\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
definition 0.0.1
A triangulated category is an additive category \(\mathcal C\), along with the structure of
- an additive automorphism \(\Sigma: \mathcal C\to \mathcal C\), called the shift functor and
- a collection of triangles, which are triples of objects and morphisms written as
\[A\xrightarrow{f} B \xrightarrow{g} C\xrightarrow{h} \Sigma A.\]
Denote by \(X[n]=\Sigma^nX\).
This data is required to satisfy the axioms for a triangulated category,
- [TR1], concerning which triangles must exist:
- The triangle \(X\xrightarrow{\id} X\to 0 \to \Sigma X\) is an exact triangle
- For every morphism \(f:X\to Y\) there exists an object (called the cone) so that \(X\to Y \to \cone(f)\) is an exact triangle
- Every triangle which is isomorphic to an exact triangle is exact.
- [TR2], concerning the interchange between exact triangles and suspension. If \(X\xrightarrow{f} Y \xrightarrow{g} Z\xrightarrow{h} X[1]\) is an exact triangle, then so are \(Y\to Z\to X[1]\to Y[1]\) and \(Z[-1]\to X\to Y\to Z\).
- [TR3] Given a commutative square, if we complete the rows to exact triangles, then there exists a morphism between the third objects making everything commute.
- [TR4] The octahedral axiom, which states that given exact triangles
\begin{align*}
X\xrightarrow{f} Y \xrightarrow{g} Z'\xrightarrow{h} X[1]\\
Y\xrightarrow{i} Z \xrightarrow{j} X'\xrightarrow{k} Y[1]\\
X\xrightarrow{i\circ f} Z \xrightarrow{l} Y'\xrightarrow{m} Z[1]
\end{align*}
There exists a triangle \(Z'\to Y'\to X'\to Z'[1]\).
making the diagram of these triangles commute.