For instance, the André–Quillen homology of a ring is a "non-abelian homology", defined and studied in this way. Both the algebraic K-theory and the André–Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set. Simplicial methods are often useful when one wants to prove that a space is a loop space. The basic idea is that if formula_32 is a group with classifying space formula_33, then formula_32 is homotopy equivalent to the loop space formula_35. If formula_33 itself is a group, we can iterate the procedure, and formula_32 is homotopy equivalent to the double loop space formula_38. In case formula_32 is an abelian group, we can actually iterate this infinitely many times, and obtain that formula_32 is an infinite loop space. Even if formula_41 is not an abelian group, it can happen that it has a composition which is sufficiently commutative so that one can use the above idea to prove that formula_41 is an infinite loop space. In this way, one can prove that the algebraic formula_43-theory of a ring, considered as a topological space, is an infinite loop space. In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry. Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, and information about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicial sets satisfying one additional condition, the weak Kan condition.