Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
This realization consists of geometric simplices, glued together according to the rules of the simplicial set.
Indeed, one may view a simplicial set as a purely combinatorial construction designed to capture the essence of a topological space for the purposes of homotopy theory.
[1] Simplicial sets are used to define quasi-categories, a basic notion of higher category theory.
A simplicial set is a categorical (that is, purely algebraic) model capturing those topological spaces that can be built up (or faithfully represented up to homotopy) from simplices and their incidence relations.
This is similar to the approach of CW complexes to modeling topological spaces, with the crucial difference that simplicial sets are purely algebraic and do not carry any actual topology.
While algebraic topologists largely continue to prefer CW complexes, there is a growing contingent of researchers interested in using simplicial sets for applications in algebraic geometry where CW complexes do not naturally exist.
Simplicial sets can be viewed as a higher-dimensional generalization of directed multigraphs.
Unlike directed multigraphs, simplicial sets may also contain higher simplices.
A 2-simplex, for instance, can be thought of as a two-dimensional "triangular" shape bounded by a list of three vertices A, B, C and three arrows B → C, A → C and A → B.
In general, an n-simplex is an object made up from a list of n + 1 vertices (which are 0-simplices) and n + 1 faces (which are (n − 1)-simplices).
The map sn,i assigns to each n-simplex the degenerate (n+1)-simplex which arises from the given one by duplicating the i-th vertex.
This description implicitly requires certain consistency relations among the maps dn,i and sn,i.
So the identities provide an alternative way to define simplicial sets.
A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC([n]) is the set of all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from i to j whenever i ≤ j.
Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C: a0 → a1 → ... → an.
The degeneracy maps si lengthen the sequence by inserting an identity morphism at position i.
Here SYn consists of all the continuous maps from the standard topological n-simplex to Y.
(In many texts, it is written instead as hom([n],-) where the homset is understood to be in the opposite category Δop.
[2]) By the Yoneda lemma, the n-simplices of a simplicial set X stand in 1–1 correspondence with the natural transformations from Δn to X, i.e.
There is a functor |•|: sSet → CGHaus called the geometric realization taking a simplicial set X to its corresponding realization in the category |•|: sSet → CGHaus of compactly-generated Hausdorff topological spaces.
Intuitively, the realization of X is the topological space (in fact a CW complex) obtained if every n-simplex of X is replaced by a topological n-simplex (a certain n-dimensional subset of (n + 1)-dimensional Euclidean space defined below) and these topological simplices are glued together in the fashion the simplices of X hang together.
To define the realization functor, we first define it on standard n-simplices Δn as follows: the geometric realization |Δn| is the standard topological n-simplex in general position given by The definition then naturally extends to any simplicial set X by setting where the colimit is taken over the n-simplex category of X.
With the model structure in place, a homotopy theory of simplicial sets can be developed using standard homotopical algebra methods.
Furthermore, the geometric realization and singular functors give a Quillen equivalence of closed model categories inducing an equivalence between the homotopy category for simplicial sets and the usual homotopy category of CW complexes with homotopy classes of continuous maps between them.
In this work, which earned him a Fields Medal, Quillen developed surprisingly efficient methods for manipulating infinite simplicial sets.
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.
is an abelian group, we can actually iterate this infinitely many times, and obtain that
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
In recent years, simplicial sets have been used in higher category theory and derived algebraic geometry.