# On the definition of triangulated categories

Triangulated categories were introduced in the 1960s by Grothendieck and Verdier in order to develop homological algebra in the framework of derived categories. An example of a triangulated category is the category $\mathbf K(\mathscr C)$ of complexes in an additive category $\mathscr C$ with morphisms the homotopy classes of morphisms of complexes.

In 1991, Neeman published a paper (“Some new axioms for triangulated categories”) proposing a variant of the definition which is slightly stricter: in the $\mathbf K(\mathscr C)$, the distinguished triangles are those $X\xrightarrow f Y \to C_f$ which are explicitly given by the construction of the cone $C_f$.

This definition does not seem to be the one adopted in Neeman's book (Triangulated categories, Princeton University Press, 2001).

Here is one interest of Neeman's definition: when a triangulated category $\mathscr T$ is endowed with a $t$-structure, one gets a canonical functor $\mathbf D(\mathscr C)\to \mathscr T$, where $\mathbf D(\mathscr C)$ is the derived category of the heart $\mathscr C$ of the given $t$-structure. In Verdier's framework, the existence functor lies in additional (possibly nonexistent) choices. An example of this situation is given by Beilinson's theorem (“On the derived category of perverse sheaves”, 1990) that on a scheme the derived category of perverse $\ell$-adic sheaves is equivalent to the constructible derived category.

I am on the verge of teaching a basic introductory course on perverse sheaves which will force me to give one definition. Which one would experts recommend that one chooses?

Edit: I had initially thought that the definition in Neeman's book was the one in his paper, but this does not seem to be the case.

• Neeman's paper was an attempt to "improve" the notion of a triangulated category. This attempt was not really successful; so I would suggest you to ignore it in your course. – Mikhail Bondarko Dec 14 '16 at 9:08
• I would give the standard definition of a triangulated category and just mention that there are several notions of "enhanced" triangulated category that fix pathologies like the one you mention (in particular stable derivators, dg-categories, stable oo-categories). For the purpose of studying perverse sheaves it would seem to be enough to know that the functor $D^b(C)\to T$ exists in the case that $T$ is itself the derived category of an abelian category, with a nonstandard $t$-structure. In this case the construction in BBD using the filtered derived category is nice and conceptual. – Dan Petersen Dec 14 '16 at 14:11