# Pushout of simplicial sets

1. Sep 3, 2009

### heras1985

Hi everyone,

I have some problems with the pushout construction for Simplicial Sets

The definition of pushouts is as follows:
Let X, Y, Z simplicial sets and f: Z -> X and g: Z -> Y simplicial morphisms, then the pushout is the quotient of the disjoint union of X and Y with the equivalence relation that for all z \in Z f(z) ~ g(z).

Then, for instance:
Let Z={*}, X=S^3 (the 3-sphere (1 simplex in dimension 0 (*_3) and 1 simplex in dimension 3 (s3))), Y=S^4 (the 4-sphere (1 simplex in dimension 0(*_4) and 1 simplex in dimension 4 (s4))), f(*) =s3 and g(*)=s4,
then, what is the final simplicial set?
it wil have two simplex in dimension 0 (*_3 and *_4) but what's wrong with the other two simplex? and with the face operator?