system associated to Y or the Y simplicial system in A.

ii So far the only condition used has been that Y is pointed. For the existence of cylinder and path objects as well as for the equivalence of right and left homotopy, we will need that Y realizes products on standard models.

In order to develop path objects [X.sup.K] and cylinders objects X [cross product] K for the models in A, and since one of the properties desired is the equality

[Hom.sub.Y] (X [[cross product].sub.Y] K,Y) [congruent to] [([Hom.sub.Y](X, Y)).sup.K]

(see in the introduction: Simplicial system), we notice that

[Hom.sub.Y][(X,Y).sup.K] = [S.sub.[DELTA]x[bar.K]] [S.sub.Yx[bar.X]](Y)

Therefore it would be helpful to have a relation between composition of singular functors and singular functors of composite functors. That we do next.

Let Y, Z : [DELTA] [right arrow] A be cosimplicial objects of A. One has the composition [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is again a model in A. Then: There exits a natural isomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In fact, one can see that the isomorphism of adjointness of the pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] applied level wise to Z gives an isomorphism

[[eta].sub.n] : [[DELTA].sup.[omicron]]S(([R.sub.Y]([Z.sup.n]),X) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[DELTA].sup.[omicron]]5([Z.sup.n], [S.sub.Y](X))

with inverse, say [[rho].sub.n]. It follows from naturality of the singular and realization functors that [eta] is a simplicial map and is natural on X.

9. Existence of Cylinder and Path Objects

Let X [member of] A and K [member of] [[DELTA].sup.[omicron]]S. There exists a pair (X [[cross product].sub.Y] K, [alpha]) with X [[cross product].sub.Y] K in A and [[alfa].sub.K] : K [right arrow] [Hom.sub.Y] (X [[cross product].sub.Y] K), which induces a natural isomorphism,

[a.sup.*.sub.K] : [Hom.sub.Y](X [[cross product].sub.Y] K, Y) [congruent to] [[[Hom.sub.Y](X, Y)].sup.K]

Remark:

The isomorphism

[Hom.sub.Y] (X [[cross product].sub.Y] K,Y) [congruent to] [[[Hom.sub.Y](X,Y)].sup.K]

has not been proved to be the one used by Quillen [QD67] within the framework of the simplicial structure of a simplicial category, but it works remarkably well and we saw no reason to insist on Quillen's isomorphism. The same remark is valid in the case of path objects, next.

Let X [member of] A and K [member of] [[DELTA].sup.[omicron]] S. There exists a pair ([X.sup.K] (rel.Y), [Beta]) where [X.sup.K] (rel.Y) is an object of A and

[beta]:K [right arrow] [Hom.sub.Y]([X.sup.K](rel.Y), X)

is a simplicial map, which induces a natural isomorphism

[[beta].sup.*.sub.K] : Horny (Y, (rel.Y)) [right arrow] [HomY(Y,X))K

The following easy to check formulas are useful in the study of relations between the Y simplicial system, Y homotopy, and Y pre model structures.

Proposition:

i [[cross product].sub.Y] is "associative" in the following sense:

X [[cross product].sub.Y] (K [cross product] L) = (X [[cross product].sub.Y] K) [[cross product].sub.Y] L

ii ([X.sup.K] [(rel.Y).sup.L](rel.Y) [congruent to] [X.sup.KxL] (rel.Y).

iii [S.sub.Y] ([Hom.sub.Y](X, K)) [congruent to] [S.sub.T] (K) where T = ([R.sub.Y] [omicron] Y) x [bar.X].

iv For each n, [S.sub.Y] ([Hom.sub.Y][(Y, X)).sub.n] = [[DELTA].sup.[omicron]]S(Y, [X.sup.N]) where N = [R.sub.Y] ([Y.sup.n]).

V [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

10. Relative Subdivisions

In this parts we dealt with the relation among homotopies induces by models. Since there is little difference when using [DELTA] or other category [delta] we use the later most of the time.

Kan [KD55,561,562] and other authors [FR68] have given characterizations of functors [[DELTA].sup.[omicron]]S [right arrow] [[DELTA].sup.[omicron]]S called "subdivision functors" which are distinguished, among other things because the diagram below commutes up to homotopy equivalence. [parallel]: [[DELTA].sup.[omicron]]S [right arrow] Top denotes Milnor's geometric realization [MJ57].

Here we extend the concept of subdivision and give techniques to build subdivision functors in a more general context. First we work with the standard scheme category [DELTA] and then, in a further generalization, any category [delta] is used as scheme. Thus instead of simplicial objects we work with general pre sheaves.

For the first part, instead of the geometric realization functor we use the realization of a model M : [DELTA] [right arrow] T op, and instead of normal homotopy we use that of a homotopy system [gamma]. If Sd : [[DELTA].sup.[omicron]]S [right arrow] [[DELTA].sup.[omicron]]S denotes the desired subdivision functor, [R.sub.M] : [[DELTA].sup.[omicron]]S [right arrow] T op the realization induced by M, and "[??]" the homotopy equivalence induced by [gamma], then one would like to have that [R.sub.M](Sd(X)) [??] [R.sub.M](X) for each simplicial set X.

If A and B are categories, and a functor R : A [right arrow] B admits right adjoint S : B [right arrow] A, we say that (R, S) is an adjoint pair. As for notation, [DELTA]B denotes the category of the cosimplicial objects of B and [[DELTA].sup.[omicron]]B the simplicial ones. Also, if X is a simplicial object of B we denote X(n) - [X.sub.n], and if w : [n] [right arrow] [n] is an arrow of [DELTA], then X(w) = [w.sup.*]. For the cosimplicial case the notation will be respectively [X.sup.n] and [w.sup.*]. Finally, if A is an object of B then the constant functor of value A will be denoted by A when is considered as a cosimplicial object of B and A when considered as simplicial object of B.

11. Singular Functors and Realizations

According to the theory of adjoint pairs, a functor Y : [DELTA] [right arrow] B defines a functor S y : B [right arrow] [[DELTA].sup.[omicron]]S given on the objects by [([S.sub.Y](A)).sub.n] = B([Y.sup.n],A) and if w : [n] [right arrow] [n] then ([S.sub.Y](A))(w) is given by composition with Y(w). Now, if f is a morphism of B then [S.sub.Y](f) is given level wise by composition with f.

When [beta] is a co complete category [S.sub.Y] admits a left ad-joint denoted [R.sub.Y] : [[DELTA].sup.[omicron]]S [right arrow] B. The categories S (sets), [[DELTA].sup.[omicron]]S and Top are co complete and therefore singular functors admit realizations. We now characterize them.

