For $\mathcal{C}$ topologically enriched category or a similar category “of spaces” (such as SSet or Infinity-Grpd), the hom-object $\mathcal{C}(X,Y)$ for any two objects $X$ and $Y$ is called the hom-space.
For enrichment in Set, a discrete hom-space is an ordinary hom-set.
hom-space, derived hom-space