Homology, Homotopy and Applications

Volume 17 (2015)

Number 1

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

Pages: 1 – 33

DOI: https://dx.doi.org/10.4310/HHA.2015.v17.n1.a1


Emily Riehl (Department of Mathematics, Harvard University, Cambridge, Masachusetts, U.S.A.)

Dominic Verity (Centre of Australian Category Theory, Macquarie University, Sydney, Australia)


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 admits any (co)limits 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 of this type, so these results specialise to (co)completeness results for quasi-categories of algebras. The second half of this paper establishes a further result in the quasi-categorical setting: proving, in analogy with the classical categorical case, that the monadic forgetful functor of the quasi-category of algebras for a homotopy coherent monad creates all limits that exist in the base quasi-category, regardless of whether its functor part preserves those limits. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasi-categories of algebras.


quasi-category, weighted limit, homotopy limit, Eilenberg-Moore object, completeness and co-completeness

2010 Mathematics Subject Classification

18A05, 18D20, 18G30, 18G55, 55U10, 55U35, 55U40

Published 18 May 2015