Given a model Y : [DETA] [right arrow] S (respectively Z : [DELTA] [right arrow] Top) then its induced realization is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [??] denotes the union (respectively the topological sum), [X.sub.n] x [Y.sup.n] the set theoretical product (respectively [X.sub.n] x [Z.sup.n] denotes the topological product taking [X.sup.n] as a discreet space), and the quotient is the set theoretical (resp. topological) quotient for the equivalence relation t that for each w : [n] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [n], x in [X.sup.n], and y in [Y.sup.n] (respectively z in [Z.sup.n]) identifies the couples ([w.sup.*](x),y) and (x,[w.sup.*],y) (respectively ([w.sup.*](x),z) and (x,[w.sup.*],z)). Also, if f : X [right arrow] K, then [R.sub.Y] (f)([x,y]) = [[f.sub.n](x),y] (respectively [R.sub.Z](f)([x,z]) = [[f.sub.n](x),z]), where [x,y] denotes the class of equivalence of the couple (x,y).

zOne proofs that [R.sub.Y] (respectively [R.sub.Z]) is a covariant functor and that is left adjoint of [S.sub.Y] (respectively [S.sub.Z]).

In the case of Y : [DELTA] [right arrow] [[DELTA].sup.[omicron]]S we denote, for each [Y.sup.n] in [[DELTA].sup.[omicron]]S, [Y.sup.n](m) = [Y.sup.n.sub.m]. Thus in [Y.sup.n.sub.m], n varies contra variantly and m co variantly. Is easy to see that, for each m, [Y.sub.m] defines a set theoretical model [Y.sub.m] : [DELTA] [right arrow] S given by [Y.sub.m][n] - [Y.sup.n.sub.m] and for each w : [n] [right arrow] [m], [Y.sub.m] (w) = [(Y(w)).sub.m].

For each Y : [DELTA] [right arrow] [[DELTA].sup.[omicron]]S an adjoint functor of [S.sub.Y], say [R.sub.Y], is given on the objects by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and ifr : [n] [right arrow] [m] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [Y.sub.r] : [Y.sub.n] [right arrow] [Y.sub.m] is the cosimplicial function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the natural transformation induced by [Y.sub.r] on the set theoretical realization functors. On the morphisms, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The proof is easy but long. A known fact, worthwhile to clarify is the following: given two adjoint pairs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists a one to one and onto function G : Trans(S,S') [right arrow] Trans(R',R), where Trans denotes the class of natural transformations of the first functor in the second. On the other hand if Y, Y' : [DELTA] [right arrow] B are two models of B and a : Y [right arrow] Y' is a cosimplicial morphism then it induces a natural transformation [S.sub.a] : [S.sub.Y'] [right arrow] [S.sub.Y], given on an object A of B by [([S.sub.a] : (A)).sub.n](T) = T [omicron] [a.sup.n] for each T in [([S.sub.Y'] (A)).sub.n]. Here we have denoted G (a) = [R.sub.a]. In previous results this mechanism was responsible for the equality [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Notice that in the set theoretical and topological cases [R.sub.a] is given by [R.sub.a] (X)[x, y] = [x, a(y)], while in the simplicial one is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following result will be very useful in what follows:

Let B be a co complete category, Y : A [right arrow] B a model on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] adjoint pair (notice that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a model on B). Then there exists a natural isomorphism P : [R.sub.IoJ] [congruent to] I o [R.sub.Y].

The proof follows from the fact that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is an adjoint pair and [S.sub.IoY] is naturally isomorphic to [S.sub.Y] [omicron] J. In particular when B is S, [[DELTA].sup.[omicron]]S or T op, and I : B [right arrow] B denotes the functor "product by A", for a A fixed in B, then I admits an adjoint when B is 5 and [[DELTA].sup.[omicron]]S, but when B is Top it happens only for some A. In such a case, for a model Y : [DELTA] [right arrow] B one will have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is precisely Y x [??]. The proposition affirms then that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

naturally.

12. Homotopies on Models

A homotopy system on a category has an extension to a system on the category of its models. Indeed if [GAMMA] = (I, [d.sup.0], [d.sup.1], s) is a homotopy system in a category B then [gamma] induces on [DELTA]B the system [DELTA][gamma] = (AJ, Ado, Ad1, As) as follows: AI : [DELTA]B [right arrow] [DELTA][beta] is the normal extension of I to [DELTA]B: if Y it belongs to [DELTA]B then [DELTA]/[(Y).sup.n] = I([Y.sup.n]) and [DELTA]I(Y)(w) = I(w). The natural transformations are given by the equalities [([DELTA][d.sup.i](Y)).sup.n] = [d.sup.i]([Y.sup.n]) and [([DELAT]s(Y)).sup.n] = s([Y.sup.n]), i = 0,1.

Given two models Y, Z over B and two cosimplicial morphisms F,G : Y [right arrow] Z then it is clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if, and only if, for each n there exists [h.sup.n] ([gamma]-homotopy) of [F.sup.n] in [G.sup.n] such that for each w : [n] [right arrow] [to], Z(w)[h.sup.n] = [h.sup.m]I(Y(w)).

Let us see now that homotopies and homotopy equivalences on models induce homotopies and homotopy equivalences on the realizations. We first advance some observations.

Given a homotopy system [gamma] on B, then to the natural tranformacion [d.sup.i] : [1.sub.B] [right arrow] I there corresponds, in a bi univocal form, a natural transformation [D.sup.i] : J [right arrow] [1.sub.B], when (I, J) is an adjoint pair. Similarly, if we denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the transformation that in X has as morphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the corresponding transformation on the singular functors is given for each A in B by [S.sub.Y]([D.sup.i](A)) : [S.sub.Y](J(A)) [right arrow] [S.sub.Y](A) and we denote it by [S.sub.Y][D.sup.i]. Other two corresponding natural transformations (natural isomorphisms) that will use are p' : [S.sub.I[omicron]Y] [right arrow] [S.sub.Y] [omicron] J with p : I [omicron] [R.sub.Y] [right arrow] [R.sub.Y[omicron]I]. Finally, there exists a cosimplicial morphism Y [right arrow] Y [omicron] I that in the level n is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will denote it by dy. We have then the corresponding natural transformations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The following fact will be of great help in what follows: For i - 0,1 the following diagram of natural transformations commutes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let B be a category and [gamma] a homotopy system whose cylinder I admits right adjoint J. Let Y and Z be models on B. Then:

i For each pair of simplicial morphisms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then for each simplicial set X one has that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], naturally on X. In particular,

ii If Y is [DELTA][gamma]-homotopically equivalent to Z, then for each simplicial set X, [R.sub.Y](X) is [gamma]-homotopically equivalent to [R.sub.Z](X), naturally on X.

