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

TL;DR

This paper introduces the effective model structure on simplicial objects in a category with finite limits, generalizing the Kan--Quillen model structure, and explores its properties and the associated $$-category.

Contribution

It constructs a new model structure called the effective model structure and characterizes the resulting $$-category, extending the theory of simplicial objects and their homotopical properties.

Findings

The effective model structure is left and right proper.

The associated $$-category has finite limits and colimits satisfying descent.

The $$-category is locally Cartesian closed when $\u001e$ is.

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.