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A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories $$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$ containing all the isomorphisms, such that the following conditions are satisfied:

  1. $\mathcal{C}$ has all finite limits.
  2. $\mathcal{W}$ has the 2-out-of-3 property.
  3. The subcategories $\mathcal{F}$ and $\mathcal{F}\cap\mathcal{W}$ are closed under base change.
  4. Every morphism $f: X \to Y$ can be factored as $X \xrightarrow{g} Z \xrightarrow{h} Y$ where $g \in \mathcal{W}$ and $h \in \mathcal{F}$.

The homotopy category $Ho(\mathcal{C})$ of a weak fibration category $(\mathcal{C},\mathcal{W},\mathcal{F})$ is obtained from $\mathcal{C}$ by formally inverting the morphisms in $\mathcal{W}$. The weak fibration category is called saturated if every map in $\mathcal{C}$ that maps to an isomorphism in $Ho(\mathcal{C})$, is in $\mathcal{W}$.

I have two questions:

  1. Does there exist a small weak fibration category $(\mathcal{C},\mathcal{W},\mathcal{F})$ such that $\mathcal{W}$ is closed under retracts, but $\mathcal{C}$ is not saturated?

  2. Does there exist a small weak fibration category $(\mathcal{C},\mathcal{W},\mathcal{F})$ such that $\mathcal{W}$ is closed under retracts, but $\mathcal{W}$ does not satisfy 2-out-of-6?

Of course, if the answer to 2. is yes then the answer to 1. is also yes.

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It is a result of Cisinski that in a fibration category the three conditions you mention (saturation, 2-out-of-6, weak equivalences closed under retracts) are all equivalent. See Theorem 7.2.7 in this paper.

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