Pushout of simplicial sets

[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?

Thank you in advance

 

[jvicens]

for any help!

Hello,

Thank you for reaching out with your question about the pushout construction for Simplicial Sets. The pushout construction is an important tool in algebraic topology, and I can understand how it can be confusing at first.

First of all, let's clarify the definition of the pushout in this context. The pushout is not just the quotient of the disjoint union of X and Y, but it is the quotient of the disjoint union of X and Y with respect to the equivalence relation defined by the maps f and g. This means that the equivalence relation is not just "for all z \in Z f(z) ~ g(z)", but it is "for all z \in Z, the images of z under f and g are identified in the pushout". In other words, we are identifying the images of the same point in Z under both f and g.

Now, let's look at the example you provided. In this case, Z is a singleton set, so there is only one point in Z, denoted by *. This means that the equivalence relation in the pushout is simply identifying the images of * under f and g. In your example, f(*) = s3 and g(*) = s4, so the equivalence relation is identifying s3 and s4 in the pushout. This means that the final simplicial set will have two simplices in dimension 0, as you correctly noted, but they will be identified as the same point. This is because in the pushout, we are not just looking at individual simplices, but we are also considering their relations with each other.

As for the other two simplices, they are not "wrong" per se, but they are not included in the pushout because they are not identified by the equivalence relation. This is because they are not in the image of * under either f or g. As for the face operator, it is defined in the pushout in the same way as it is defined in simplicial sets, but taking into account the identified points.

I hope this helps clarify the pushout construction for Simplicial Sets. If you have any further questions, please don't hesitate to ask. Best of luck with your studies!