  For a category $\mathcal E$ with finite limits and well-behaved countable
coproducts, we construct a model structure, called the effective model
structure, on the category of simplicial objects in $\mathcal E$, generalising
the Kan--Quillen model structure on simplicial sets. We then prove that the
effective model structure is left and right proper and satisfies descent in the
sense of Rezk. As a consequence, we obtain that the associated
$\infty$-category has finite limits, colimits satisfying descent, and is
locally Cartesian closed when $\mathcal E$ is, but is not a higher topos in
general. We also characterise the $\infty$-category presented by the effective
model structure, showing that it is the full sub-category of presheaves on
$\mathcal E$ spanned by Kan complexes in $\mathcal E$, a result that suggests a
close analogy with the theory of exact completions.
