Weak complicial sets I. Basic homotopy theory

Ordinal subdivision and special pasting in quasicategories

“…Extra material has been added in more recent versions, and proofs simplified, including some for which thanks are due to a referee. Many of the results make striking analogies with more recent results of Verity, [19,20]. His work also contains a separate analysis of ordinal subdivision in strict ω-categories and elsewhere, results from this current paper are linked with an analysis of shuffles in his context.…”

“…This, in turn, links with the emerging theory of quasicategories, cf. Joyal [13] and Verity's theory of (weak) complicial sets [19,20].…”

Homotopy Theory of Higher Categories

“…This started with Street's proposal in [201], of a definition of weak n-category as a simplicial set satisfying a certain variant of the Kan condition where one takes into account the directions of arrows, and also using the idea of "thinness". His suggestion, in retrospect undoubtedly somewhat similar to Joyal's iteration of the notion of quasicategory, wasn't worked out at the time, but has recieved renewed interest, see Verity [212] for example.…”

Enriched categories and models for spaces of evolving states

“…The first would seem to be a Kan complex, whilst the second is a 'weak Kan complex' or 'quasi-category', cf. the papers by Joyal, [30], on quasi-categories, and Verity on complicial sets [31][32][33], which are models for weak infinity categories. (For an introduction to some of the types of weak infinity category including quasi-categories and information on the weakening of S-categories, known as Segal categories, the reader is referred to the notes, [34].…”