# Reference/Definition of Homotopy in an Abstract Category

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

• What do you not understand in this definition ? Apr 28, 2016 at 17:35
• @CaptainLama Mostly notational I think. First of all, what is $M$? Secondly what is $C(X)$? Maybe once I understand that, I can figure out what $i$, $j$, and $h$ are. Apr 28, 2016 at 17:38
• @CaptainLama I was trying to prove the claim in the next paragraph that when $X$ and $Y$ are cofibrant and fibrant, respectively, homotopy gives an equivalence relation on $Hom(X,Y)$ Apr 28, 2016 at 17:41
• I think $M$ might be a typo for $C$. And as I understand it $C(X)$ is any object that fits in a diagram such as indicated. You can understand the definition as "there exist an objetc $C(X)$ and maps $i$, $j$, $h$ such that...". You can see that $C(X)$ is supposed to imit $I\times X$ in classical homotopy, and $i$ and $j$ would be the two inclusions $X\to X\times I$ at each end. Apr 28, 2016 at 17:45
• @CaptainLama OK. I had thought that and that's how I was treating it. He, and most people, just typically use $i$ and $j$ to mean cofibrations or fibrations and $C(X)$ means a whole bunch of different stuff depending on the settting. Thanks for the classical example. I thought about that for a second, but kept getting confused somehow... Apr 28, 2016 at 17:48

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).