Section 13.8 (05RN): The homotopy category

Let $\mathcal{A}$ be an additive category. The homotopy category $K(\mathcal{A})$ of $\mathcal{A}$ is the category of complexes of $\mathcal{A}$ with morphisms given by morphisms of complexes up to homotopy. Here is the formal definition.

Definition 13.8.1. Let $\mathcal{A}$ be an additive category.

We set $\text{Comp}(\mathcal{A}) = \text{CoCh}(\mathcal{A})$ be the category of (cochain) complexes.
A complex $K^\bullet $ is said to be bounded below if $K^ n = 0$ for all $n \ll 0$.
A complex $K^\bullet $ is said to be bounded above if $K^ n = 0$ for all $n \gg 0$.
A complex $K^\bullet $ is said to be bounded if $K^ n = 0$ for all $|n| \gg 0$.
We let $\text{Comp}^{+}(\mathcal{A})$, $\text{Comp}^{-}(\mathcal{A})$, resp. $\text{Comp}^ b(\mathcal{A})$ be the full subcategory of $\text{Comp}(\mathcal{A})$ whose objects are the complexes which are bounded below, bounded above, resp. bounded.
We let $K(\mathcal{A})$ be the category with the same objects as $\text{Comp}(\mathcal{A})$ but as morphisms homotopy classes of maps of complexes (see Homology, Lemma 12.13.7).
We let $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, resp. $K^ b(\mathcal{A})$ be the full subcategory of $K(\mathcal{A})$ whose objects are bounded below, bounded above, resp. bounded complexes of $\mathcal{A}$.

It will turn out that the categories $K(\mathcal{A})$, $K^{+}(\mathcal{A})$, $K^{-}(\mathcal{A})$, and $K^ b(\mathcal{A})$ are triangulated categories. To prove this we first develop some machinery related to cones and split exact sequences.

The Stacks project

13.8 The homotopy category

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