# Categorical semantics 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)?

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The title of the question has a typo which should be fixed. –  Michael A Warren Feb 7 '12 at 17:04

The categorical semantics of W-types, as initial algebras, have been studied in the following paper of Moerdijk and Palmgren: "Wellfounded trees in categories", Annals of Pure and Applied Logic 104(2000), 189 - 218.

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And for M-types (as terminal coalgebras) there is the dual paper "Non-well-founded trees in categories" by van den Berg and de Marchi, APAL 146 (2007) 40–59. –  Mike Shulman Feb 7 '12 at 21:37