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 Top is a simplicial topological space.

• A simplicial object in Diff is a simplicial manifold.

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

• A simplicial object in the category of topological groups is a simplicial topological group.

• A simplicial object in Lie algebras is a simplicial Lie algebra.

• A simplicial object in Ring is a simplicial ring.

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

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

• A cosimplicial object in sSet is a cosimplicial simplicial set (equivalently a simplicial object in cosimplicial sets).

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

Category of simplicial objects

For $D$ a category, we write $D^{{\Delta }^{\mathrm{op}}}$ for the functor category from ${\Delta }^{\mathrm{op}}$ to $D$: its category of simplicial objects.

Definition

Let $D$ be a category with all limits and colimits. This implies that it is tensored over Set

$$\cdot :D\times \mathrm{Set}\to D.$$\cdot : D \times Set \to D \,.

This induces a functor

$${\cdot }^{{\Delta }^{\mathrm{op}}}:D^{{\Delta }^{\mathrm{op}}}\times \mathrm{sSet}\to D^{{\Delta }^{\mathrm{op}}}$$\cdot^{\Delta^{op}} : D^{\Delta^{op}} \times sSet \to D^{\Delta^{op}}

which we shall also write just ”$\cdot$”.

For $X,Y\in D^{{\Delta }^{\mathrm{op}}}$ write

$$D^{{\Delta }^{\mathrm{op}}}(X,Y):={\mathrm{Hom}}_{D^{{\Delta }^{\mathrm{op}}}}(X\cdot \Delta [•],Y)\in \mathrm{sSet}$$D^{\Delta^{op}}(X,Y) := Hom_{D^{\Delta^{op}}}(X \cdot \Delta[\bullet], Y) \in sSet

and for $X,Y,Z\in D^{{\Delta }^{\mathrm{op}}}$ let

$$D^{{\Delta }^{\mathrm{op}}}(X,Y)\times D^{{\Delta }^{\mathrm{op}}}(Y,Z)\to D^{{\Delta }^{\mathrm{op}}}(X,Z)$$D^{\Delta^{op}}(X,Y) \times D^{\Delta^{op}}(Y,Z) \to D^{\Delta^{op}}(X,Z)

be given in degree $n$ by

$$(X\cdot \Delta [n]\to Y,Y\cdot \Delta [n]\to Z)\mapsto (X\cdot \Delta [n]\to X\cdot \Delta [n]\times \Delta [n]\to Y\cdot \Delta [n]\to Z).$$(X \cdot \Delta[n] \to Y, Y \cdot \Delta[n] \to Z) \mapsto ( X \cdot \Delta[n] \to X \cdot \Delta[n]\times \Delta[n] \to Y \cdot \Delta[n] \to Z) \,.

Proposition

With the above definitions $D^{{\Delta }^{\mathrm{op}}}$ becomes an sSet-enriched category which is both tensored as well as cotensored over $\mathrm{sSet}$.

Definition

We may regard the category of cosimplicial objects $D^{\Delta }$ as an $\mathrm{sSet}$-enriched category using the above enrichment by identifying

$$D^{\Delta }\simeq ({D^{\mathrm{op}}}^{{\Delta }^{\mathrm{op}}}{)}^{\mathrm{op}}.$$D^{\Delta} \simeq ({D^{op}}^{\Delta^{op}})^{op} \,.

References

• Peter May, Simplicial objects in algebraic topology , University of Chicago Press, 1967, (djvu)