On the construction of limits and colimits in $\infty$-categories

Abstract

In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneously. In this paper, we establish criteria implying that an $\infty$-category - for instance, a quasi-category, a complete Segal space, or a Segal category - is complete and cocomplete, admitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly that the modules that express the universal properties of (co)limits of diagrams of these shapes are reconstructable as limits of the modules that express the universal properties of (co)limits of the restricted diagrams. We also prove that the Yoneda embedding preserves and reflects limits in a suitable sense, and deduce our main theorems as a consequence.

57 pages; a continuation of the program developed in the papers arXiv:1306.5144, arXiv:1310.8279, arXiv:1401.6247, arXiv:1506.05500, arXiv:1507.01460, arXiv:1706.10023, and arXiv:1808.09834, as summarized in arXiv:1608.05314; v3 peer-reviewed, completely re-organized, and much improved (thanks to the referee); v4 final journal version