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?