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.

## Chain complexes form a triangulated category

Proposition: For the category of chain complexes of R-modules, with homotopy equivalence classes of chain maps as morphism, we have a triangulated category structure.

^{i}=A

^{i+1}. We further call a triangle (u,v,w) exact if it is isomorphic to a triangle (u',v',d) on the complex triple (A',B',cone(u)). That is, the exact triangles are (up to isomorphism) the triangles that go through a mapping cone.

### Digression: The cone of a chain map

Suppose [tex]f\colon B^*\to C^*[/tex] is a chain map. We define the mapping cone cone(f) as the complex [tex]B[-1]\oplus C[/tex], with degree n part [tex]B^{n+1}\oplus C^n[/tex] and with differential [tex]d(b,c)=(-d(b),d(c)-f(b))[/tex].

^{n}

For the other direction, suppose we can extend f to a chain map [tex](c,c')\mapsto f(c')-sc[/tex]. Then the calculation above shows that thus df(c')-fd(c')+f(c)-dsc-sdc=0. But f still is a chain map, so the df(c')-fd(c') portions disappears. And the remaining equation says that for any c in C, we need f(c)=(ds+sd)(c). Thus the s used in extending f to a chain map from the cone is precisely a contracting homotopy.

### Return to the triangles

This last section gives the prototype for our exact triangles. Anything isomorphic to this is exact. Suppose [tex]u\colon A\to B[/tex] is a chain morphism. Then this fits into the triangle [tex](u,v,\delta)[/tex] for [tex]v\colon b\mapsto (0,b)[/tex] and [tex]\delta\colon(a,b)\mapsto-a[/tex]. Furthermore, since the mapping cone of the identity map is split exact, this splitting map gives a null homotopy of any map to that cone. So the cone is isomorphic to the zero object in this category, and thus the identity map has the required embedding. By defining exact triangles as all triangles isomorphic to a triangle with a cone, it's clear that isomorphism of triangles preserves exactness.

Rotation of triangles follows after working out that the maps appearing when matching the rotation to what it should be are chain homotopy equivalences. I'm not going to do this here.

The existence of morphisms follows since the construction of a mapping cone is natural.

And right about now, I'm slowly growing bored by the verification of axioms. The octahedral axiom seems (by a quick readthrough) to be a diagram chase to find the right homotopies and the right maps to get to the end.

## 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.