Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

Understand

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape.

• We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting these (co)limits and that they are preserved by the functors in the limit cone.
• In particular, the Bousfield-Kan homotopy limit of a diagram of quasi-categories admit any limits or colimits existing in and preserved by the functors in that diagram.
• In previous work, we demonstrated that the quasi-category of algebras for a homotopy coherent monad could be described as a weighted limit with projective cofibrant weight, so these results immediately provide us with important (co)completeness results for quasi-categories of algebras.