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## Multi angle  Type theories and polynomial monads Awodey, Steve (Auteur de la Conférence) | CIRM (Editeur )

A system of dependent type theory T gives rise to a natural transformation p : Terms $\to$ Types of presheaves on the category Ctx of contexts, termed a "natural model of T". This map p in turn determines a polynomial endofunctor P : $\widehat{Ctx}$ $\to$ $\widehat{Ctx}$ on the category of all presheaves. It can be seen that P has the structure of a monad just if T has $\Sigma$-types and a terminal type, and that p is itself a P-algebra just if T has $\Pi$-types. I will explain this rather unexpected connection between type theories and polynomial monads, and will welcome any insights from the other participants regarding it. A system of dependent type theory T gives rise to a natural transformation p : Terms $\to$ Types of presheaves on the category Ctx of contexts, termed a "natural model of T". This map p in turn determines a polynomial endofunctor P : $\widehat{Ctx}$ $\to$ $\widehat{Ctx}$ on the category of all presheaves. It can be seen that P has the structure of a monad just if T has $\Sigma$-types and a terminal type, and that p is itself a P-algebra just if ...

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