Now that we have a general formula for homotopy limit and colimit functors in any simplicial model category, we should take a moment to see what these objects look like in particular examples, in particular, for traditional topological spaces. This demands that we delve into a topic that is perhaps overdue. In addition to the category of simplicial sets, several choices are available for a simplicial model category of spaces. In Section 6.1, we discuss the point-set topological considerations that support the definitions of two of the aforementioned convenient categories of spaces: k-spaces and compactly generated spaces. In Section 6.2, we use these results to list several simplicial model categories of spaces, paying particular attention to their fibrant and cofibrant objects, which feature in the construction of homotopy limits and colimits. The short Section 6.3 concludes this background segment with some important preparatory remarks.

Finally, in Sections 6.4 and 6.5, we turn to examples, describing the spaces produced by the formulae of Corollary 5.1.3 and considering what simplified models might be available in certain cases. A particularly intuitive presentation of the theory of homotopy limits and colimits, including several of the examples discussed in those sections, can be found in the unfinished yet extremely clear notes on this topic by [18]. We close this chapter in Section 6.6 with a preview of Part II. We show that our preferred formulae for homotopy limits and colimits are isomorphic to certain functor cotensor and functor tensor products.