The effective model structure and $\infty$-groupoid objects

Abstract

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.

arXiv comment67 pages, reflects the final journal version, two additional minor typos fixed

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