The effective model structure and -groupoid objects

Abstract

AbstractFor 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.