*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 \(X\xrightarrow{f} Y \xrightarrow{i} \text{cone}(f)\).

- 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.

## 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 \[A\xrightarrow{f} B \xrightarrow{g} C\xrightarrow{h} \Sigma A.\]

- [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 \[\text{cone}(f):=\left(M^{i+1}\oplus N^i, d=\begin{pmatrix}- d_M^{i+1} & 0 \\ - f^{i+1} & d_N^i\end{pmatrix}\right).\]

*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 \[F(A)\to F(B)\to F(C).\]

*long exact sequence*\[\cdots F(C[-1])\to F(A)\to F(B)\to F(C)\to F(A[1])\to \cdots \]