This result assures that if the cylinder I of a homotopy system [gamma] on B has right adjoint J, then the homotopy system induced by [gamma] on the models, which we have denoted [DELTA][gamma], has a direct effect on the realizations defined by the models. Indeed the proposition assures that if the models are homotopically equivalent then their realizations produce homotopically equivalent images for the original system on B. In particular if we consider on Top the normal homotopy and if we take as Y the cosimplicial space of the topological simplexes [[DELTA].sup.n] then Ry is the geometric realization of Milnor. If Z is another model on Top, homotopically equivalent to the first one, then for each simplicial set X one has that [valor absolute de X] [congruent to] [R.sub.Z](X), where [congruent to] denotes the normal homotopy equivalence on Top.

Notice that although each [DELTA]n is homotopically equivalent to a point not all realization of simplicial sets are so. This happens because the homotopy equivalence in question on the models is not simplicial and therefore is not an equivalence of the extended homotopy. Naturally, if a model is level wise homotopically equivalent to a point, for a given homotopy, so that the given model and the cosimplicial point are homotopically equivalent, then the relation homotopy induced by the model is trivial.

In categories with final object, an object is homotopically trivial if it is homotopically equivalent to the final object. If the concept of sub object also exists and these are preserved by morphisms, then one has that if a model is homotopically trivial then the final model object is a sub object of the given model. Said otherwise: if the final model is not a sub object of the model then the last one cannot be homotopically trivial.

In the case of S, [[DELTA].sup.[omicron]]S and Top this means that if a model doesn't have cosimplicial points then it is not homotopically trivial.

This advantage, together with the one of having Eilenberg--Zilber representations make of the models without cosimplicial points specially capable tools for good homotopy theories (see for example [RCRR811]).

Up to now we have studied conditions so that two models Y,Z: [DELTA] [right arrow] B have homotopically equivalent realizations for a homotopy system [gamma] on [beta]. Now we consider functors [[DELTA].sup.[omicron]]S [right arrow] [[DELTA].sup.[omicron]]S capable of preserving the homotopy type of the realizations.

Let Y : [DELTA] [right arrow] [[DELTA].sup.[omicron]]S be a model of [[DELTA].sup.[omicron]]S and M : [DELTA] [right arrow] B a model of B. We consider a homotopy system [gamma] = (I, [d.sup.0], [d.sup.1], s) on B for which I has a right adjoint J. In general the diagram is not commutative, but it is when Y is the model of the simplicial simplexes [DELTA][n]. However, when the diagram commutes up to a [DELTA][gamma]-homotopy equivalence, then [R.sub.M]([S.sub.Y](X)) and [R.sub.M](X) are [gamma]--homotopically equivalent on B. We develop this point next.

We will call [R.sub.Y] : [[DELTA].sup.[omicron]]S [right arrow] [[DELTA].sup.[omicron]]S a subdivision (or a subdivision of the identity) relative to the pair ([gamma], M) if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. When [DELTA] denotes the model of the [DELTA][n] simplexes, [R.sub.[DELTA]] is the identity of [[DELTA].sup.[omicron]]S up to isomorphism. Then what we are doing is to compare at the homotopy level [R.sub.Y] with [1.sub.[[DELTA].sup.[omicron]]S], a notion that allows us to assimilate [R.sub.Y](X) to a "subdivision" of X. For example when [gamma] is the homotopy system induced by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by means of the product, we can accept that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then the model y is a "subdivision" of A. It will be seen that this implies that[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which means that, at level of realizations (via [R.sub.M]), X and [R.sub.Y](X) have the same [gamma]-homotopy type. For consistency we assimilate [R.sub.Y](X) as a subdivision of [gamma], achieved upon the subdivision Y of [DELTA] limited only by homotopy. Then one goes further using any homotopy [gamma], and imposing conditions on Y so that the previous results still hold. The conditions on y and [DELTA] can hold, up to weak homotopy equivalence. That is the reason for the name of relative subdivisions.

Theorem: Let [gamma] be a homotopy system on B whose cylinder admits a right adjoint. Let M be a model on B and Y a model on [[DELTA].sup.[omicron]]S. If Ry is a subdivision relative to (7, M) then there exists a natural transformation a : [R.sub.M] [right arrow] [R.sub.M] [omicron] [R.sub.Y] such that for each X in [[DELTA].sup.[omicron]]S, [a.sub.x] : [R.sub.M](X) [right arrow] [R.sub.M] ([R.sub.Y] (X)) is a [gamma]-homotopy equivalence.

For the proof, let us remember that, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] M, there exists a transformation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that for each X, ex is a [gamma]-homotopy equivalence. The theorem will be proved if we exhibit a natural isomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], or equivalently one natural isomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], but it is obtained by means of the following chain of natural isomorphisms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A pair (Y, M), where y is a model of [[DELTA].sup.[omicron]]S and M is a model of B, is said to be a singular pair if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Is clear that if (Y, M) is a singular pair then there exists a natural isomorphism [R.sub.M] [omicron] [R.sub.Y] [right arrow] [R.sub.M]. Therefore y is a subdivision relative to ([gamma], M) for any homotopy system [gamma] on [beta]. The converse is also true since "=" is the homotopy relation of particular homotopy systems on any category. For example that is the case for any homotopy system in which [d.sup.0] = [d.sup.1].

Up to now we have defined subdivisions of the identity relative to pair ([gamma], M). Now we consider subdivisions of any models.

Let [gamma] be a homotopy system on B whose cylinder admits a right adjoint and let M be a model of B. If Y and Z are models of [[DELTA].sup.[omicron]]S we say that Y is a subdivision of Z relative to ([gamma], M), denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Notice that if the homotopy of [gamma] is transitive then the relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an equivalence relation on the class of the models of [[DELTA].sup.[omicron]]S.

The relation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] compares models through its realizations in a weak way, via the homotopy equivalence induced by [gamma] on the models of B. Also, it compares their realizations (via [R.sub.M]) by means of the homotopy equivalence of [gamma] in B. For if Y, Z, [gamma], and M are as above and if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then there exists a natural transformation a : [R.sup.M] [omicron] [R.sub.Y] [right arrow] [R.sub.M] [omicron] [R.sub.Z] such that for each X in [[DELTA].sup.[omicron]]S the morphism [a.sub.X] : [R.sub.M](X) [right arrow] [R.sub.M]([R.sub.Z](M)) is a [gamma]-homotopy equivalence.

13. Subdivisions on Pre Sheaves

It is known that a covariant functor Y: [delta] [right arrow] A, where [delta] and A are any categories, induces a covariant functor [S.sub.Y] : A [right arrow] [[delta].sup.[omicron]]S. If A is co complete then S y has a left adjoint functor, denoted here by [R.sub.Y] : [[delta].sup.[omicron]]S [right arrow] A. Generalizing the terminology of [MJ57] and [RR76] we call

i The functor Y : [delta] [right arrow] A a [delta]-model of A, or a model of A when [delta] is clear.

ii [S.sub.Y] the singular functor and [R.sub.Y] the realization functor of Y.

