Product of a finite complex and a point

[Ad123q]

I have in my algebraic topology notes, as a step in the proof of another theorem, that the product of a finite simplicial complex X with a single point (a 0-simplex) is isomorphic to the finite simplicial complex X, but I can't see why this is so.

i.e. Xx{point} isomorphic to X

Thanks

 

[Tinyboss]

I'm a little rusty on the details of forming products of simplicial complexes, but it's probably only necessary to note that taking the product of a topological object and a single point is about like forming the product of an (algebraic) group with the trivial group, or multiplying a real number by 1, in that nothing changes.

Anytime the objects in the product are pairs of objects from the factors, and the second coordinate is always the same. Then the isomorphism is just projection onto the first coordinate, which has as its inverse inclusion into the product.

 

[Landau]

I think we can generalize this to any category: if * is a terminal object and A is any object, then the product of A and * is canonically isomorphic to A. We just show that A has the universal property of the categorical product of A and *, which we know to be unique up to unique isomorphism.

Let the two projections $\pi_A:A\to A$ and $\pi_\star:A\to \star$ be the identity and the unique one, respectively. Now suppose Z is any object with arrows $p_A:Z\to A$ and $p_\star:Z\to \star$. Then there is indeed a unique arrow $u:Z\to A$ such that $\pi_A\circ u=p_A$ and $\pi_\star\circ u=p_\star$: it is of course u=p_A!

So $$A\times\star\cong A$$.