Editing Simplicial set (section)

==The standard ''n''-simplex and the category of simplices==

The '''standard ''n''-simplex''', denoted Δ<sup>''n''</sup>, is a simplicial set defined as the functor hom<sub>Δ</sub>(-, [''n'']) where [''n''] denotes the ordered set {0, 1, ... ,''n''} of the first (''n'' + 1) nonnegative integers. (In many texts, it is written instead as hom([''n''],-) where the homset is understood to be in the opposite category Δ<sup>op</sup>.<ref>{{harvnb|Gelfand|Manin|2013}}</ref>)

By the [[Yoneda lemma]], the ''n''-simplices of a simplicial set ''X'' stand in 1–1 correspondence with the natural transformations from Δ<sup>''n''</sup> to ''X,'' i.e. <math>X_n = X([n])\cong \operatorname{Nat}(\operatorname{hom}_\Delta(-,[n]),X)=
\operatorname{hom}_{\textbf{sSet}}(\Delta^n,X)</math>.

Furthermore, ''X'' gives rise to a [[category of elements|category of simplices]], denoted by <math>\Delta\downarrow{X}</math> , whose objects are maps (''i.e.'' natural transformations) Δ<sup>''n''</sup> → ''X'' and whose morphisms are natural transformations Δ<sup>''n''</sup> → Δ<sup>''m''</sup> over ''X'' arising from maps [''n''] ''→'' [''m''] in Δ.  That is, <math>\Delta\downarrow{X}</math> is a [[slice category]] of Δ over ''X''. The [[density theorem (category theory)|following isomorphism]] shows that a simplicial set ''X'' is a [[colimit]] of its simplices:<ref>{{harvnb|Goerss|Jardine|1999|p=7}}</ref>

: <math>X \cong \varinjlim_{\Delta^n \to X} \Delta^n</math>

where the colimit is taken over the category of simplices of ''X''.