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Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves

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Abstract

In this paper we prove that various quasi-categories whose objects are \(\infty \)-categories in a very general sense are complete: admitting limits indexed by all simplicial sets. This result and others of a similar flavor follow from a general theorem in which we characterize the data that is required to define a limit cone in a quasi-category constructed as a homotopy coherent nerve. Since all quasi-categories arise this way up to equivalence, this analysis covers the general case. Namely, we show that quasi-categorical limit cones may be modeled at the point-set level by pseudo homotopy limit cones, whose shape is governed by the weight for pseudo limits over a homotopy coherent diagram but with the defining universal property up to equivalence, rather than isomorphism, of mapping spaces. Our applications follow from the fact that the \((\infty ,1)\)-categorical core of an \(\infty \)-cosmos admits weighted homotopy limits for all flexible weights, which includes in particular the weight for pseudo cones.

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Notes

  1. Known \(\infty \)-cosmoi of \((\infty ,n)\)-categories include \(\theta _n\)-spaces, iterated complete Segal spaces, n-complicial sets, and n-quasi-categories.

  2. Here we show that the codomain of the comparison map in (4.1.9) is a quasi-category by applying Proposition 4.1.5 in the \(\infty \)-cosmos of quasi-categories.

  3. The simplicial computads are the cofibrant objects [12, §16.2] in the model structure on simplicial categories due to Bergner [2].

  4. For bookkeeping reasons it is convenient to adopt the convention that atomic arrows are not identities, though in a simplicial computad the identities will also admit no non-trivial factorisations. With this convention, an identity arrow factors uniquely as an empty composite of atomic arrows.

  5. For more details about the Leibniz or “pushout-product” construction see [13, §4].

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Acknowledgements

The authors are grateful for support from the National Science Foundation (DMS-1551129 and DMS-1652600) and from the Australian Research Council (DP160101519). This work was commenced when the second-named author was visiting the first at Harvard and then at Johns Hopkins, continued while the first-named author was visiting the second at Macquarie, and completed after everyone finally made their way home. We thank all three institutions for their assistance in procuring the necessary visas as well as for their hospitality. The final published manuscript benefitted greatly from the astute suggestions of an eagle-eyed referee. We are also grateful to the referee for the sequel [20] who noticed that the proof of the converse result Theorem 6.2.7 originally given there could be simplified enough so that we could include it here.

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Correspondence to Emily Riehl.

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Communicated by Nicola Gambino.

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Riehl, E., Verity, D. Recognizing Quasi-Categorical Limits and Colimits in Homotopy Coherent Nerves. Appl Categor Struct 28, 669–716 (2020). https://doi.org/10.1007/s10485-020-09594-x

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