Cubical models of \((\infty, 1)\)-categories (Q6605390)

An \textit{\((\infty,1)\)-category} is conceptually meant to be a structure consisting of objects, \(1\)-morphisms between objects, \(2\)-morphisms between \(1\)-morphisms, and so on, where all morphisms above level \(1\) are invertible and the composition and associativity of morphisms is characterized up to choices of higher equivalences. This is of course not a precise mathematical definition! Hence, significant effort has gone into developing precise mathematical definitions that can capture this concept of an \((\infty,1)\)-category, which are commonly called \textit{models of \((\infty,1)\)-category theory}. The first prominent example of such a model is the theory of \textit{quasi-categories}. Quasi-categories are simplicial sets satisfying appropriate lifting conditions [\textit{ J. M. Boardman} and \textit{R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces. Berlin-Heidelberg-New York: Springer-Verlag (1973; Zbl 0285.55012)]. Here the vertices of the simplicial set correspond to the objects, the edges give us the morphisms and the higher simplices give us the desired invertible higher morphisms and relevant composition and associativity data. Quasi-categories have proven to be an effective definition and many \((\infty,1)\)-categorical concepts have been developed in this context, first by \textit{A. Joyal} [J. Pure Appl. Algebra 175, No. 1--3, 207--222 (2002; Zbl 1015.18008)] and \textit{J. Lurie} [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)], and later on by many others.\N\NDespite the strength of quasi-category theory, it soon became apparent that certain aspects, such as constructing relevant examples of \((\infty,1)\)-categories, necessitated access to other models. As a result many other models were developed. Two prominent examples are \textit{complete Segal spaces}, due to \textit{C. Rezk} [Trans. Am. Math. Soc. 353, No. 3, 973--1007 (2001; Zbl 0961.18008)], and \textit{Kan enriched categories}, due to \textit{J. E. Bergner} [Trans. Am. Math. Soc. 359, No. 5, 2043--2058 (2007; Zbl 1114.18006)]. Part of this development included showing that these two models are equivalent to quasi-categories, meaning all these three definitions carry a \textit{ Quillen model structure} and they are all \textit{(Quillen) equivalent} to each other. For example, \textit{A. Joyal} and \textit{M. Tierney}, building on work of Rezk, proved the equivalence between complete Segal spaces and quasi-categories [Contemp. Math. 431, 277--326 (2007; Zbl 1138.55016)].\N\NIn this work the authors construct a further model of \((\infty,1)\)-categories via cubical sets, which are defined as set-valued presheaves on the box category. Concretely, in \textit{Theorem 4.2} they prove that the category of cubical sets carries a model structure and in \textit{Theorem 6.1} they prove this model structure is Quillen equivalent to the model structure for quasi-categories. Beyond constructing a new model of \((\infty,1)\)-categories their work is also a direct generalization of a result of \textit{D.-C. Cisinski} [Les préfaisceaux comme modèles des types d'homotopie. Paris: Société Mathématique de France (2006; Zbl 1111.18008)], who proved that cubical sets model homotopy types.\N\NThe authors then proceed to exhibit several benefits of this new model. One major benefit, in contrast to quasi-categories or complete Segal spaces, is a direct access to mapping spaces. Indeed, unlike the simplex category, the cube category is closed under products, making the construction of the \((\infty,1)\)-category of arrows, and hence mapping spaces, very convenient (\textit{Definition 7.1}). The authors employ this convenient definition to give an alternative proof of the fact that equivalences of \((\infty,1)\)-categories are given by fully faithful and essentially surjective functors of \((\infty,1)\)-categories (\textit{Theorem 7.10}).