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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