CHAPTER I

Model approximations and bounded diagrams

1. Notation

The symbol A denotes the simplicial category (cf. [40, section 2]), in which

the objects are the ordered sets [n] = {n • • • 0}, and the morphisms are

weakly monotone maps of sets. The morphisms of A are generated by coface maps

di : [n — 1] — [n] and codegeneracy maps Si : [n + 1] — [n] for 0 i n, subject to

well-known cosimplicial identities. A simplicial set is then a functor K : Aop — Sets

where Sets denotes the category of sets. One usually denotes the set K[n] by Kn.

A morphism between two simplicial sets is by definition a natural transformation

of functors. A simplicial set K can be interpreted as a collection of sets (Kn)no

together with face maps di : Kn — Kn-\ and degeneracy maps si : Kn — » ifn+i

which satisfy the simplicial identities. For a description of how to do homotopy

theory in the category of simplicial sets see [7], [12], [40] and [41]. In this paper

we use the symbol Spaces to denote the category of simplicial sets, and by a space

we always mean a simplicial set.

An element a G Kn is called an n-dimensional simplex of K. It is said to be

degenerate if there exists a' G i^

n

-i and 0 i n — 1 such that Sid' = a.

The standard n-simplex A[n] is an important example of a space. By def-

inition, its set of ^-simplices is given by (A[n])k := morA([fc], [n]). There is a

distinguished n-dimensional simplex i in A[n], namely the unique non-degenerate

one which comes from the identity map [n] — [n]. The assignment / i— f(i) yields

a bijection of sets

mor5paces(A[n],i;f)

— • Kn. Thus we do not distinguish between

maps A[n] — • K and n-simplices in K. If a G Kn is a simplex, we use the same

symbol a : A[n] ^ K to denote the corresponding map.

The simplicial subset of A[n] that is generated by the simplices {dit | 0 i n}

is denoted by 9A[n] and called the boundary of A[n]. The simplicial subset of 9A[n]

that is generated by the simplices {dit \ i ^ k} is denoted by A[n, k] and called a

horn . There are obvious inclusions A[n, k] C 9A[n] C A[n].

Let C be a category and / be a small category. By Fun(I,C) we denote the

category whose objects are functors indexed by / with values in C, and whose

morphisms are natural transformations. For any object X G C, there is a constant

diagram X : / — C with value X. This assignment defines a functor C — Fun(I, C).

Its left adjoint is called the colimit and is denoted by colimi : Fun(I,C) — C. Its

right adjoint is called the limit and is denoted by h'raj : Fun(I,C) — » C. If this left

(respectively right) adjoint exists for any small category /, we say that C is closed

under colimits (respectively limits).

Let F : / — C be a functor. The object colimjF is equipped with a natural

transformation F — colimiF, from F to the constant diagram with value colimiF.

This natural transformation has the following universal property. For an object

X G C, any natural transformation F — X factors uniquely as F — colimiF — X.

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