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.