Holmstrom Simplicial object

[CDATAFor skeleta, see Goerss-Jardine chapter VII.]

Beatrice R Gonzalez on simplicial descent categories. Discusses homotopical structures on a category and on the category of simplicial objects in a category. http://front.math.ucdavis.edu/0808.3684

A simplicial object in a category $C$ is a functor $\Delta^{op} \to C$. A cosimplicial object is a functor $\Delta \to C$.

Here $\Delta$ is the category of objects $[n] = \{0,1,\ldots, n\}$ and morphisms the weakly order-preserving maps, i.e. maps $f$ such that $x \leq y$ implies $f(x) \leq f(y)$. Note that every such map can be factored uniquely as an injective map followed by a surjective map. The category $\Delta$ is generated by the injective maps $d^i$ and the surjective maps $s^i$ satisfying the cosimplicial identities.

A simplicial object is given by objects $K_n$ for each non-negative $n$, together with face maps $d_i: K_n \to K_{n-1}$ ($n \geq 1, \ 0 \leq i \leq n$) and degeneracy maps $s_i: K_{n-1} \to K_{n}$ ($n \geq 1, \ 0 \leq i \leq n-1$), which must satisfy the simplicial identities:

• $d_i d_j = d_{j-1} d_i \quad (i j)$
• $d_i s_j = s_{j-1} d_i \quad (i j)$
• $d_i s_j = id \quad (i = j, j+1)$
• $d_i s_j = s_j d_{i-1} \quad (i > j)$
• $s_i s_j = s_j s_{i-1} \quad (i > j)$

A map of simplicial objects is equivalent to a collection of maps $f_n: K_n \to L_n$ commuting with the face and degeneracy maps.

Notes from Goerss and Schemmerhorn: Description and handling of skeletons, using Kan extensions. One considers a complete and cocomplete category, and simplicial objects in this category. Latching and matching objects. Applications to lifting problems, by induction. When our category has enough colimits, have a notion of geometric realization from $\Delta^{op} \mathcal{C}$ to $\mathcal{C}$. If $\mathcal{C}$ is a model category, this can be made into a Quillen functor. Description of the (Reedy?) model category structure on $\Delta^{op} \mathcal{C}$. If $X \in \mathcal{C}$ is cofibrant, then it is also cofibrant in the Reedy model structure. This is not true for fibrant. More details on this.

nLab page on Simplicial object