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Keywords:
Higher topos theory; Parametrised higher category theory; Internal higher category theory
Higher topos theory; Parametrised higher category theory; Internal higher category theory
Summary:
We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to establish the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.
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