2024/02/02 by Raffael Stenzel, Stenzel, Raffael
Mathematics · Medicine · #18B15 #18D20 #18D40 #18D70 #18F20 #18N40 #18N50 #18N60 #18N65 #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.2402.01396
openalex publication_date 2024/02/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define and study the (∞,2)-category Cat∞(C) of (∞,1)-categories internal to a general (∞,1)-category C via an associated externalization construction. In the first part, we show various formal closure properties of Cat∞(C) regarding limits, tensors, cotensors and internal mapping objects under the assumption of various suitable closure properties of C. In particular, we show that Cat∞(C) defines a cartesian closed full sub-∞-cosmos of the ∞-cosmos Fun(Cop,Cat∞) of C-indexed (∞,1)-categories under suitable assumptions on C. We furthermore characterize the objects of Cat∞(C) by means of a Yoneda lemma that expresses indexed diagrams of internal shape over C in terms of an (∞,1)-categorical totalization. In the second part, we relate the general theory developed to this point to results in the model categorical literature. We show that every model category \mathbbM gives rise to a ``hands-on'' ∞-cosmos Cat∞(\mathbbM) directly by restriction of the Reedy model structure on \mathbbM^Δop. We then define a corresponding right derived model categorical externalization functor, and use it to show that the (∞,1)-categorical and the model categorical constructions correspond to one another whenever \mathbbM is a suitable model category.