equivalences in/of $(\infty,1)$-categories
Homotopy weighted colimits (alias weighted homotopy colimits) are the analog of weighted colimits in homotopy theory.
For the special case of model categories, we can define homotopy weighted colimits as follows.
Fix a monoidal model category $V$, a $V$-enriched model category $C$, and a small $V$-enriched category $J$.
For simplicity, assume all enriched hom objects of $J$ are cofibrant. If this is not the case, we can first cofibrantly replace $J$ in the Dwyer-Kan model structure on enriched categories.
We have a left Quillen bifunctor
given by the ordinary weighted colimit functor.
The homotopy weighted colimit can then be defined as the left derived Quillen bifunctor of the weighted colimit functor.
See Section 9.2 in
and for simplicially based theories,
(That article uses the older terminology of ‘indexed colimits’ rather than the `weighted' one.)
Other references:
Lukáš Vokřínek, Homotopy weighted colimits, arXiv:1201.2970.
Nicola Gambino, Weighted limits in simplicial homotopy theory, Journal of Pure and Applied Algebra 214:7 (2010), 1193–1199. doi.
Michael Shulman, Homotopy limits and colimits and enriched homotopy theory, arXiv:math/0610194.
Sergey Arkhipov, Sebastian Ørsted, Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories (arXiv:1807.03266).