Quasi-categories and Kan complexes
It is well known that the simplicial nerve of a (small) category is a Kan complex, that is, every horn fills. Likewise in the nerve of a (small) category, horns fill provided that the missing face is not the zeroth or the last face of the hoped-for simplex. Such objects in other contexts were introduced in the 1970s by \textit{J. M. Boardman} and \textit{R. M. Vogt} [``Homotopy invariant algebraic structures on topological spaces'', Lect. Notes Math. 347 (1973; Zbl 0285.55012)] where they were called weak Kan complexes. They have also occured in studies of homotopy coherence [\textit{J.-M. Cordier}, Cah. Top. Géom. Différ. 23, 93-112 (1982; Zbl 0493.55009), \textit{J.-M. Cordier} and \textit{T. Porter}, Math. Proc. Camb. Philos. Soc. 100, 65-90 (1986; Zbl 0603.55017) and Topol. Appl. 28, 255-275 (1988; Zbl 0655.55008)]. This paper is an introduction to some of their theory viewed from a slightly different perspective, namely that arbitrary weak Kan complexes are generalisations of categories, hence the suggested renaming as quasi-categories. This change of perspective suggests that concepts of, for instance, limit can be defined internally within the language of quasi-categories. Such generalisations are simple, but effective, and clarify links between the homotopy theoretic aspects of the quasicategory as simplicial set and the categorical aspects of that quasi-category as a generalised category.
scientific article; zbMATH DE number 1839068