\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip: triangulated categories
Broadly speaking, many of the structural results which make algebra such a useful tool come from the identifications between maps between objects, and the objects themselves. For example, in the setting of abelian groups, we can associate to every morphism \(f: A\to B\) a kernel, image, and cokernel. Because this language is so useful, there is a whole language of abelian categories whose morphisms come with the data of kernels, images, and cokernels.
In many categories there is not a reasonable way to construct the ``kernel'' of a morphism. For example, given a continuous map \(f: X\to Y\) between two topological spaces, we have no reasonable definition of a kernel of a continuous map. There is, however, the notion of a cone \(\operatorname{cone}(f)\), which remembers which points in \(Y\) came from the image of \(f\).
definition 0.0.1
Given \(f: X\to Y\) a continuous map, the cone of \(f\) is the space
\[\text{cone}(f):= (X\times I)\cup Y / \sim,\]
where the equivalence identifies the points \((x_1, 0)\sim (x_2, 0)\) and \((x, 1)\sim f(x)\).
We have continuous maps \(X\xrightarrow{f} Y \xrightarrow{i} \text{cone}(f)\).
In the homotopy category of pointed spaces, the mapping cone takes a special meaning. Let \(f: (X,x)\to (Y, y)\) be a pointed map. From this we can form \((Z, z):=(\cone(f), y)\) which is again a pointed space. We now address some relations between spaces \((X, x), (Y, y)\) and \((Z, z)\).
- Observe that the composition \(i\circ f: (X, x)\to (Z, x))\) is homotopic to the constant map. In the homotopy category of pointed spaces, we can therefore write \(i\circ f \sim 0\), where \(0: (X, x)\to (Z, x)\) is the map factoring through a point.
- We can additionally look at the cone
\[(Y, y)\xrightarrow{i}(Z, z).\]
This second cone can be rewritten in terms of the data \(f, X\) and \(Y\) as
\[ (Y\times J)\cup ((X\times I)\cup Y / \sim\]
where the relations are
\[(x_1, 0)\sim (x_2, 0) , (x, 1)\sim f(x)) , (y_1, 0)\sim (y_2, 0), (y, 1)\sim y.\]
This is homotopic to the suspension \(\Sigma X\) allowing us to write the ``long exact sequence''
\[(X, x)\to (Y, y)\to (Z, z)\to (\Sigma X, x)\to (\Sigma Y, y)\to \cdots\]
in the homotopy category of pointed spaces.
A triangulated category is a set of axioms for a category which encapsulates many of the properties of the cone construction.
definition 0.0.2
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.
definition 0.0.3
Given two cochain complexes \(M^\bullet\) and \(N^\bullet\) and a chain map \(f: M^\bullet\to N^\bullet\), the cone of \(f\) is a new chain complex
\[\text{cone}(f):=\left(M^{i+1}\oplus N^i, d=\begin{pmatrix}- d_M^{i+1} & 0 \\ - f^{i+1} & d_N^i\end{pmatrix}\right).\]
We observe that when \(X, Y\) are simplicial spaces and \(f:X\to Y\) is a simplicial map that simplicial cochains topological mapping cone (definition 0.0.1) are the cone-cochains,
\[C^\bullet(\cone(f))=\cone(f^*:C^\bullet(X)\to C^\bullet(Y)).\]
This does not end up giving a triangulated category.
However, the category of chain complexes with morphisms modulo chain homotopies is a triangulated category. This is called the homotopy category.
Triangulated categories capture many important aspects of homological algebra for chain complexes through the study of cohomological functors.
definition 0.0.4
Let \(\mathcal C\) be a triangulated category, and \(\mathcal A\) be an abelian category.
A cohomological functor is a functor \(F: \mathcal C\to \mathcal A\) which sends exact triangles
\[A\to B\to C\to A[1]\]
to exact sequences
\[F(A)\to F(B)\to F(C).\]
From this, we see that cohomological functors associate to exact triangles in \(\mathcal C\) a long exact sequence
\[\cdots F(C[-1])\to F(A)\to F(B)\to F(C)\to F(A[1])\to \cdots \]
proposition 0.0.5
Let \(\mathcal A\) be an abelian category, and \(\mathcal K(\mathcal A)\) the homotopy category, the functor
\[H^0: \mathcal K(\mathcal A)\to \mathcal A\]
is an example of a cohomological functor.
The idea of proof is to observe that every exact triangle in the homotopy category is homotopic to one of the form \(A\xrightarrow{f} B \to \cone(f)\), which is an exact sequence of chain complexes. The long exact sequence of cohomology groups arising from the snake lemma then proves the proposition.