Is the classifying space of a group infinite-dimensional? According to the textbook by Allen Hatcher, the classifying space B$\mathscr{C}$ of a category ${\mathscr C}$ is constituted with a set of simplices of which $n$-simplices are the strings of morphisms. Since a group $G$ can be taken as the set of morphisms of a category $\mathscr{C}$ with a single object, the textbook states that for this category, B$G$=B$\mathscr{C}$. Due to the definition of classifying spaces of categories, B$\mathscr{C}$ should be always infinite-dimensional. However, I saw on other materials, B$\mathbb{Z}=S^1$ which is only one-dimensional. Are the definitions different somewhere?

If a group is finite, any CW model of its classifying space must be infinite dimensional: <