Monoidal Model Categories
对于环上的模范畴进行推广, 环范畴的推广是 monoidal category, 模范畴的推广是 model category.
mondial model category 的例子:
- Simplicial sets
- Pointed simplicial sets
- chain complexes of modules over a commutative ring
- chain complexs of comodules over a commutative Hopf algebra
拓扑空间范畴不是 monoidal model category. 但是 $k$-spaces 和 compactly generated topological spaces 是 monoidal model category.
4.1 Closed Monoidal Categories and Closed Modules
| Algebra | Category |
|---|---|
| rings | monoidal categories |
| commutative rings | symmetric monoidal categories |
| modules over a ring | modules over a monoidal category |
| algebras over a ring | algebras over a monoidal category |
| central and commutative algebras over a commutative ring | central and symmetric algebras over a symmetric monoidal category |
一个范畴 $\mathcal{C}$ 上的 monoidal structure 是
- 一个 tensor product bifunctor $$\begin{equation} \mathcal{C}\times\mathcal{C} \xrightarrow{\otimes} \mathcal{C}, \end{equation}$$
- 一个 unit onject $S∈\mathcal{C}$,
- 一个 natural associativity isomorphism $$\begin{equation} a: (X⊗Y)⊗Z \xrightarrow{\cong} X⊗(Y⊗Z), \end{equation}$$
- 一个 natural left unit isomorphism $$\begin{equation} l: S⊗X \xrightarrow{\cong} X, \end{equation}$$
- 一个 natural right unit isomorphism $$\begin{equation} r: X⊗S \xrightarrow{\cong} X, \end{equation}$$
满足交换图:
设 $\mathcal{C},\mathcal{D},\mathcal{E}$ 是范畴. 一个从 $\mathcal{C}×\mathcal{D}$ 到 $\mathcal{E}$ 的 adjunction of two variables 是一个五元组 $(\otimes,\mathrm{Hom}_r,\mathrm{Hom}_l,φ_r,φ_l)$, 其中
- $\otimes: \mathcal{C}×\mathcal{D} \to \mathcal{E}$ 是双函子,
- $\mathrm{Hom}_r: \mathcal{D}^{op}×\mathcal{E} \to \mathcal{C}$ 是双函子,
- $\mathrm{Hom}_l: \mathcal{C}^{op}×\mathcal{E} \to \mathcal{D}$ 是双函子,
- $φ_r$ 和 $φ_l$ 是自然同构 $$\begin{equation} \mathcal{C}(C, \mathrm{Hom}_r(D,E)) \xrightarrow[\cong]{\varphi_r^{-1}} \mathcal{E}(C \otimes D, E) \xrightarrow[\cong]{\varphi_\ell} \mathcal{D}(D, \mathrm{Hom}_\ell(C,E)). \end{equation}$$