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Models of dependent type theories are contextual categories with some additional structure. We prove that if a theory $T$ has enough structure, then the category $Ttext{-}mathbf{Mod}$ of its models carries the structure of a model category. We also show that if $T$ has $Sigma$ types, then weak equivalences can be characterized in terms of homotopy categories of models.

Model structures on categories of models of type theories