Tags: #todo

homotopy category

The homotopy category of \(\mathsf{C}\): same objects, \({\mathsf{ho}}\mathsf{C}(x, y) := \pi_0 \mathsf{C}(x, y]\).

Define \({\mathsf{ho}}\mathsf{C}\) as the universal category equipped with a functor \(\mathsf{C} \to {\mathsf{ho}}\mathsf{C}\) sending weak equivalences to isomorphisms. Morphism \begin{align*} \mathsf{C}(x, y) = {\mathsf{ho}}\mathsf{C}(RQx, RQy) \end{align*}