Computation on homotopy colimit cocomplete triangulated categories

Question

I have a couple of questions about dealing with homotopy (co)limits cocomplete triangulated categories.

Question I:The first one concerns a comment by Peter Arndt in this discussion about derived categories regarding posibibility to calculate the homotopy colimit when working with nice enough category. Peter wrote:

I also find this a very enlightening view point, but just for the record: Ho(co)lims in cocomplete triangulated categories are MUCH easier to compute by completing the right map to an exact triangle than by going via a simplicial (or any other) enrichment...

Where I can look up the theoretical background explaining that applying successively these steps we indeed obtain an object homotopic to homotopical (co)limit. In other words why this cooking recipe work?

Question 2: searching for an answer for my first question I found in this paper on Homotopy limits in triangulated categories by Bökstedt & Neeman an approach by so called 'Totalization of a complex'.

The steps in the construction look quite similar to the step Peter described and the constructed object is also described as homotopical colimit.

Question: How close the construction in the paper is to that one in first question. The principal aspect that confuses me is that the construction in the paper (as well the paper) not explicitly working with simplicial enrichments of homs.

Is using simplicial enrichment a more 'modern' approach to obtain the same object? And how would it flow into the construction?

Answer 1

Where I can look up the theoretical background explaining that applying successively these steps we indeed obtain an object homotopic to homotopical (co)limit. In other words why this cooking recipe work?

The recipe under discussion computes the homotopy colimit of a sequence $X_0→X_1→X_2→⋯$ as the homotopy cofiber of the shift map $⨁_{i≥0}X_i→⨁_{i≥0}$. The shift map is the difference of the identity map and the map induced by transition maps to the next degree. The homotopy cofiber of this difference can be computed as the homotopy coequalizer of the two maps under consideration. The latter homotopy coequalizer of two maps between direct sums (i.e., homotopy coproducts) can be rewritten as the homotopy colimit of a single diagram indexed by a category $I$. The latter category $I$ has a canonical functor $I→\{0→1→2→⋯\}$, which is a homotopy final functor (the comma categories can be easily checked to be contractible), hence the induced map on homotopy colimits is a weak equivalence.

The criterion for homotopy finality can be found, e.g., in Lurie's Higher Topos Theory (Proposition 4.1.1.8), in Cisinski's book, and in many other places.

How close the construction in the paper is to that one in first question. The principal aspect that confuses me is that the construction in the paper (as well the paper) not explicitly working with simplicial enrichments of homs. Is using simplicial enrichment a more 'modern' approach to obtain the same object? And how would it flow into the construction?

We don't see enrichments because the diagrams involved are extremely special: they are sequences $X_0→X_1→X_2→⋯$ in which there are no nontrivial (homotopy) commutativity (or coherence) relations. In this (very special) case one can show that a sequential diagram $X_0→X_1→X_2→⋯$ in a triangulated category is the same data as a weak equivalence class of sequential diagrams in a stable model category that models the triangulated category. This is part of the reason why one can compute the homotopy colimit inside the triangulated category in this (very special) case.

Any time there is a nontrivial commutativity (coherence) involved (e.g., when computing the homotopy colimit of a simplicial diagram), the whole machinery of triangulated categories breaks down. Indeed, it is not even possible to say what a (homotopy coherent) simplicial object in a triangulated category is, since the necessary data of coherences is simply not present in a strict functor from Δ^op to the triangulated category, and the relevant information is altogether missing from the triangulated category. This is part of the reason why constructing enhancements of triangulated categories is often necessary. But then again, one might as well work with the original stable model (or relative) category. For more information, see the homotopy theory FAQ.