nLab
simplicial object

Idea

A simplicial object $X$ in a category $C$ is a collection $\{X_n{\}}_{n\in \mathbb{N}}$ of objects in $C$ that behave as if $X_n$ were an $n$-dimensional simplex internal to $C$.

Definition

A simplicial object in a category $C$ is a functor ${\Delta }^{\mathrm{op}}\to C$, where $\Delta$ is the simplicial indexing category.

A cosimplicial object in $C$ is similarly a functor out of the opposite category, $\Delta \to C$.

Accordingly, simplicial and cosimplicial objects in $C$ themselves form a category in an obvious way, namely the functor category $[{\Delta }^{\mathrm{op}},C]$ and $[\Delta ,C]$, respectively.

Remark

A simplicial object $X$ in $C$ is often specified by the objects, $X_n$, which are the images under $X$, of the objects $[n]$ of $\Delta$, together with a description of the face and degeneracy morphisms, $d_i$ and $s_j$, which must satisfy the simplicial identities.

Examples

• A simplicial object in Set is a simplicial set.

• A simplicial object in a category of presheaves is a simplicial presheaf.

• A simplicial object in Diff is a simplicial manifold.

• A cosimplicial object in the category of rings (algebras) is a cosimplicial ring (cosimplicial algebra).

• A simplicial object in the category Grp of groups is a simplicial group. See also Dold-Kan correspondence.

• A simplicial object in a category of simplicial objects is a bisimplicial object.

• The bar construction produces a simplicial object from a monad and an algebra over that monad.