CHAPTER 1
Simplicial Operators and Simplicial Sets
1. Simplicial Operators
DEFINITION 1 (the category A+). Let A
+
denote the skeletal category of finite
ordinals and order preserving functions. In other words, A
+
has:
Objects ordered sets [n] = {0 1 n} one for each n 1,
Maps a: [n] - [m] which are order preserving functions from the or-
dered set [n] to the ordered set [ra],
Composition simply that of functions, which is well defined since the
composite of order preserving maps is again order preserving.
We use the notation id[n] to denote the identity function on [n]. The maps of A
+
are often referred to as simplicial operators, for reasons which will become clear.
NOTATION
2 (the Topologist's A). The category A
+
is known as the Alge-
braist's A since, as recalled later on, this category "classifies" the algebraic theory
of monoids. However, for much of this work we will be interested in studying simpli-
cial operators from the topological perspective. To this end, we will often restrict
our attention to A the full subcategory of A
+
whose objects are the non-zero
ordinals [n] for n 0. Following the usual tradition, we will usually refer to this
category as the Topologist's A.
Under the topological interpretation of A, the object [n] is considered to be a
combinatorial rendering of the standard n-dimensional simplex. This explains our
rather peculiar, but nonetheless entirely standard, use of the notation [n] to denote
the (n + l)
t h
ordinal.
NOTATION
3 (faces and degeneracies). The following standard notation and
nomenclature will be used throughout:
The injective maps in A
+
are referred to as face operators.
For each integer n £ N and each j G [n] define the simplicial operator
Sf : [n - 1] ^ [n] by
53r(i)
= l '
i H i
'
v y
| i + 1 otherwise.
This is called the j t h elementary face operator of [n].
The surjective maps in A
+
are referred to as degeneracy operators.
For each integer n G N and each j £ [n] define the simplicial operator
r? : [n + 1] ^ [n] by
a3'^
n(i)
= i
i lii^Ji
' \ i 1 otherwise.
This is called the j
t h
elementary degeneracy operator of [n].
l
Previous Page Next Page