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The category $(A\downarrow \mathcal{M}\downarrow B)$ is not necessarily a model category, as it can fail to be complete or cocomplete (or both). For example, $(\{*\} \downarrow \mathsf{Set} \downarrow \varnothing)$ is not a model category: it is empty! There is no map from a singleton to the empty set. However one can define, for every $f : A \to B$, the ...


In the case that all isomorphisms in the model category are fibration, cofibration, and weak equivalences, (trivial cofibrations, fibrations) and (fibrations, trivial cofibrations) form factorization systems in the sense of Borceux: A factorization system in a category $\mathbf B$ is defined as a pair $(E, F)$ of classes of arrows in $\mathbf B$ such ...

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