2017/06/30 by Emily Riehl, Riehl, Emily, Dominic Verity +1 · 1 citation
Mathematics · Medicine · #18G55 #55U35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.1706.10023
openalex publication_date 2017/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the "comprehension construction". Its input is a cocartesian fibration p \colon E → B between ∞-categories together with a third ∞-category A. The comprehension construction then defines a map from the quasi-category of functors from A to B to the large quasi-category of cocartesian fibrations over A that acts on f \colon A → B by forming the pullback of p along f. To illustrate the versatility of this construction, we define the covariant and contravariant Yoneda embeddings as special cases of the comprehension functor. We then prove that the hom-wise action of the comprehension functor coincides with an "external action" of the hom-spaces of B on the fibres of p and use this to prove that the Yoneda embedding is fully faithful, providing an explicit equivalence between a quasi-category and the homotopy coherent nerve of a Kan-complex enriched category.