Model Structure

A way to present an $(\infty ,1)$ category¹ from just a category.

What do you need to add?

• $(C,W)$ where $W$ is a wide subcategory.
  • With isos and 2/3 — of triangles — ?
  • UP $\infty$ -functor $\eta :C\to L(C)$, $C\to D$, then $\exists !L(C)\to D$. Difficult to calculate the maps $\mathrm{Map}(\eta X,\eta Y)$

3 classes of maps (C, W, F)

• C - cofibrations
• W - wide functors
• F - fibrations

Serre Fibration Hurewicz Fibration

Examples:

• Set - (Inj, All, Surj) which generalizes to terminal infinity category
• StrCat : (Inclusion on objects $\uplus$ ff, equiv of cat, isofibrations)

In Simplicial Set we cannot prove constructively that: the fibration ${\Pi }_g(f)\in \mathcal{F}$. — check this

Cofibrantly Generated $(\mathcal{L},\mathcal{R})$ iff $\mathcal{R}=S^{?}$ for some small set $S\subseteq \mathcal{L}$ if both $(C\cap W,F)$ & $(C,W\cap F)$ are model structures ??

BCH Cube Category

Footnotes

1. Weak Infinity Category ↩