methods for manipulating infinite simplicial sets. These methods were used in other areas on the border between algebraic geometry and topology. 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 is a group with classifying space , then is homotopy equivalent to the loop space . If itself is a group, we can iterate the procedure, and is homotopy equivalent to the double loop space . In case is an abelian group, we can actually iterate this infinitely many times, and obtain that is an infinite loop space. Even if 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 is an infinite loop space. In this way, one can prove that the algebraic -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. See also Delta set Dendroidal set, a generalization of simplicial set Simplicial presheaf Quasi-category Kan complex Dold–Kan correspondence Simplicial homotopy Simplicial sphere Abstract simplicial complex Notes References (An elementary introduction to simplicial sets). Further reading May, J. Peter. Simplicial Objects in Algebraic Topology, University of Chicago Press 1967 Algebraic topology Homotopy theory Functors