# On the equivalence of two different notions of the join of two topological spaces

I've seen two different notions of the join of two topological spaces $$X$$ and $$Y$$, namely:

$$X*Y := \left(X \times Y \times I \right)/ \sim$$ where $$\sim$$ is the equivalence relation generated by $$(x',y,0)\sim (x'',y,0)$$ and $$(x,y',1)\sim (x,y'',1)$$. In other words, the pushout of the diagram $$CX\times Y \leftarrow X\times Y\to X \times CY$$ where $$CZ$$ denotes the cone of $$Z$$.

and,

For the cone of a topological space $$Z$$ let $$h_Z$$ be the height map on $$CZ$$, namely $$h_Z(z,t) = 1- t$$ for $$(z,t)\in CZ$$. Then, the join $$X*Y$$ is defined as the subspace of $$CX\times CY$$ given by $$\{(x,t) \times (y,s)\mid h_X(x,t) + h_Y(y,s) = 1\}$$.

The latter is obviously associative, while the former fails to be associtaive in some pathological cases. In which subcategory of spaces do they coincide? (I hope that at least on compactly generated spaces, but I wasn't able to prove it).

Update: Okay, I believe that some ideas are clearer now.

It is enough to show that the second construction satisfies the universal property of the quotient given in the first construction. Denote by $$X *' Y$$ the second construction of the join. Consider the map $$q\colon X\times Y \times I\to X *' Y$$ given by $$(x,y,t) \mapsto ((x,1-t),(y,t))$$. I believe the trick here is encoded in the fact that this map is a quotient map. Observe that $$q$$ can be realized as the composition $$X\times Y \times I \xrightarrow{id\times \Delta} X\times Y\times I \times I = (X\times I)\times (Y\times I) \xrightarrow{\sigma \times id} (X\times I)\times (Y\times I)\xrightarrow{q_X\times q_Y} X*'Y,$$ where $$q_X$$ and $$q_Y$$ denote the quotient maps to $$CX$$ and $$CY$$ respectively and $$\sigma(x,t) = (x, 1-t)$$. Thus $$q$$ is a quotient map in the case where $$q_X\times q_Y$$ is a quotient map (which I believe is true in the category of compactly generated spaces). The fact that $$q$$ factors uniquely those maps which are constant in the equivalence classes defining $$X* Y$$ is more or less clear.