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The triple product:
This is probably the formal linear combination approach you mentioned.
The triple product:
- We start with three spaces $X$, $Y$ and $Z$.
- We construct a quotient of $X\times Y\times Z \times \Delta$, with $\Delta$ a $2$-simplex.
- We take a quotient where at each vertex two of the three spaces are identified to points.
This is probably the formal linear combination approach you mentioned.
Added double $$ for latex
KaTeX parse error: Undefined control sequence: \Y at position 9: X \times\̲Y̲ ̲\times Z\times …: X \times\Y \times Z\times R
where $R = [0,1]\times [0, 1]$