In this post, we will quickly discuss the (basic) definition of triangulated categories and why there exist non-functorial cones in these categories. (The whole motivation, at least pointed in, to pass to some stable homotopy theory, like $\infty$-category, is built somewhat around this failure of triangulated categories.) We will discover more about these in the coming posts.
A triangulated category is an additive category with a shift functor that satisfies some axioms (see below). It is represented by $(\mathcal{T},[1],\Delta)$ where $[1]$ is a shift-functor (an autoequivalence $[1]: \mathcal{T} \to \mathcal{T}$) and $\Delta$ are distinguished triangles. A distinguished triangle is of the form
$$X \to Y \to Z \to X[1]$$
and the triangulated category $\mathcal{T}$ satisfies Verdier's axioms (of which an important one is the octahedral axiom, TR4). A good example of a triangulated category is the derived category of an abelian category. Anyhow, I will stop this definition by mentioning that we can see these distinguished triangles as a replacement of short exact sequences in abelian categories. For more on these triangulated categories, please see.
Our goal here is to see why there is no canonical functorial definition of (co)limits in a triangulated category. When working with some nice definition like $Qcoh(X_i) = A_i-mod$ where we have taken an open cover on $X$, we want the gluing to be defined. This motivates one to develop dg-enhancements, stable $\infty$-category, and so on (for some other day) to recover 'functorial cones'. However, I must mention that there exists a notion of homotopy (co)limits (see this work). In a distinguished triangle diagram, those come at the third vertex ($Z$). I do not know much about this theory, but they face certain restrictions too (like totalization, non-functoriality, and so on). Anyway, let us proceed with our problem. We will be mostly dealing with Stevenson's argument.
The argument is as follows. If an idempotent complete triangulated category $\mathcal{T}$ which admits a functorial cone, then it is a semisimple abelian category. A functorial cone is a functor defined as $\text{cone}: Mor(\mathcal{T}) \to \mathcal{T}$ where $Mor(\mathcal{T})$ is a category with objects $f: X \to Y$ for $X, Y$ in $\mathcal{T}$ (also called arrow category). For any morphism $f$, we can write a distinguished triangle
$$X \xrightarrow{f} Y \xrightarrow{g} cone(f) \xrightarrow{h} X[1]$$
and this triangle satisfies Verdier's axioms. Any morphism between two morphisms $f, f'$ is a commutative square. And there exists a coherent choice, which means that two cones of the same morphism are equivalent up to canonical isomorphism. For a non-functorial cone, cones are defined up to non-unique isomorphisms. Hence, there does not exist a coherent choice for a cone that is natural for the commutative square of morphisms.
Given a functorial cone in a category, this implies the existence of functorial weak colimits, and if a category also admits split idempotents, there exist actual colimits. Let us say our $\mathcal{T}$ is such a category. The functorial cone provides a functorial weak cokernel. In $\mathcal{T}$ for a functorial cone, there would exist an actual cokernel. Similarly, the category has an actual kernel as well. This makes $\mathcal{T}$ into an abelian category (it was already defined to be an additive category). Now, in the triangulated category $\mathcal{T}$, all monomorphisms and epimorphisms split, which makes $\mathcal{T}$ into a semi-simple abelian category. This is proposition 3.1, Stevenson. Without idempotents, one would still arrive at the same conclusion.
The only problem is that there are only trivial examples of triangulated categories that are semi-simple abelian categories. Hence, non-trivial triangulated categories cannot admit functorial cones. For example, the derived category of bounded coherent sheaves on a variety, the stable homotopy category of spectra, and so on are some examples that are not semi-simple categories. This brings us to conclude that a triangulated category admits non-functorial cones. To restore functoriality, one would need more than a triangulated structure, and such motivations will be discussed in due course.
The only problem is that there are only trivial examples of triangulated categories that are semi-simple abelian categories. Hence, non-trivial triangulated categories cannot admit functorial cones. For example, the derived category of bounded coherent sheaves on a variety, the stable homotopy category of spectra, and so on are some examples that are not semi-simple categories. This brings us to conclude that a triangulated category admits non-functorial cones. To restore functoriality, one would need more than a triangulated structure, and such motivations will be discussed in due course.