So far we have studied the repercussions on the functors [R.sub.Y] and [R.sub.Z] produced by the existence of a homotopy equivalence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], induced in [delta]A by a preset homotopy of A when [delta] is the category [DELTA]. The homotopy induced in [DELTA]A imposes conditions of naturality that occasionally can be very restrictive. One can see however that if one considers naturality on a restricted number of arrows of [DELTA], then the theory still works. This is equivalent to change the category [DELTA] restricting arrows.

For generalization we take any unrestricted category [delta] and work as follows: first we give the extension of a homotopy system of A to [delta]A ([delta]A the category of covariant functors [delta] [right arrow] A and natural transformations as morphisms) and we show that if Y and Z are models of A, homotopically equivalent for that extension, then for each pre sheaf X of [[delta].sup.[omicron]]S, it happens that in A, [R.sub.Y](X) is homotopically equivalent to [R.sub.Z](X), naturally on X. Further we fix a realization [R.sub.M] : [[delta].sup.[omicron]]S [right arrow] A and generalize the case of [KD55,561,562] to give conditions on two functors F, G : [[delta].sup.[omicron]]S [right arrow] [[delta].sup.[omicron]]S so that the realizations of F(X) and G(X) for [R.sub.M] are naturally homotopically equivalent. Those said conditions define a relationship on the functors [[delta].sup.[omicron]]S [right arrow] [[delta].sup.[omicron]]S that will be called of relative subdivision, extending the terminology of [KD57]. In general a standard homotopy in [[delta].sup.[omicron]]S doesn't exist. So we fix a homotopy system there, another in A, and we give conditions relating these two systems by means of the functor [R.sub.Y]: [[delta].sup.[omicron]]S [right arrow] A induced by Y : [delta] [right arrow] A and we say that "Y carries the system of 5o S into the one of A".

If both Y and Z carry the system of [[delta].sup.[omicron]]S into the one of A and if Y is homotopically equivalent to Z (for the homotopy of [[delta].sup.[omicron]]S extended from A) then for each A of A, [S.sub.Y](A) is homotopically equivalent to [S.sub.Z](A) in [[delta].sup.[omicron]]S. When [delta] = A and homotopy is the normal one in [[DELTA].sup.[omicron]]S and Y [??] Z then the homologies induced by Y and Z in A are isomorphic.

With normal changes the proofs for [DELTA] can be adapted to the new generalization.

In what follows A denotes a co complete category. For two models Y, Z : [delta] [right arrow] A and a morphism of models f : Y [right arrow] Z (natural transformation) we denote by [S.sub.f] : [S.sub.Z] [right arrow] [S.sub.Y] the transformation induced by f on singular functors.

We dealt now with the construction of a natural transformation [R.sub.f] : [R.sub.y] [right arrow] [R.sub.z].

Recall by [RCRR812] that if we denote by [[PHI].sub.Y] : 1 [right arrow] [S.sub.Y] [omicron] [R.sub.Y] and [PSI].sub.Y]: [R.sub.Y] [omicron] [S.sub.Y] [right arrow] 1 adjointness transformations for ([R.sub.Y], [S.sub.Y]), by [[PSI].sub.Z], [[phi].sub.z], those of ([R.sub.Z], [S.sub.Z]) and by

A([R.sub.Y](X),A) [flecha diestra y siniestra] [[delta].sup.[omicron]]S(X, [S.sub.Y] (A))

[alpha] [right arrow] [[alpha].sup.Y]

[[beta].sub.Y] [flecha siniestra] [beta]

(respectively [alpha] [right arrow] [[alpha].sup.Z], [beta] [right arrow] [[beta].sub.z]) the adjointness isomorphisms, then

[R.sub.f](X) = [[[S.sub.f]([R.sub.Z](X)) [omicron] [[PSI].sub.Z](X)].sub.y] : [R.sub.Y] (X) [right arrow] [R.sub.Z] (X)

The extension of homotopy systems on A to systems on [delta]A is the following one: If [eta] = (I, [d.sup.0], [d.sup.1], s) is a homotopy system on A then [delta][eta] - ([delta]I, [delta][d.sup.0], [delta][d.sup.1], [delta]s) denotes the system in [delta]A with cylinder [delta]I(Y) = I [omicron] Y and [delta]I[([lambda]).sub.x] = 7(AX) for each morphism [lambda] : Y [right arrow] Z in [delta] A, where [delta][d.sup.i](Y) : Y [right arrow] [delta]I (Y) is given by [delta][d.sup.i][(Y).sub.x] = [d.sup.i](Y(x)) and [delta]s(Y) by [delta]s[(Y).sub.x] = s(Y(x)), [member of] [delta].

The [delta]n homotopy is a natural extension of the homotopy [eta] because if Y, Z are objects of [delta] A and F, G : Y [right arrow] Z morphisms, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is homotopic to G via [delta][eta]) if, and only if, there exists a family [h.sub.X] : I(Y(X)) [right arrow] Z(X) of [eta]-homotopies [F.sub.X] ~[right arrow] [G.sub.x] such that if w : X [right arrow] X' is a morphism of [delta], then Z[(w).sub.0][h.sub.X] = [h.sub.X'] [omicron] I(Y(x)).

Since the homotopy relation of a homotopy system is not generally an equivalence relation we fix and keep an order that we exemplify for the system [delta][eta]: If Z, Y are two models of A (on [delta]) then Z is [delta][eta] homotopically equivalent to Y if there exist morphisms F : Z [right arrow] Y and G: Y [right arrow] Z such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If the cylinder of [eta] = (I, [d.sup.0], [d.sup.1], s) admits right adjoint J : A [right arrow] A then J is completed in a right homotopy system denoted [[eta].sup.*] = (J, [D.sup.0], [D.sup.1], S), for which [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In fact if we denote the adjointness transformations by [PHI] : [1.sub.A] [right arrow] J o 7 and [PSI] : I [omicron] J [right arrow] For [alpha] : I(A) [right arrow] B, [[alpha].sup.*] = J([lafa]) [omicron] [[PHI].sub.A] and for [beta] : A [right arrow] J(B), [[beta].sup.*] = [[PSI].sub.B] [omicron] I([beta]), then to the transformations [d.sup.i] (i = 0,1) there correspond the transformations [D.sup.i.sub.A] : J [right arrow] [1.sub.A] given for each A in A by [D.sup.i.sub.A] = [PSI].sub.A] [omicron] [d.sup.i] J(A) or [d.sup.i] = Di with the notation of [RCRR811].

