## Categories in Algebra, Geometry and Mathematical Physics

### Conference in honour of Ross Street's sixtieth birthday

#### July 11-16 2005, Macquarie University, Sydney

Fri, 15 July: 14:45 - 15:25

##### Triangulated Derivators

###### Maltsiniotis, Georges (Université Paris 7)

The aim of the theory of derivators is to introduce the ``good'' framework for homotopical and homological algebra. Triangulated derivators are more particularly concerned with homological algebra. Soon after the introduction of triangulated categories by Verdier, it was observed that this notion was insufficient to take hold of all the structures carried by the derived category of an abelian one. In particular the derived category with its structure of triangulated category does not satisfy a universal property. Moreover homotopical limits or colimits in a triangulated category are defined only up to a non unique isomorphism. In particular the cone is not functorial.

In his thesis Bernhard Keller proved a universal property of the derived category of an abelian, or even an exact, category by considering ``towers of triangulated categories''. Instead of looking only to the derived

category of the abelian one, he considers the derived categories of the categories of cubical diagrams. More generally, fix for example a Grothendieck category $\mathcal{A}$. For every small category $I$, the category of presheaves on $I$ with values in $\mathcal{A}$ is still a Grothendieck category, and one can consider its derived category $\mathbf{D}(I)$. The derived category of $\mathcal{A}$ is just $\mathbf{D}(e)$ ($e$ being the final category). For every functor $u:I\to J$ between small categories the inverse image of presheaves induces a functor $u^*:\mathbf{D}(J)\to\mathbf{D}(I)$, and every natural transformation $\alpha:u\to v$ induces a natural transformation $\alpha^*:v^*\to u^*$. This defines a contravariant $2$-functor from the $2$-category $\mathcal{C}at$ of small categories to the $2$-category $\mathcal{C}AT$ of all categories. The idea of Grothendieck is that it is this $2$-functor, considered as a structure on the derived category $\mathbf{D}(e)$, that carries all ``the triangulated structure'' of the derived category.

A triangulated derivator is a 2-functor $\mathbf{D}$ defined on a full $2$-subcategory $\mathcal{D}iag$ of $\mathcal{C}at$ (satisfying some stability properties) with values in $\mathcal{C}AT$ satisfying a list of axioms. Some of these axioms are similar to those defining the notion of ``homotopy theory'' of Alex Heller. The most important is the one saying that for every functor $u:I\to J$ in $\mathcal{D}iag$, the functor $u^*:\mathbf{D}(J)\to\mathbf{D}(I)$ has a left adjoint $u^{}_!$ and a right adjoint $u^{}_*$. Some other axioms are close to those considered by Jens Franke in his study of systems of triangulated diagram categories. The basic theorem of the theory of triangulated derivators is that for every category $I$ in $\mathcal{D}iag$, $\mathbf{D}(I)$ is a triangulated category, and that for every functor $u:I\to J$ in $\mathcal{D}iag$, $u^*$ defines a triangulated functor. Bernhard Keller has proved that to every exact category $\mathcal{E}$ one can associate a triangulated derivator $\mathbf{D}_{\mathcal{E}}$ defined on finite direct categories such that $\mathbf{D}_{\mathcal{E}}(e)$ is the derived category of $\mathcal{E}$.

One can define a K-theory of triangulated derivators. It is conjectured that if $\mathcal{E}$ is an exact category, the K-theory of the triangulated derivator $\mathbf{D}_{\mathcal{E}}$ coincides with the one of the exact category $\mathcal{E}$, and that the K-theory of triangulated derivators satisfies some additivity and localization theorems. Amnon Neeman has a plan for proving the additivity conjecture.