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Logical Structure on Inverse Functor Categories
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Inspired by recent work on the categorical semantics of dependent type theories, we investigate the following question: When is logical structure (crucially, dependent-product and subobject-classifier structure) induced from a category to categories of diagrams in it? Our work offers several answers, providing a variety of conditions on both the category itself and the indexing category of diagrams. Additionally, motivated by homotopical considerations, we investigate the case when the indexing category is equipped with a class of weak equivalences and study conditions under which the localization map induces a structure-preserving functor between presheaf categories.
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Cited by 1 Pith paper
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Pushforwards in Inverse Homotopical Diagrams
If every weak equivalence out of an index object is either initial among the maps out of it, or all maps out of it are weak equivalences, then homotopical inverse diagrams are closed under pushforwards along Reedy fibrations.
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