Properties of the Combinatorial Category

This introductory chapter is concerned with the combinatorial properties of pure

simplical complexes, and maps between them which never collapse any simplices. In this

category, Hom is a functor, and has all the usual properties. There is also the functor B,

composed of all the colorings of a complex which satisfies Hom(X,BY)

~

Hom(Y,BX).

Similarly, the collection of all automorphisms of a pure simplical complex is functorial.

In addition to these basic operations, we also study the join, wreath products, composi-

tion products, direct and inverse limits.

1. Hom and Cartesian Product

In this section we establish general properties of the Hom functor. A pure n-

complex is an n-complex with the property that every simplex is contained in an n-

simplex. Let X and Y be pure n-complexes. A simplicial map f : X

-+

Y is non-

degenerate iff maps all n-simplices onto n-simplices. Equivalently, no edge (!-simplex)

is mapped onto a point. Hom(X,Y) has as vertices all non-degenerate maps from X to

Y. A set of maps {

I

0

• • •

In }

is defined to be ann-simplex of Hom(X,Y) ifF for all z in

X,

{I

0

(z) · ·

·I

n(z) }

is ann-simplex of Y.

The Cartesian product

X:JI=

Y of two pure n-complexes X and Y has vertices

{ (z,11)l zEX , 1/E

Y }. The simplices are of the form

{(z,11)I11Ea}

for all vertices

z

of X

and simplices

a

of Y and

{(z,11)l zEr}

for all vertices 11 of Y and simplices

T

of X. We

write these simplices as

z:JI=a

and r.fl:ll· If X and Y are !-complexes (i.e. graphs ) then

this is the usual definition of Cartesian product.

The following result gives the basic functorial properties of Hom and Cartesian pro-

duct.

http://dx.doi.org/10.1090/conm/103