For a model Y : [delta] [right arrow] A, one has another, model [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] whose realization functors Ri0y and singular [S.sub.I[omicron]Y] are related with [R.sub.Y] and [S.sub.Y] in the following way.

The adjoint pairs (I [omicron] [R.sub.Y], [S.sub.Y] [omicron] J) and ([R.sub.I[omicron]Y], [S.sub.I[omicron]Y]) are equivalent. That is to say that there exists a natural isomorphism [bar.[rho]Y] : [S.sub.I[omicron]Y] [right arrow] [omicron] J (and, equivalently, a natural isomorphism [rho]Y : I [omicron] [R.sub.Y] [right arrow] [R.sub.I[omicron]Y]).

In fact [([[bar.[rho]].sub.y](A)).sub.X] is the isomorphism of A given by the following chain of natural isomorphisms: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [rho]Y is the isomorphism induced by [[bar.[rho]].sub.y] at realizations.

The effect on the realizations, of homotopy among models, is the following one:

Let Y, Z : [delta] [right arrow] A be two models and F, G : Y [right arrow] Z morphisms of A. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then there exists a natural transformation H : I [omicron] [R.sub.Y] [right arrow] [R.sub.Z] such that for each X in [[delta].sup.[omicron]]S, [H.sub.X] defines a homotopy [R.sub.F](X) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [R.sub.G](X).

That is so because a [delta][eta] homotopy of F into G, say H produces a commutative diagram, where of course [d.sup.i][R.sub.Y] (i = 0,1) is the transformation given by [d.sup.i] [R.sub.Y] (X) = [d.sup.i] ([R.sup.Y] (X)). One takes H = [R.sub.H] [omicron] [rho]y. As a consequence one also has that if Y and Z are models of A and Y is [delta][eta] homotopically equivalent to Z then there exists a natural transformation [lambda] : [R.sub.Y] [right arrow] [R.sub.Z] such that for each X in [[delta].sup.[omicron]]S, [[lambda].sub.Y] : [R.sub.Y](X) [right arrow] [R.sub.Z](X) is a 77 homotopy equivalence.

When A = Top (or Kelly) and [delta] = [DELTA] for a model Y : [DELTA] [right arrow] Top the realizations [R.sub.Y](X) will be called Y - CW complexes. They are thought of as complexes with (in general non Euclidean) cells [Y.sup.n]. The last result establishes a sufficient condition so that the Y-CW complex [R.sub.Y](X) is homotopically equivalent to the CW complexes [absolute value of X] ([parallel] Milnor's realization of [MJ57]).

In the topological case Y - CW complexes are a particular case of cellular complexes with non Euclidean cells [RCRR812] studied by the author, professor Carlos Ruiz at National University of Colombia, and Joaquin Luna [RCLJ82]. However, for any co complete category A and any model Y : S [right arrow] A on A the concept of Y-complex exist. In fact

the sub category of y-complexes of A is precisely [R.sub.Y]([[delta].sup.[omicron]]Conj), [R.sub.Y] : [[delta].sup.[omicron]]Conj [right arrow] A.

Moving to the case of relative subdivisions among functors [[delta].sup.[omicron]]S [right arrow] [[delta].sup.[omicron]]S we think of an object of [[delta].sup.[omicron]] S as a set theoretical skeleton that serves as a pattern to patch the pieces provided by y(X) of A to obtain an object of A. Up to now we have "modified" a model Y to obtain another Z, in such a way that for each skeleton X the obtained objects, one via patching objects Y(X) and the other via patching objects Z(X), were homotopically equivalent. Now we are interested in how to modify the skeleton X into another X' so that for Y : [delta] [right arrow] A fixed, the objects obtained of them, patching Y(X) and y(X'), are homotopically equivalent in A.

We consider the process X [??] X' as a functor [[delta].sup.[omicron]] S [right arrow] [[delta].sup.[omicron]]S which is the result of a "modification" of the (identity) functor X [??] X. Accepting that, it is clear that a more general situation arises considering two functors F,G : [[delta].sup.[omicron]]S [right arrow] [[delta].sup.[omicron]]S for which we want to decide in which sense G(X) is a "modification" of F(X), in such a way that G(X) and F(X) are homotopically equivalent.

We use the way suggested by the precedent theory namely, we consider the realization functors [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] A and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] A, provided F and G each admits right adjoint. In such a case F and G are "modelable". In other words they are realizations induced by models and we can concentrate on the models that define them. We will use the term "subdivision" rather than "modification" following the case exposed by Kan [KD57].

Let us fix a model M : [delta] [right arrow] A and a system 77 in A with right adjoint [[eta].sup.*]. Let Y,Z : [delta] [right arrow] [[delta].sup.[omicron]]S be models of [[delta].sup.[omicron]]S. We will say that Z is a subdivision of Y relative to the couple ([eta], M) if [R.sub.M] [omicron] Y is [[delta].sub.[eta]] homotopically equivalent to [R.sub.M] [omicron] Z. We denote it by y [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Z.

If the homotopy [eta] is transitive then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is an equivalence relation on the models of [[delta].sup.[omicron]]S. With no conditions on [eta] one has the wanted result: If Y [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Z, then there exists a natural transformation L : [R.sub.M] [omicron] [R.sub.Y] [right arrow] [R.sub.M] [omicron] [R.sub.Z] such that for each object of [[delta].sup.[omicron]]S, [L.sub.X] : [R.sub.M]([R.sub.Y](X)) [right arrow] [R.sub.M]([R.sub.Z](X)) is a [eta] homotopy equivalence.

When F and G admit right adjoint it will be said that G is a subdivision of F relative to ([eta], M) if F [omicron] [delta] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] G [omicron] [delta] ([delta] : [delta] [right arrow] [[delta].sup.[omicron]]S the natural inclusion). Since F [omicron] [delta] (respectively G [omicron] [delta]) is the model defining F (respectively G), what we have is that if G is a subdivision of F relative to ([eta], M) then for each X in [[delta].sup.[omicron]]S, [R.sub.M](F(X)) is homotopically equivalent to [R.sub.M](G(X)) naturally on X. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we will simply say that G is a "subdivision functor module ([eta], M)".

Notice that so far we have obviated the use of homotopy systems in [[delta].sup.[omicron]]S. However there exists an intermediate step giving a homotopy system in [[delta].sup.[omicron]] S that is carried by M into [eta] (see below). If Y is homotopically equivalent to Z for the extension of the homotopy from [[delta].sup.[omicron]]S then Y [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Z. In such a case, a sufficient condition for [R.sub.Z]([[delta].sup.[omicron]]S [right arrow] [[delta].sup.[omicron]]S) to be a subdivision of [R.sub.Y] relative to ([eta], M) is that Y [equivalent] Z(mod [delta]n).

