\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\QQ{{\mathbb Q}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \)

Symplectic snippets

triangulated Category

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.

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