Let $\mathcal{D}\_{i}$ be a family of triangulated categories, labelled by a countable poset $I$ with a lowest element. Suppose further that for $i\leq j$, we have exact functors $F\_{i,j}: \mathcal{D}\_{i} \to \mathcal{D}\_{j} $ (and $F\_{i,i}=Id$).


Is the 2-(co)limit $\operatorname{colim}\_{I}\mathcal{D}\_{i}$ a triangulated category?
If not, are there further conditions on the categories or the functors that would ensure it is?


Thanks!



2-limits of triangulated categories
This is the kind of problem where working with stable $\infty$-categories is much easier than triangulated categories. Lurie considers the $\infty$-category of $\infty$-categories $Cat\_\infty$, and the subcategory $Cat\_\infty^{Ex}$ of stable $\infty$-categories and exact functors, and shows that $Cat\_\infty^{Ex}$ is closed under small limits and small filtered colimits (Higher algebra, §1.1.4). And the argument is easy, given the $\infty$-categorical formalism. So the answer is positive if the poset $I$ is directed and your diagram can be enhanced to a diagram of stable $\infty$-categories.


In the classical triangulated world this kind of question is much thornier. One example is the definition of the derived category of $\ell$-adic sheaves in étale cohomology. Right at the start of Deligne's "Weil II" (1980), he defines the "derived category" of $\mathbf Z\_\ell$-sheaves as the 2-limit of the derived categories of $\mathbf Z/\ell^n$-sheaves. It turns out that the 2-limit is triangulated in this case, but it's not automatic, the argument is delicate and uses finiteness properties of the Hom-sets. See (1.1.2)(d) of Deligne. Compare with the much more powerful $\infty$-categorical results!


