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>>148
> https://en.wikipedia.org/wiki/Simplicial_set
> Simplicial set
> From Wikipedia, the free encyclopedia
> 
> In mathematics, a simplicial set is a construction in categorical homotopy theory that is a pure algebraic model of the notion of a "well-behaved" topological space.
> Historically, this model arose from earlier work in combinatorial topology and in particular from the notion of simplicial complexes. Simplicial sets are used to define quasi-categories, a basic notion of higher category theory.
> 
> History and uses of simplicial sets
> 
> Simplicial sets were originally used to give precise and convenient descriptions of classifying spaces of groups. This idea was vastly extended by Grothendieck's idea of considering classifying spaces of categories, and in particular by 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.
> Later these methods were used in other areas on the border between algebraic geometry and topology. For instance, the Andre-Quillen homology of a ring is a "non-abelian homology", defined and studied in this way.
> 
> Both the algebraic K-theory and the Andre-Quillen homology are defined using algebraic data to write down a simplicial set, and then taking the homotopy groups of this simplicial set. Sometimes one simply defines the algebraic K {\displaystyle K} K-theory as the 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.

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