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We introduce the notion of weighted limit in an arbitrary quasi-category,suitably generalizing ordinary limits in a quasi-category, and classicalweighted limits in an ordinary category. This is accomplished by generalizingJoyal's approach: we identify a meaningful construction for the quasi-categoryof weighted cones over a diagram in a quasi-category, whose terminal object isthe weighted limit of the considered diagram. When the quasi-category arises asthe homotopy coherent nerve of a category enriched over Kan complexes, we usetechniques by Riehl-Verity to show that the weighted limit agrees with thehomotopy weighted limit in the sense of enriched category theory, for whichexplicit constructions are available. When the quasi-category is complete,tensored and cotensored over the quasi-category of spaces, we discuss apossible comparison of our definition of weighted limit with the approach byGepner-Haugseng-Nikolaus.

Weighted limits in an $(\infty,1)$-category