A simplicial object in a category is a contravariant functor from the simplicial category to . Such a functor is uniquely specified by the morphisms and , which satisfy
In particular, a simplicial set is a simplicial object in . Equivalently, one could say that a simplicial set is a presheaf on . The object of a simplicial set is a set of -simplices, and is called the -skeleton.
![$ X(\delta^n_i)\colon X([n]) \to X([n-1])$](http://images.planetmath.org:8080/cache/objects/3368/l2h/img5.png "$ X(\delta^n_i)\colon X([n]) \to X([n-1])$"){kind=small}
![$ X(\sigma^n_i)\colon X([n]) \to X([n+1])$](http://images.planetmath.org:8080/cache/objects/3368/l2h/img6.png "$ X(\sigma^n_i)\colon X([n]) \to X([n+1])$"){kind=small}
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![$ X([n])$](http://images.planetmath.org:8080/cache/objects/3368/l2h/img20.png "$ X([n])$"){kind=small}
