# nLab simplicial mapping complex

Contents

### Context

#### Mapping space

internal hom/mapping space

# Contents

## Idea

The simplicial mapping complex or simplicial hom complex of two simplicial set is the collection of maps, left homotopies and higher homotopies between them, all itself organized into a simplicial set.

More formally, SimplicialSets is a Cartesian monoidal category and the corresponding internal hom is what is traditionally known as the simplicial mapping complex.

## Definition

Since SimplicialSets is the category of presheaves over the simplex category, its internal hom has the general form discussed at closed monoidal structure on presheaves:

###### Definition

For $X, Y \,\in\, sSet$, their internal hom or simplicial mapping complex is the simplicial set

$[X,Y]_\bullet \;=\; Hom_{sSet} \big( X \times \Delta[\bullet], \, Y \big) \;\;\; \in sSet \,.$

whose

• component set in degree $k$ is the hom-set of simplicial sets

• from the product of simplicial sets of $X$ with the standard n-simplex $\Delta[n]$

• to $Y$,

• and whose face maps $d_i$ and degeneracy maps $s_i$ are given by precomposition with $id_X \times \delta_i$ and $id_X \times \sigma_i$, respectively ($\delta_i$ and $\sigma_i$ denoting the generating morphisms of the simplex category).

## Examples

###### Example

(simplicial mapping complex between nerves of groupoids) For $N(\mathcal{G}_i)$ the nerves of groupoids, their simplicial mapping complex is isomorphic to the nerve of the functor groupoid from $\mathcal{G}_1$ to $\mathcal{G}_2$:

$\big[ N(\mathcal{G}_1), \, N(\mathcal{G}_2) \big] \;\; \simeq \;\; N \big( Func(\mathcal{G}_1, \, \mathcal{G}_2) \big) \;\;\; \in \; SimplicialSets$

Last revised on July 12, 2021 at 04:09:32. See the history of this page for a list of all contributions to it.