Added a reference to A general formulation of homotopy limits.
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Corrected obsolete links to updated ones in
I, localizers, (pdf);
II, model categories, (pdf);
IV, summary on derivators (pdf)
]]>I have added to this addition (here) a link to Bousfield-Kan map and also to the entry’s own section “Resolved (co)ends”.
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In older literature, such as the original work of Bousfield–Kan \cite{BousfieldKan}, the homotopy limit of a diagram is defined as the weighted limit
where the weight sends , the nerve of the comma category of .
Such a functor must be derived in order to get the correct (homotopy invariant) notion, which in this case amounts to replacing with an objectwise weakly equivalent diagram valued in Kan complexes.
At some point (when?) a shift in terminology happened, and in the modern parlance homotopy limits are commonly assumed to be derived.
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