Simplicial Set - Formal Definition

Formal Definition

Using the language of category theory, a simplicial set X is a contravariant functor

X: Δ → Set

where Δ denotes the simplex category whose objects are finite strings of ordinal numbers of the form

n = 0 → 1 → ... → n

(or in other words non-empty totally ordered finite sets) and whose morphisms are order-preserving functions between them, and Set is the category of small sets.

It is common to define simplicial sets as a covariant functor from the opposite category, as

X: Δop → Set

This definition is clearly equivalent to the one immediately above.

Alternatively, one can think of a simplicial set as a simplicial object (see below) in the category Set, but this is only different language for the definition just given. If we use a covariant functor X instead of a contravariant one, we arrive at the definition of a cosimplicial set.

Simplicial sets form a category usually denoted sSet or just S whose objects are simplicial sets and whose morphisms are natural transformations between them. There is a corresponding category for cosimplicial sets as well, denoted by cSet.

These definitions arise from the relationship of the conditions imposed on the face maps and degeneracy maps to the category Δ.