The Stacks project

Definition 13.3.2. A triangulated category consists of a triple $(\mathcal{D}, \{ [n]\} _{n\in \mathbf{Z}}, \mathcal{T})$ where

  1. $\mathcal{D}$ is an additive category,

  2. $[1] : \mathcal{D} \to \mathcal{D}$, $E \mapsto E[1]$ is an additive auto-equivalence and $[n]$ for $n \in \mathbf{Z}$ is as discussed above, and

  3. $\mathcal{T}$ is a set of triangles (Definition 13.3.1) called the distinguished triangles

subject to the following conditions

  1. Any triangle isomorphic to a distinguished triangle is a distinguished triangle. Any triangle of the form $(X, X, 0, \text{id}, 0, 0)$ is distinguished. For any morphism $f : X \to Y$ of $\mathcal{D}$ there exists a distinguished triangle of the form $(X, Y, Z, f, g, h)$.

  2. The triangle $(X, Y, Z, f, g, h)$ is distinguished if and only if the triangle $(Y, Z, X[1], g, h, -f[1])$ is.

  3. Given a solid diagram

    \[ \xymatrix{ X \ar[r]^ f \ar[d]^ a & Y \ar[r]^ g \ar[d]^ b & Z \ar[r]^ h \ar@{-->}[d] & X[1] \ar[d]^{a[1]} \\ X' \ar[r]^{f'} & Y' \ar[r]^{g'} & Z' \ar[r]^{h'} & X'[1] } \]

    whose rows are distinguished triangles and which satisfies $b \circ f = f' \circ a$, there exists a morphism $c : Z \to Z'$ such that $(a, b, c)$ is a morphism of triangles.

  4. Given objects $X$, $Y$, $Z$ of $\mathcal{D}$, and morphisms $f : X \to Y$, $g : Y \to Z$, and distinguished triangles $(X, Y, Q_1, f, p_1, d_1)$, $(X, Z, Q_2, g \circ f, p_2, d_2)$, and $(Y, Z, Q_3, g, p_3, d_3)$, there exist morphisms $a : Q_1 \to Q_2$ and $b : Q_2 \to Q_3$ such that

    1. $(Q_1, Q_2, Q_3, a, b, p_1[1] \circ d_3)$ is a distinguished triangle,

    2. the triple $(\text{id}_ X, g, a)$ is a morphism of triangles $(X, Y, Q_1, f, p_1, d_1) \to (X, Z, Q_2, g \circ f, p_2, d_2)$, and

    3. the triple $(f, \text{id}_ Z, b)$ is a morphism of triangles $(X, Z, Q_2, g \circ f, p_2, d_2) \to (Y, Z, Q_3, g, p_3, d_3)$.

We will call $(\mathcal{D}, [\ ], \mathcal{T})$ a pre-triangulated category if TR1, TR2 and TR3 hold.1

[1] We use $[\ ]$ as an abbreviation for the family $\{ [n]\} _{n\in \mathbf{Z}}$.

Comments (0)

There are also:

  • 6 comment(s) on Section 13.3: The definition of a triangulated category

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0145. Beware of the difference between the letter 'O' and the digit '0'.