# Contents

## Idea

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

## Properties

### Simplicial enrichment

When $\mathcal{C}$ has finite limits and finite colimits, then $\mathcal{C}^{\Delta}$ 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 $\mathcal{C}$, we can make $\mathcal{C}^{\Delta}$ into a simplicially enriched category using the end formula

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

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

## References

Last revised on February 10, 2015 at 11:56:24. See the history of this page for a list of all contributions to it.