Let $\mathscr{C}$ be a complete and cocomplete category, and let $W$ be the collection of weak equivalences relative to some model on $\mathscr{C}$. We can form the homotopy category by localizing at weak equivalences: $$Ho(\mathscr{C}) = W^{-1}\mathscr{C}$$
My question is seemingly very simple; how do we define homotopy in an abstract category? I understand it for chain complexes and topological spaces, etc. But I am having trouble understanding it in general.
I was reading through Toen's Article on derived algebraic geometry and it occurred to me that I do not understand his definition on pg. 21
Calling this category the homotopy category at this level of generality is a bit misleading. The sense in which it is a homotopy-category-as-in-morphisms-up-to-homotopy comes from, for example, taking the simplicial localization first. This is a simplicially enriched category which presents an $\infty$-category. It has a notion of homotopy between maps given by the 1-simplices in its mapping spaces. Any $\infty$-category has a homotopy category given by applying $\pi_0$ to its mapping spaces (so taking maps up to homotopy), which I think in the case of the simplicial localization recovers the ordinary localization (maybe with some additional hypotheses).