14. Isomorphic Homologies

Returning to the case [DELTA] [right arrow] A, the singular functor Sy : A [right arrow] [[DELTA].sup.[omicron]]S gives place in an obvious way to a homology on A induced by Y. With the same homotopy concept among models, Y, Z : A [right arrow] A one can compare Sy with [S.sub.Z] using the standard homotopy of [[delta].sup.[omicron]]S . The process is also generalizable as we will see later. In this paragraph we consider the effect produced by homotopy equivalencies of models on the respective singular functors.

We consider in A a homotopy system [eta] = (I, [d.sup.0], [d.sup.1],s), with right adjoint [[eta].sup.*] = (,J, [D.sup.0], [D.sup.1]S). There are then adjoint pairs A [?? A [??] A and A [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Once fixed adjointness transformations there exist isomorphisms Trans([1.sub.A], I) [right arrow] Trans(J, [1.sub.A]) and Trans(I, [1.sub.A]) [right arrow] Trans([I.sub.A], J). The first one sends if in [D.sub.i] (i = 0,1) and the second sends s in S. If one also has an adjoint pair A [??] B [??] A, with fixed adjointness transformations then there are two adjoint pairs in which we are interested namely A [??] B [??] A and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] together with the following natural transformations and notations:

* R([d.sup.i.sub.A]) : R(A) [right arrow] R(I(A)), A [member of] A, i = 0,1, denoted [Rd.sup.1].

* R([s.sub.A]) : R(I(A)) [right arrow] R(A), A [member of] A, denoted Rs.

* [D.sup.i.sub.S](B) : J(S(B)) [right arrow] S(B), B [member of] B, i = 0,1, denoted [D.sup.i.sub.S].

* [S.sub.S](b) : S(B) [right arrow] J(S(B)), B [member of] B, denoted [S.sub.S].

One can verify that if the adjointness transformations of (RI, JS) are those obtained from the pairs (I, J) and (R, S), the isomorphism Trans(R, RI) [right arrow] Trans(JS, S) then sends [Rd.sup.i] in [D.sup.i.sub.S] (i = 0,1) and Trans(RI,R) [right arrow] Trans(S, JS) sends [R.sub.S] in [S.sub.S.] Similarly if [beta] [??] A [??] B is an adjoint pair then also is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and we have transformations and notations as follows:

* [d.sup.i.sub.H](B) : I(H(B)) - H(B), B [member of] B, i = 0,1 denoted [d.sup.i.sub.H].

* [S.sub.H(B)] : H(B) [right arrow] I(H(B)), B [member of] B, denoted [S.sub.H].

* T([D.sup.i]) : T(J(A)) T(A), A [right arrow] A, i = 0,1 denoted T[D.sup.i]

* T([S.sub.A]) : T(A) [right arrow] T(J(A)), A [member of] A, denoted TS.

with the obvious correspondences among them. Using the notation of [RCRR811] for two adjoint pairs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that assigns r [right arrow] [bar.r] by the isomorphism Trans([F.sub.1], [F.sub.2]) [right arrow] Trans([G.sub.1], [G.sub.2]) one has the following group of formulas:

i [bar.[Rd.sup.i]] =[D.sup.i.sub.S].

ii [bar.[R.sub.S]] = [S.sub.S].

iii [bar.[d.sup.i]] = T[D.sup.i].

iv [bar.[.sub.H]] = TS.

In our case we are using models Y : [delta] [right arrow] A each one with a adjoint pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a homotopy system [eta] = (I, [d.sup.0], [d.sup.1],s) in [[delta].sup.[omicron]]S with right adjoint [[eta].sup.*] = (J, [D.sup.0], [D.sup.1], S) and the system [eta] in A with adjoint [[eta].sup.*].

We will say that Y realizes [eta] into [eta] (or equivalently [[eta].sup.*] realizes into [[eta].sup.*]) if a natural transformation [epsilon] : [R.sub.Y]I [right arrow] I[R.sub.Y] exists such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for i = 0,1.

As has been seen in [RCRR811], if X : [delta] [right arrow] [[delta].sup.[omicron]] S denotes the inclusion functor, then the previous definition can be given equivalently, restricting I[R.sub.Y] and [R.sub.Y]I to X([delta]). That is to say demanding that for each x of 6 a morphism ey(x) : I(Y(x)) [right arrow] [R.sub.Y](I(Y(x)) exists such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] naturally on x.

When one uses [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the normal system in [[DELTA].sup.[omicron]]S, [Y.sup.0] final object of A and [eta] the system induced by product with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] one has a natural transformation, [[epsilon].sub.K] [R.sub.Y](K x A) [right arrow] [R.sub.Y](K) x [R.sub.y](A) [equivalent] [R.sub.Y](K) x [Y.sup.1], the first part of which is induced by the projections and the second by the natural isomorphism [R.sub.Y] ([DELTA][n]) [equivalent] [Y.sup.n] connecting the normal cylinder in [[DELTA].sup.[omicron]]S with the one of [R.sub.Y](K) in A. One also has the following commutative diagram for the inclusion [d.sup.i.sub.K] : K [right arrow] K x A,

It shows that [R.sub.Y] also connects the inclusions of objects into their cylinders. This is the situation we just generalized. In [RCRR811] conditions were given so that e becomes an isomorphism, case in which [R.sub.Y] respects cylinders together with the inclusions and therefore transmits homotopies. Our condition here is then weaker than the one used normally for transmission of homotopies.

Let us suppose now that F,G : Y [right arrow] Z are morphisms between models Y, Z : [delta] [right arrow] A. For each A, object of A, one has two morphisms [S.sub.F](A), [S.sub.G](A) : [S.sub.Z](A) [right arrow] Sy(A). Let us see the implication derived on them by the fact that F and [member of] are homotopic via [delta][eta].

Let us suppose that Y realizes [eta] into [eta]. Let F, G : Y [right arrow] Z be two morphisms. If F is [delta][eta]-homotopic to G, then a natural transformation [S.sub.Z](A) [right arrow] J(Sy(A)) exists that is a right (or [[eta].sup.*]-) homotopy of [S.sub.F](A) into [S.sub.G](A), for each A.

The proof is long and will be omitted.

Let Y,Z: [DELTA] [right arrow] A be two models of A which realize [eta] into [eta]. If Y is [delta][eta] homotopically equivalent to Z then a natural transformation A : Sy [right arrow] [S.sub.Z] exists such that [[lambda].sub.A] is a [eta] (equivalently [[eta].sup.*]) homotopy equivalence.

