Page 114 - Textos de Matemática Vol. 45

106 Chapter 9. commutative square in E like the following right hand side one:
Appendix
Rn[f].... Rn (x)
n ′ R[f]....
....R[f]oo
p0 //X f //Y s0 //
p1
R(x) xy
p′0 // ....R[f] s′0 //X′
p′ 1
//Y′.
f′
′ oo
determines a simplicial map (x, R(x), ..., Rn(x)) between the respective associ- ated simplicial objects.
A n-simplicial (or n-truncated simplicial) object, for n ≥ 1, has a simi- lar definition except that everything is only defined up to n. Hence we get a category denoted by SimplnE.
Given a split epimorphism (f,s): X Y, the associated simplicial ob- ject is endowed with a further family of degeneracy maps sn : R[f ]n−1 → R[f ]n satisfying the same simplicial identities and making it an augmented split sim- plicial object:
oos0
oos0
p0 //
R[f] oo s0 X oo f
.... R[f]n
TT oo
sn−1
.... R[f]2
SS oo
RR
p1
// s
R[f]n−1 ....
sn s2s1
// Y
starting with s1 : X → R[f ] defined by s1 (x) = (sf (x), x).
9.5 Internal preorders, categories and monoids
A 1-truncated simplicial object is just a reflexive graph:
d0 // X1oos0 //X0
d1
The object X0 is called the “object of objects”, while the object X1 is called the “object of arrows”; the map d0 is called the “domain” map, while the map d1 is called the “codomain” map. It is a reflexive relation as soon as the pair (d0,d1) is jointly monomorphic. If E is a variety of universal algebra, an inter- nal reflexive relation is just a reflexive relation which is compatible with all the operations.
s1