Index

With the slider all the way to the right, this shows a complex $K$. With the slider all the way to left, we see the barycentric subdivision $K'$. Sliding the slider shows the effect of the map $|\mu|\colon|K'|\to|K|$ and the straight line homotopy between $\beta$ and $|\mu|$.

Note that the $2$-simplices of $K'$ have different colours. The red ones get squashed by $|\mu|$ to a single point, the blue ones get squashed by $|\mu|$ onto a line, and the green ones get stretched by $|\mu|$ to cover a $2$-simplex of $K$. Each $2$-simplex of $K$ is divided into two red $2$-simplices, three blue $2$-simplices and a single green $2$-simplex of $K'$.