Homotopical inverse diagrams in categories with attributes
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.
- Publication:
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arXiv e-prints
- Pub Date:
- August 2018
- DOI:
- 10.48550/arXiv.1808.01816
- arXiv:
- arXiv:1808.01816
- Bibcode:
- 2018arXiv180801816K
- Keywords:
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- Mathematics - Logic;
- Mathematics - Category Theory;
- 03B15 Higher-order logic and type theory (primary);
- 03G30 Categorical logic;
- topoi;
- 18C50 Categorical semantics of formal languages
- E-Print:
- v3: various minor revisions for publication version