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For a category
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
, 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
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
spanned by Kan complexes in
, a result that suggests a close analogy with the theory of exact completions.
We provide a constructive version of the notion of sheaf models of univalent type theory. We start by relativizing existing constructive models of univalent type theory to presheaves over a base category. Any Grothendieck topology of the base category then gives rise to a family of left-exact modalities, and we recover a model of type theory by localizing the presheaf model with respect to this family of left-exact modalities. We provide then some examples.
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