The effective model structure and ∞-groupoid objects

For a category 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 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 ∞ -category has finite limits, colimits satisfying descent, and is locally Cartesian closed when E is but is not a higher topos in general. We also characterise the ∞-category presented by the effective model structure, showing that it is the full sub-category of presheaves on E spanned by Kan complexes in E, a result that suggests a close analogy with the theory of exact completions.