t-cat

The t-cat of a topological space $X$ is the minimal number of open sets that cover $X$ such that each open set in the cover has the homotopy type of the unit circle $S^1$. This means that for each open set $U$, the inclusion $U\stackrel{i}{\hookrightarrow }X$ is homotopic to some factorization $U\stackrel{a}{\to }{S}^1\stackrel{b}{\to }X$, i.e.

$$i\simeq b\circ a.$$

When $X$ is manifold, this is related to the round complexity of $X$.