One predominant tendency in the algebra/category theory camp is to seek
out the minimal set of conditions needed to be able to perform a certain
technique, and then codifying this into a specific axiomatic system.
Thus, you only need to verify the axioms later on in order to get
everything else for free.

One such system is the theory of *triangulated categories*. This pops up
in homological algebra; where you like to work with Tor and Ext - both
of which turn out to be derived functors, generalizing the tensor
product and the homomorphism set respectively. With the construction of
the derived category, we can find a category, in which the tensor
product in that category is our Tor, and the hom sets is our Ext.

Once that entire yoga is worked through, you could start backtracking,
and pulling out all properties you used. Minimizing this set of
properties in one specific way leads to the concept of a triangulated
category, and in this post, I intend to retrace, using the definition of
a triangulated category, and looking more specifically on the case of
the derived category of the category of chain complexes of modules over
a fixed ring [tex]R[/tex] for a canonical example.

## Triangulated category

Weibel gives a definition, due to Verdier, of a triangulated category.
It is an additive category - meaning that each Hom(A,B) is an abelian
group, and the group operation distributes over composition of morphisms
- equipped with an automorphism T called the translation functor, and
with a distinguished family of triangles (u,v,w) - by which we mean
[tex]u\colon A\to B[/tex], [tex]v\colon B\to C[/tex] and
[tex]w\colon C\to TA[/tex]. We further expect these to fulfill the
following axioms

Every morphism u can be embedded into an exact triangle (u,v,w).
Furthermore, (1,0,0) is exact and exactness is closed under
isomorphisms of triangles.

If (u,v,w) is an exact triangle, then so is (v,w,-Tu) and
(-T:sup:-1w,u,v).

If (u,v,w) and (u',v',w') are exact triangles with f and g such that
gu=u'f, then there is a morphism h such that (f,g,h) is a morphism of
triangles. In clear text, this means that the following diagram has
the dotted arrow, such that everything commutes:

[tex]\begin{diagram}
A &\rTo^u& B &\rTo^v& C &\rTo^w& TA \\
\dTo^f && \dTo^g && \dDotsto^h && \dTo^{Tf} \\
A' &\rTo^{u'}& B' &\rTo^{v'}& C' &\rTo^{w'}& TA'
\end{diagram}[/tex]

Note that the leftmost square commutes by the condition on f and g.

Suppose we have exact triangles through the triples A,B,C', A,C,B'
and B,C,A'. Then these determine an exact triangle on C',B',A'. This
is, due to one of the ways to visualize it, called the octahedral
axiom.

Now, for an exact triangle (u,v,w), we can form the following diagram

[tex]\begin{diagram}

A &\rTo^1& A &\rTo^0& 0 &\rTo^0& TA \\

\dEq && \dTo^u && \dDotsto && \dEq \\

A &\rTo^u& B &\rTo^v& C &\rTo^w& TA

\end{diagram}[/tex]

where the morphism [tex]0\to C[/tex] exists due to the axiom (3).
Thus, since the diagram commutes, vu=0.

By forming similar diagrams for all the rotations, we also get wv=0
and (Tu)w=0. Thus, exact triangles behave as expected, if we view them
as representatives for long exact sequences.

## The funky stuff

So, what are these things good for? The first meatier construction we'll
see is that of a long exact sequence in cohomology. For a triangulated
category **K** and an abelian category *A*, we say that an additive
functor H from **K** to *A* is a cohomological functor whenever all
exact triangles have long exact sequences in cohomology. The cohomology
functor is the canonical example.

Having this machinery, we will - in future posts - strive to construct a
*derived category*, in which the quasi-isomorphisms are inverted by a
localisation process; and thus the objects are isomorphic when they have
induced isomorphisms on cohomology. Work in the derived category is very
reminiscent of work with derived functors - with Ext and Tor.