nLab
cosimplicial object

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

Revised on February 10, 2015 11:56:24 by Urs Schreiber (195.113.30.252)