15. Examples

Let Y : [DELTA] [right arrow] A be a cosimplicial model of A where A is co complete, closed for finite products and with final object *. Let us suppose that [Y.sup.0] = *. Then one has in A a homotopy system [eta] = (I, [d.sup.0], [d.sup.1], s) where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let us suppose now that (-) x [Y.sub.1] has right adjoint. It is usually denoted [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] If [eta] denotes the normal system in [[DELTA].sup.[omicron]]S then Y realizes [eta] in [eta]. If a priori Y is [DELTA][eta] equivalent to Z then a cosimplicial morphism exists F : Z [right arrow] y, equivalence of [DELTA][eta] homotopy, for which [d.sub.i] [omicron] [F.sup.0] = [F.sup.1] [omicron] [d.sup.i].

This in turn implies at long last the following commutative diagram

Where Pr denotes the map ([R.sub.Z](k\), iso [omicron] [R.sub.Z](tt2)). Therefore one has the following:

Let A be a co complete category, closed for finite products, with final object *, and let Y : [DELTA] [right arrow] A be such that [y.sup.0] = * and (-) x [y.sup.1] has right adjoint. Let [eta] be the system induced by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and Z : [DELTA] [right arrow] A any model of A. If Y is [DELTA][eta] homotopically equivalent to Z then a natural transformation A : Sy [right arrow] [S.sub.Z] exists such that for each A of A, [[lambda].sub.A] : Sy(A) [right arrow] [S.sub.Z](A) is a homotopy equivalence in [[DELTA]sub.[omicron]]S.

Now we consider examples of homotopically equivalent models. There is a class of models in Top for which the homotopy extended from Top preserve details from the original that can be helpful. As an illustration let Y be a model of Top. We say that Y is convex if for each n, [Y.sup.n] is a convex subspace of a vector topological space (on M) and for each w : [n] [right arrow] [m], [w.sup.*] : [Y.sup.n] [right arrow] [Y.sup.m] is a lineal function. Then

i If y is a convex model, Z is any model of Top and F, [member of] : Z [right arrow] Y are any two cosimplicial continuous functions, then F is homotopic to [member of] by the homotopy of [DELTA]Top, extension of the normal homotopy of Top.

ii If y and Z are convex models of Top admitting cosimplicial functions Y [right arrow] Z and then Y is homotopically equivalent to Z for the homotopy of [DELTA]Top extension of that of Top.

In what follows the homotopy of [DELTA]Top (respectively of [DELTA]Kelly) extension of the normal homotopy of Top (respectively of Kelly) will be called the normal homotopy of [DELTA]Top (respectively of [DELTA]Kelly).

We have that if y is a convex model of Top (respectively Kelly) with cosimplicial points, then Y is nullhomotopic, that is to say, homotopically equivalent (for the normal homotopy) to a cosimplicial point.

The converse is true for any not necessarily convex space. In fact, if Y has cosimplicial points and Z doesn't have, then Y and Z are not homotopically equivalent.

For the model of the complexes [[DELTA].sup.n] of Top (or Kelly) its realization is the geometric realization [MJ57]. It is clear that [absolute value of K] is not in general equivalent (homotopically) to the discreet space [absolute value of -] [omicron] K, which happens to be the realization [R.sub.Y](K) for Y a cosimplicial point, even thought each [[DELTA].sup.n] is homotopically null.

Our definition of homotopy among models is somehow taking in consideration these facts. For convex spaces of the category Kelly the relationship is very precise as we will see. But for the general case the existence of a natural transformation A : [R.sub.Y] [right arrow] [R.sub.Z] such that for each X in [[DELTA].sup.[omicron]]S, [[lambda].sub.X] is a [eta] homotopy equivalence doesn't seem to imply that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] unless [[lambda].sup.-1.sub.X] form a natural transformation at least for X = [DELTA][n], n = 0,1, * * *.

Recall that for Y : [DELTA] [right arrow] A, RC(Y) ("The right cut of Y", see [RR76]) is the cosimplicial object of A defined this way: [[RC(Y)].sup.n] = [Y.sup.n+1], the [d.sup.i] : RC[(Y).sup.n] [right arrow] RC[(Y).sup.n+1] are the same [d.sup.i] : [Y.sup.n+1] [right arrow] [Y.sup.n+2] but only for i = 0,1,..., n + 1. Similarly for the [s.sup.j] : RC[(Y).sup.n] [right arrow] RC[(Y).sup.n-1] are the same [s.sup.j] : [Y.sup.n+1] [right arrow] [Y.sup.n] but only for j = 0,...n - 1. It is easy to see that RC : [DELTA]A [right arrow] [DELTA]A is a covariant functor and that d : 1 [right arrow] RC given for [[d(y)].sup.n] = [d.sup.n+1] : [Y.sup.n] [right arrow] [Y.sup.n+1] is a natural transformation. If Y : [DELTA] [right arrow] Top (respectively Kelly) is convex then so is RC(Y). Therefore if Y : [DELTA] [right arrow] Top is a convex model and there exists a continuous cosimplicial function RC(Y) [right arrow] Y then Y and RC(Y) are homotopically equivalent for the normal homotopy in [DELTA]Top.

Taking the case of [DELTA] one has that:

[d.sup.i] : RC[([DELTA]).sup.n] [right arrow] RC[([DELTA]).sup.n+1]

([x.sub.0],..., [x.sub.n]) [right arrow] ([x.sub.0],..., [x.sub.i-1], [x.sub.i],..., [x.sub.n+1])

for i - 0,..., n + 1

[s.sup.j] : RC[([DELTA]).sup.n] [right arrow] RC[([DELTA]).sup.n+1] ([x.sub.0],..., [x.sub.n+1]) [right arrow] ([x.sub.0],..., [x.sub.j] + [x.sub.j+1],..., [x.sub.n+1])

for j = 0,..., n + 1

RC([DELTA]) is not homotopically equivalent to [DELTA] because the first one has the cosimplicial point [P.sup.n] = (0.....0,1) [member of] [R.sup.n+2].

That example shows that, in general, do not exist simplicial morphisms RC(Y) [right arrow] (Y) and also that RC([DELTA]) is homotopically null.

Now RC([DELTA]) - P is a cosimplicial space and d : [DELTA] [right arrow] RC([DELTA]) - P since [P.sup.n] [not equal to] [d.sup.n]([[DELTA].sup.n]). Further RC([DELTA]) - P is convex since (0,..., 0,1) is a vertex of [[DELTA].sup.n+1]. Finally the [f.sup.n-1] : ([[DELTA].sup.n] - [P.sup.n-1]) [right arrow] [[DELTA].sup.n-1] that maps ([x.sub.0],..., [x.sub.n]) [right arrow] ([n-1.summation over (i=0)][x.sub.i]) ([x.sub.0],..., [x.sub.n-1]) is a cosimplicial continuous function (RC([DELTA]) - P) [right arrow] ([DELTA] - [P.sup.n]).

