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There are essentially two ways to impose extentionality on a type theory (I know, it is not very fashionable to impose extentionality these days, but please, bear with me) you can either have a "propo …

These question of existence of free monad are not "derailling" the discusion. They are the whole point of the discusion. Let me clarify : If I'm not mistaken, we have the following: Theorem: Let $V$ b …

I suspect there is no good answer to the question: The type theoretic results you are mentioning definitely have a category theoretic interpretation in fact their proof using gluing is already very ca …

In an informal sense, the answer "should be yes", in the sense that if one ignore type theory and work with an $\infty$-topos one can make sense of the construction $List(A)$ either by the usual unive …

I would say that the question is not even well defined. Saying that Martin löf type theory with extensional identity types is the internal language of cartesian closed categories with natural number …

Let me first clarify some confusion in the comments to the original question. To be clear : I'm not at all saying the persons making them were confused, as far as I can tell all the comments were corr …

One can read at several places that Martin-löf type theory should be the internal language of a locally Cartesian closed infinity category, and that the univalence axiom should distinguished infinity …