## 13.8 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.

1. We set $\text{Comp}(\mathcal{A}) = \text{CoCh}(\mathcal{A})$ be the category of (cochain) complexes.

2. A complex $K^\bullet$ is said to be bounded below if $K^ n = 0$ for all $n \ll 0$.

3. A complex $K^\bullet$ is said to be bounded above if $K^ n = 0$ for all $n \gg 0$.

4. A complex $K^\bullet$ is said to be bounded if $K^ n = 0$ for all $|n| \gg 0$.

5. 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.

6. 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).

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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).