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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

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  • $\begingroup$ What do you not understand in this definition ? $\endgroup$ Apr 28, 2016 at 17:35
  • $\begingroup$ @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. $\endgroup$ Apr 28, 2016 at 17:38
  • $\begingroup$ @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 $Hom(X,Y)$ $\endgroup$ Apr 28, 2016 at 17:41
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    $\begingroup$ 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. $\endgroup$ Apr 28, 2016 at 17:45
  • $\begingroup$ @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... $\endgroup$ Apr 28, 2016 at 17:48

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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).

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    $\begingroup$ The truncation of the simplicial localisation always recovers ordinary localisation. (This is an instance of the fact that the left adjoint of the composite is the composite of the left adjoints.) $\endgroup$
    – Zhen Lin
    Apr 28, 2016 at 20:10

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