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9.4. Simplicial objects
subject to the following identities: didj+1 =djdi, i≤j
sj+1si =sisj, i≤j
105
p0 oo s0
R[f ]n−1
p0
oos0 p0 //
p1 // R[f] oo s0 X // Y oo KK // f
pn
disj =sj−1di, i<j
disj =1, i=j,j+1 disj =sjdi−1, i>j+1
Any map f : X → Y in E determines a simplicial object by means of its iterated kernel equivalence relations:
.... R[f ]n
oo JJ sn−1
....
....
R[f ]2
Given any (n + 1)-uple (x0, x1, ..., xn), we denote by (x0, x1, ..., xˆi, ..., xn) the n-uple where xi is erased. In the category Set, the set R[f]n is the set of (n+1)- uples (x0, x1, ..., xn) of elements of X such that f(x0) = f(x1) = ... = f(xn). The face map pi is defined by pi(x0,x1,...,xn) = (x0,x1,...,xˆn−i,...,xn) and thedegeneracymapsi bysi(x0,x1,...,xn)=(x0,x1,...,xn−i,xn−i,...,xn).
A morphism of simplicial objects is a graded set of maps fn : Xn → Yn, n ∈ N which makes the following diagram commutative:
d0d0 d0
oos0 oo s0 oos0 d0
s1 p2
p1
// //
d1
.... Xn+1 dn+1
Xn Xn−1 .... .... X2 d1 // X1 oo s0
X0
ooLLooKK ooLL
sn
sn−1 dn
s1 d2
fn+1
.... Yn+1
sn sn−1 s1 dn+1 dn d2
fn
d0d0 d0
s0
fn−1
f2 f1
f0
oos0 YLLn
oo
oooo oo
oo s0 d0
// Y
0 Yn−KK1 .... .... Y2 d1 // LL1 oo s
Y
// 0
d1
This way, we get the category SimplE of simplicial objects in E. Any