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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!

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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!

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  • $\begingroup$ Thanks Dan that's very useful- I was trying to see if I could get away without using stable $\infty$-categories. I think in my setup (finitely presented modules over a commutative ring), the argument of Deligne also goes through- but I'll have a think if I can enhance that to the stable $\infty$-case. $\endgroup$
    – ThomasZ
    Commented Dec 30, 2020 at 12:19
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    $\begingroup$ Note though that Deligne takes a 2-limit and you're taking a 2-colimit! And his argument uses that the rings $\mathbf Z/\ell^n$ are finite... $\endgroup$
    – Dan Petersen
    Commented Dec 30, 2020 at 14:15
  • $\begingroup$ Even if the diagram can be lifted to $\infty$-categories, it is not true in general that the homotopy category of the limit of $\infty$-categories is equivalent to the limit of homotopy categories. $\endgroup$
    – D.-C. Cisinski
    Commented Nov 6, 2021 at 15:11
  • $\begingroup$ In fact, in the case where the indexing category is $\mathbb{N}^{op}$, there is a $lim^1$ obstruction against this equivalence, which vanishes for constructible $\ell$-adic sheaves in the context considered by Deligne (we do not only need to work with $\mathbb{Z}/\ell^n$-linear coefficients, but also with schemes of finite type over a finite field or over a separably closed field to ensure that all cohomology groups are finite). $\endgroup$
    – D.-C. Cisinski
    Commented Nov 6, 2021 at 15:11

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