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I am following Verdier's notion of derived functors as Kan extensions along the localization $K(\mathcal{A}) \to D(\mathcal{A})$ of the homotopy category of complexes to the derived category.

In the treatment I am reading, the fact that $RF$ is triangulated follows from the explicit construction and noting that the additive functor $F$ must preserve cones.

I am wondering if there is a more intrinsic reason that $RF$ must be triangulated. It does not seem like the Kan extension should have any way of ``knowing'' about the triangulated structure and therefore it seems surprising from the definition that such an object ought to respect the triangulated structure.

Because $F$ preserves cones, it is clear that $K(F) : K(\mathcal{A}) \to K(\mathcal{A})$ is triangulated. So is there a general reason that the Kan extension of such a triangulated functor is again triangulated?

Is there more interplay between the Kan extension and the exact triangles that I am missing?

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    $\begingroup$ Are you sure Verdier uses Kan extensions? Those categories don't have colimits. $\endgroup$ Commented Jul 10, 2021 at 7:39
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    $\begingroup$ @FernandoMuro Derived functors in the sense of Verdier are Kan extensions by definition, but not necessarily pointwise or absolute. As it turns out they are often absolute Kan extensions and therefore have a colimit universal property. $\endgroup$ Commented Jul 10, 2021 at 8:25
  • $\begingroup$ @zhenlin it must be indexed by a category with final object then. The derived category is a localization of the homotopy category and the universal property of a localization yields the exactness of derived functors. That's the easiest way to see it IMHO. $\endgroup$ Commented Jul 10, 2021 at 16:09

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$\def\C{\mathcal{C}} \def\A{\mathcal{A}} \def\S{\mathcal{S}} \def\ind{\operatorname{Ind}} \def\D{\mathcal{D}} \def\t{\operatorname{t}} \def\op{\mathrm{op}} \def\indlim#1{\underset{\underset{#1}{\longrightarrow}}{“\operatorname{lim}”\,}} \def\s{\mathrm{s}} \def\t{\mathrm{t}}$It is true in full generality that derived triangulated functors are triangulated. This is [D, Proposition 1.2.2.ii] and [SP, Tag 05SC]. Here is a proof idea not found in neither of these references, which leverages the content of [KS, Sect. 7.4].

Let $\C$ be a category. The “free filtered cocompletion” of $\C$ (also known as ind-completion) is a category $\ind(\C)$ that has all filtered colimits and that comes equipped with a fully faithful embedding $\iota_\C:\C\to\ind(\C)$ satisfying Definition 2 from here. Thus, if $F:\C\to \A$ is a functor, there is a unique-up-to-isomorphism functor $IF:\ind(\C)\to\ind(\A)$ preserving filtered colimits and such that $\iota_\A\circ F\cong IF\circ\iota_\C$ [KS, Proposition 6.1.9]. Let $\S$ be a left multiplicative system in $\C$ [SP, Tag 04VC] and write $Q:\C\to\C_\S$ for the localization functor of $\C$ with respect to $\S$. Let $\alpha_\S:\C_\S\to\ind(\C)$ be the left Kan extension of $\iota_\C$ along $Q$. Consider the diagram: $$ \require{AMScd} \begin{CD} \C@>F>>\A@=\A\\ @VQVV@.@VV{\iota_\A}V\\ \C_\S@>>{\alpha_\S}>\ind(\C)@>>IF>\ind(\A) \end{CD} $$ Then $$ \tag{[KS, eq. (7.4.3)]}\label{equality} R_\S(\iota_\A\circ F)\cong IF\circ\alpha_\S, $$ where $R_\S$ means “taking left Kan extension along $Q$” (it is what Kashiwara and Schapira refer to as the right localization of a functor [KS, Definition 7.3.1]). A proof of \ref{equality} may be read in [G, p. 162].

[KS, Definition 7.4.2]. The functor $F$ is right localizable at $X\in\C$ if the image of $Q(X)$ through \ref{equality} lies in the essential image of $\iota_\A$.

It turns out that $F$ is right localizable at all objects $X\in\C$ if and only if the left Kan extension of $F$ along $Q$ exists and is absolute [KS, Proposition 7.4.4] (in Kashiwara and Schapira notation, the latter is “the right localization $R_\S F$ exists and is universal”)

With these ideas on mind, suppose we knew:

  1. $\ind(\C)$ is triangulated whenever $\C$ is triangulated,
  2. $IF,\iota_\A,\alpha_\S$ are triangulated functors respectively if $F$ is triangulated functor, $\A$ is a triangulated category and $\S$ is a saturated multiplicative system compatible with the triangulated structure in $\C$,

Then we would immediately conclude that universally right localized triangulated functors are triangulated, for $R_\S(\iota_\A\circ F)$ would be the composite of triangulated functors (i.e., if $F$ is a triangulated functor of triangulated categories and $R_\S F$ exists and is universal, then $R_\S F$ is triangulated too). The problem is that point 1 is not true in general [ref] (so point 2 has no meaning). There is a workaround:

Definition. A proto-triangulated category $(\D,T,\t(\D))$ is an additive category $\D$, an auto-equivalence $T:\D\to\D$ and a class $\t(\D)$ of triangles in $(\D,T)$, called the proto-triangles.

(This terminology is made up.) In other words, a proto-triangulated category is an additive category with translation with a class of distinctive triangles.

Definition. A proto-triangulated functor between proto-triangulated categories $(\D,T)$ and $(\D',T')$ is an additive functor $F:\D\to\D'$ along with a natural isomorphism $\xi:FT\cong T'F$ such that a proto-triangle in $\D$ is sent to a proto-triangle in $\D'$. A morphism of proto-triangulated functors is a 2-morphism in the 2-category of categories with translation [KS, Definition 10.1.1.iii].

If there is no risk of confusion, we might just say “triangulated functor” for a proto-triangulated functor. Here's the key insight: if $\C$ is a triangulated category, one endows $\ind(\C)$ with the proto-triangulated structure of those triangles that are a filtered colimit of distinguished triangles in $\C$. With this proto-triangulated structure, it turns out that $\alpha_\S$ and $IF$ are triangulated functors and that a proto-triangle in the essential image of $\iota_\A$ is a distinguished triangle in $\A$. With these facts, one can easily prove that right derived triangulated functors are triangulated. I wrote all the details in Derived Triangulated Functors are Triangulated.

References

[KS]. M. Kashiwara, P. Schapira, Categories and Sheaves

[SP]. The Stacks Project Authors, The Stacks Project

[D]. P. Deligne. “Cohomologie à supports propres”. In: Théorie de Topos et Cohomologie Étale des Schémas (SGA4). Vol. III, Exp. XVII. Lecture Notes in Math. 305. Springer, 1971, pp. 250-461

[G]. P.-Y. Gaillard, About “Categories and Sheaves” https://vixra.org/abs/1602.0067

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  • $\begingroup$ I wrote a follow-up to this question here, with the proof of “existence of an absolute left Kan extension along the localization functor implies existence of an absolute left extension in the 2-category of triangulated categories.” This relates the two possible definitions of the derived functor found in the literature. $\endgroup$ Commented Oct 22, 2024 at 12:12

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