In mathematics, especially algebraic topology, the homotopy dimension of a topological space does not have a fixed meaning. However, it can refer to the minimum dimension of a CW complex that is homotopy equivalent to the space. the Lebesgue covering dimension of the space homotopy dimension introduced by Lurie in his Higher Topos Theory for an ∞-topos.
Further reading Dimension of a homotopy type Homotopy dimension of a mapping 次元論 in Japanese homotopy dimension in nLab (this article is about homotopy dimension in Lurie' higher topos theory)