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Ordinary inductive types is initial algebras for free monads. However, HITs are not initial algebras for endofunctors but presented monads.

From nLab, initial algebra of a presentable (infinity,1)-monad and blog comment, Peter and Mike constructed models for HITs in their paper.

Where is the reference or link for this paper?

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https://arxiv.org/abs/1705.07088 . I've updated the nLab page.

There are also some slides available at my web page.

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