nLab
cosimplicial object

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 $\Delta$.

Related concepts

• cosimplicial abelian group

• cosimplicial algebra

References

• Daniel Quillen, Chapter I, Axiomatic homotopy theory in: Homotopical Algebra, Lecture Notes in Mathematics 43, Springer 1967(doi:10.1007/BFb0097438)

• Dai Tamaki, Akira Kono, Appendix A in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Aldridge Bousfield, Cosimplicial resolutions and homotopy spectral sequences in model categories (arXiv:0312531)