Quasi-algebras I

Denis-Charles Cisinski – 10 September 2008

The notions of regular localizers and homotopy sketches will be recalled. These are convenient tools to produce natural model structures on presheaf categories associated to nice monads (e.g. monads with arities and parametric right adjoint monads as considered by M. Weber). We will recover as an example of this general process the Joyal model structure for quasi-categories and the Rezk model structure for complete Segal spaces, as well as the Quillen equivalences relating them. This also allows us to consider in the same way the homotopy theory of "quasi-n-categories" or of "quasi-multicategories" (a.k.a. dendroidal inner Kan complexes in the sense of I. Weiss and I. Moerdijk).