Therefore one can also "extend" [DELTA] to a model W with Wn = Rn+1 co faces and co degeneracies defined identically as those of [DELTA], thus [DELTA] forming a cosimplicial subspace of W. But W also has a cosimplicial point, namely 0. Therefore it is homotopically null and cannot exist continuous cosimplicial functions W [right arrow] [DELTA]. Contrary to the previous case W - 0 is not a cosimplicial space. But if we consider [H.sup.n] = (([x.sub.0],..., [x.sub.n])| [n.summation over (i=0)] [x.sub.i] = 0) then W - H is a cosimplicial space and 0 [subset or equal to] H [subset or equal to] W. Now the function W - H [right arrow] [DELTA], ([x.sub.0],..., [x.sub.n]) [right arrow] ([n.summation over (i=0)][x.sub.i]) ([x.sub.0],..., [x.sub.n]) is continuous cosimplicial. However we cannot apply results for convex models since W - H is not convex (it is not connected). In this case it is clear that W - H is not homotopically equivalent to [DELTA] since for each n [not equal to] 0, [(W - H).sup.n] is not homotopically null while [[DELTA].sup.n] is, and clearly from the definition one has that a necessary condition for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is that for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The model H will be called by obvious reasons the model of the hyperplane of W. We notice that [H.sup.0] = {0} and [H.sup.1] is the straight line at 135[degrees] that goes trough the origin. Also [d.sup.0] = [d.sup.1] : [H.sup.0] [right arrow] [H.sup.1], which implies the interesting fact that this model's homotopy is the equality and of course, homotopy equivalencies are the homeomorphisms.

If we consider Ti([k).sup.n]= {([x.sub.0],..., [x.sub.n]) [member of] [R.sup.n+1] | [SIGMA][x.sub.i] - k} with co faces and co degeneracies like the ones of W, then for each n, H[(k).sup.n] is homeomorphic to [H.sup.n]. However H(k) doesn't have cosimplicial points if k [not equal to] 0. This can be seen from the fact that if H[(k).sup.0] has cosimplicial points then has only one, since H[(k).sup.0] has one element, which most belong to the cosimplicial point. But [d.sup.0](k) = (0,k) and [d.sup.1](k) = (0,k). As (0, k) = (k, 0) [left and right arrow] k = 0, then H(k) doesn't have cosimplicial points if k [not equal to] 0 although [R.sub.H] and [R.sub.H](k) are level wise homeomorphic. Since [partial derivative][DELTA] is realized by [R.sub.H] into ((0,0)) [subset or equal to] ((x,y) [member of] [R.sup.2] | x + y = 0} and by [R.sub.H](k) into the subspace ((0, k), (k, 0)) of ((x, y) [member of] [R.sup.2] | x + y = k} then it is clear that the realizations are not homotopically equivalent. Apart from this H(k) ~ ~H(1) if k [not equal to] 0 [not equal to] l.

The question on whether two models can have homotopically equivalent realizations without being themselves homotopically equivalent is partially clarified for convex models of the category Kelly with next result which improves the late version: Let Y, Z be two convex models of Kelly. The following statements are equivalent:

i Y is homotopically equivalent to Z for the normal homotopy of [DELTA]Kelly.

ii There exist natural transformations [lambda] : [R.sub.Y] [right arrow] [R.sub.Z] and [rho] : [R.sub.Z] [right arrow] [R.sub.Y].

We notice that [DELTA], RC([DELTA]) - P, W and H(k) are models of the category Kelly. Since in this category and in that of the simplicial sets the functor (-) x A has right adjoint [().sup.A] for each A, then in particular the cylinder functors (-) x [DELTA] in [[DELTA].sup.[omicron]]S and (-) x I have right adjoint. It is also known that Kelly is a co complete category. At the level of realizations one thus has that

i There exists a natural transformation

[lambda] : [R.sub.RC([DELTA])] [right arrow] [R.sub.P] such that for each simplicial

set X, [[lambda].sub.x] : [R.sub.RC([DELTA])](X) [right arrow] [R.sub.P](X) is a homotopy equivalence.

ii There exists a natural transformation [eta] : [R.sub.W] [right arrow] [R.sub.P](X) such that for each simplicial set X, [[eta].sub.x] : [R.sub.W](X) [right arrow] [R.sub.P](X) is a homotopy equivalence.

iii There exists a natural transformation [rho] : [R.sub.RC([DELTA])-P] [right arrow] [R.sub.[DELTA]] = [parallel] that for each simplicial set X, [[rho].sub.x] : [R.sub.RC([DELTA])-P](X) [right arrow] [absolute value of X] s a homotopy equivalence.

Because of parts i and ii for each simplicial set the spaces [[pi].sub.0](X) (discreet), [R.sub.RC([DELTA])](X) and RW(X) are homotopically equivalent, naturally on X.

As for the singular functors since the homotopy of [DELTA] in Kelly is the normal homotopy then by using relations already given for singular functors one has that: there exists a transformation [epsilon] : [R.sub.RC([DELTA])-P] [right arrow] [S.sub.[DELTA]] - Sing such that for each space A of Kelly, [[epsilon].sub.A] : [R.sub.RC([DELTA])-P](A) [right arrow] Sing A is a homotopy equivalence in [[DELTA].sup.[omicron]]S.

Under the context of the theory here developed parts i and iii above are not extendable to singular functors since cosimplicial points do not carry the homotopy of [[DELTA].sup.[omicron]] S into that of Kelly. In fact suppose the opposite and consider the diagrams (i = 0,1).

When X is a simplicial point [R.sub.P](X) and [R.sub.P](X x [DELTA]) have a single point. Therefore [[epsilon].sub.X][R.sub.P]([d.sup.0.sub.x]) - [e.sub.X][R.sub.p]([d.sup.1.sub.X]), or equivalently (*,0) = (*,i),* [member of] [R.sub.P](X).

In the same way W and [R.sub.C]([DELTA]) are homotopically equivalent but the theory cannot be applied at level of singular functors. Indeed by a similar argument to the one just given it follows that W and RC([DELTA]) do not carry the homotopy of [[DELTA].sup.[omicron]] S in that of Kelly.

References

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[DFDJ] F. Bauer & J. Dugundji, Categorical Homotopy and Fibrations, Trans. Am. Math. Soc. 140 (1969), 239-256.

[CJ90] J. Cintura, Closed structures on reflective subcategories of the category of topological spaces, Theory and Appl of Categories 37 (1990), 237-247.

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Roberto Ruiz S. (1)

To the memory of Professor Jairo Charris Castaneda

(1) Departamento de Matematicas, Universidad del Valle, Cali, Colombia. E-mail: robruizs@yahoo.com AMS Classification 2000: 18G55
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