Fibrations and Yoneda's lemma in an ∞-cosmos

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• 作者：Emily Riehl^a ; ^{eriehl@math.jhu.edu" class="auth_mail" title="E-mail the corresponding author} ; Dominic Verity^b ; ^{dominic.verity@mq.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：221
• 期：3
• 页码：499-564
• 全文大小：1102 K

文摘

We use the terms ∞-categories and ∞-functors to mean the objects and morphisms in an ∞-cosmos  : a simplicially enriched category satisfying a few axioms, reminiscent of an enriched category of fibrant objects. Quasi-categories, Segal categories, complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916301001&_mathId=si1.gif&_user=111111111&_pii=S0022404916301001&_rdoc=1&_issn=00224049&md5=251887652d2c22ffc8624720e51a4c1e" title="Click to view the MathML source">θ_n${\theta }_n$-spaces, and fibered versions of each of these are all ∞-categories in this sense. Previous work in this series shows that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our definitions are 2-categorical in natural, making no use of the combinatorial details that differentiate each model.

In this paper, we introduce cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. Cartesian fibrations form a cornerstone in the abstract treatment of “category-like” structures a la Street and play an important role in Lurie's work on quasi-categories. After setting up their basic theory, we state and prove the Yoneda lemma, which has the form of an equivalence between the quasi-category of maps out of a representable fibration and the quasi-category underlying the fiber over its representing element. A companion paper will apply these results to establish a calculus of modules between ∞-categories, which will be used to define and study pointwise Kan extensions along ∞-functors.