Algebraic models for higher categories

We introduce the notion of algebraic fibrant objects in a general model category and establish a (combinatorial) model category structure on algebraic fibrant objects. Based on this construction, we propose algebraic Kan complexes as an algebraic model for ∞∞-groupoids and algebraic quasi-categories as an algebraic model for (∞,1)(∞,1)-categories. We furthermore give an explicit proof of the homotopy hypothesis.