cosimplicial object




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.


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.


Last revised on January 3, 2021 at 02:21:00. See the history of this page for a list of all contributions to it.