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### Abstract

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.

Original language | English |
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Pages (from-to) | 1-33 |

Number of pages | 33 |

Journal | Homology, Homotopy and Applications |

Volume | 17 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2015 |

## Fingerprint Dive into the research topics of 'Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions'. Together they form a unique fingerprint.

## Projects

- 1 Finished

## Structural homotopy theory: a category-theoretic study

Street, R., Lack, S., Verity, D., Garner, R., MQRES, M., MQRES 3 (International), M. 3., MQRES 4 (International), M. & MQRES (International), M. (.

1/01/13 → 31/12/16

Project: Research