nLab
cosimplicial object

Contents

Context

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

Where a simplicial object is a functor Δ op𝒞\Delta^{op} \to \mathcal{C} out of the opposite category of the simplex category, a cosimplicial object is a functor Δ𝒞\Delta \to \mathcal{C} out of the simplex category itself.

Properties

Simplicial enrichment

When 𝒞\mathcal{C} has finite limits and finite colimits, then 𝒞 Δ\mathcal{C}^{\Delta} is canonically a simplicially enriched category with is tensored and powered over sSet. This is called the external simplicial structure in (Quillen 67, II.1.7). Review includes (Bousfield 03, section 2.10).

More generally, for any 𝒞\mathcal{C}, we can make 𝒞 Δ\mathcal{C}^{\Delta} into a simplicially enriched category using the end formula

𝒞 Δ̲(X,Y) m= [n]:Δ(𝒞(X n,Y n)) Δ n m\underline{\mathcal{C}^{\Delta}} (X, Y)_m = \int_{[n] : \Delta} (\mathcal{C} (X^n, Y^n))^{\Delta^m_n}

with composition inherited from 𝒞\mathcal{C} and Δ\Delta.

References

Last revised on July 14, 2021 at 07:07:20. See the history of this page for a list of all contributions to it.