Cylindrical Model Structure

Start with Premodel Structure, $C_t,C,F_t,F$. - $C_t,F_t$ trivial (co)fibrations.

• Add $(C_t,F),(C,F_t)$ weak factorization systems (WFS)
• • adjoint cylinder, $\mathbb{I}\otimes \_:\mathcal{C}\to \mathcal{C}$ .
• Natural transformations $S_k\otimes \_:Id\to \mathbb{I}\otimes \_$ for $k=0,1$.
  • ${\delta }_k:X\to \mathbb{I}\otimes X$.
    • $\partial :X+X\to \mathbb{I}\otimes X$
  • ${\delta }_k\hat{\otimes }\_$
    …
    Endpoing inclusion endpoints map $C_t\to C_t$ and $C\to C$.

Theorem (Sattler-Cavallo)

If we have a cylindrical premodel structure,
in which all objects are cofibrant and we have the (fibration extension property) FEP then we get a model structure $(C,F_t\circ C_t,F)$

Thm (C-S)

Type theoreitic model structure on $PSh({\square }_{01\vee ,we})=\text{test model structure}$, for any small A, $N_A:Cat\to PShA$ is adjoint to $\int \_$.

Zuillen Equiv $F:C\to D;G:D\to C$ such that $F\perp G$ adjunciton playing nicely with model structures & ind $LC\leftrightarrow LD$.