Where a simplicial object is a functor out of the opposite category of the simplex category, a cosimplicial object is a functor out of the simplex category itself.
When has finite limits and finite colimits, then 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 , we can make into a simplicially enriched category using the end formula
with composition inherited from and .
Daniel Quillen, Homotopical Algebra, Lecture Notes in Mathematics, vol. 43, Springer-Verlag, 1967
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)
Last revised on January 3, 2021 at 02:21:00. See the history of this page for a list of all contributions to it.