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arXiv · 1808.01816

Homotopical inverse diagrams in categories with attributes

Chris Kapulkin·Peter LeFanu Lumsdaine
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Abstract

We define and develop the infrastructure of homotopical inverse diagrams in categories with attributes. Specifically, given a category with attributes $C$ and an ordered homotopical inverse category $I$, we construct the category with attributes $C^I$ of homotopical diagrams of shape $I$ in $C$ and Reedy types over these; and we show how various logical structure ($\Pi$-types, identity types, and so on) lifts from $C$ to $C^I$. This may be seen as providing a general class of diagram models of type theory. In a companion paper "The homotopy theory of type theories" (arXiv:1610.00037), we apply the present results to construct semi-model structures on categories of contextual categories.

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BibTeXRIS

Chris Kapulkin, Peter LeFanu Lumsdaine. 2018-08-06. Homotopical inverse diagrams in categories with attributes. https://doi.org/10.1016/j.jpaa.2020.106563

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