Editing Simplicial set (section)

==History and uses of simplicial sets==
Simplicial sets were originally used to give precise and convenient descriptions of [[classifying space]]s of [[group (mathematics)|group]]s. This idea was vastly extended by [[Grothendieck]]'s idea of
considering classifying spaces of categories, and in particular by [[Daniel Quillen|Quillen]]'s work of [[algebraic K-theory]]. In this work, which earned him a [[Fields Medal]], Quillen
developed surprisingly efficient 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 cohomology|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 <math>G</math> is a group with classifying space <math>BG</math>, then <math>G</math> is homotopy equivalent to the loop space <math>\Omega BG</math>. If <math>BG</math> itself is a group, we can iterate the procedure, and <math>G</math> is homotopy equivalent to the double loop space  <math>\Omega^2 B(BG)</math>. In case  <math>G</math> is an abelian group, we can actually iterate this infinitely many times, and obtain that <math>G</math> is an infinite loop space.

Even if <math>X</math> 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 <math>X</math> is an infinite loop space. In this way, one can prove that the algebraic <math>K</math>-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-category|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.