Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products?

Question

Does the homotopy category functor $\mathsf{Top}_\ast\rightarrow \mathsf{hTop}_\ast$ create products? I know it preserves products, but it seems to actually create them.

Answer 1

No, as it does not reflect products. For example the cone $\mathbb{R} \gets \mathbb{R} \to \mathbb{R}$ over the constant diagram $\{ \mathbb{R} \quad \mathbb{R} \}$ gets sent via the localization functor to a cone isomorphic to $* \gets * \to *$ (where $*$ is a singleton). This is a product in $\mathsf{hTop}_*$ (a singleton times a singleton is a singleton). But $\mathbb{R}$ is not the product $\mathbb{R} \times \mathbb{R}$ in $\mathsf{Top}_*$...