Let \(\mathcal E\) be a countably lextensive category and let \(s\mathcal E\) be the category of simplicial objects in \(\mathcal E\). The authors construct a model structure on \(s\mathcal E\), called the effective model structure. When \(\mathcal E\) is the category of sets, the effective model structure recovers the Kan-Quillen model structure on simplicial sets. The effective model structure can be described as follows. For \(E\in\mathcal E\), consider the functor \[ \operatorname{Hom}_{\mathrm{sSet}}(E,-)\colon s\mathcal E\to \mathrm{sSet} \] sending \(X\in s\mathcal E\) to the simplicial set defined by \(\operatorname{Hom}_{\mathrm{sSet}}(E,X)_n=\operatorname{Hom}(E,X_n)\). A morphism in \(s\mathcal E\) is a fibration in the effictive model structure if its image under the above functor is a Kan fibration in \(\mathrm{sSet}\) for every \(E\in\mathcal E\). The trivial fibrations are defined analogously. The authors show that for every countably lextensive category \(\mathcal E\), the effective model structure with the above fibrations and trivial fibrations exists.
Besides the existence of the effective model structure, the main results are the following.

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The effective model structure is left and right proper and homotopy colimits in \(s\mathcal E\) satisfy descent.

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The associated \(\infty\)-category has finite limits and \(\alpha\)-small colimits satisfy descent if \(\mathcal E\) is \(\alpha\)-lextensive.

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The associated \(\infty\)-category is locally Cartesian closed if \(\mathcal E\) is.

Besides the above properties, the associated \(\infty\)-category is not a higher topos in general.
The authors characterize the associated \(\infty\)-category as the full subcategory of the \(\infty\)-category of presheaves on \(\mathcal E\) spanned by Kan complexes in \(\mathcal E\).