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Neil Strickland
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Categorical sematnicssemantics of W-types

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Russell O'Connor
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Categorical sematnics of W-types

Jacob's book titled "Categorical Logic and Type Theory" gives a nice description of Π and Σ types as adjunctions to substitution functors induced by display maps. Is there a similar categorical description of W-types (and maybe M-types while we are at it)?