nLab
simplicial mapping complex

Contents

Context

Mapping space

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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,YsSetX, Y \,\in\, sSet, their internal hom or simplicial mapping complex is the simplicial set

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

whose

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

  • and whose face maps d id_i and degeneracy maps s is_i are given by precomposition with id X×δ iid_X \times \delta_i and id X×σ iid_X \times \sigma_i, respectively (δ i\delta_i and σ i\sigma_i denoting the generating morphisms of the simplex category).

Examples

Example

(simplicial mapping complex between nerves of groupoids) For N(𝒢 i)N(\mathcal{G}_i) the nerves of groupoids, their simplicial mapping complex is isomorphic to the nerve of the functor groupoid from 𝒢 1\mathcal{G}_1 to 𝒢 2\mathcal{G}_2:

[N(𝒢 1),N(𝒢 2)]N(Func(𝒢 1,𝒢 2))SimplicialSets \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.