From: "András Kovács" <"putta..."@gmail.com>
To: Homotopy Type Theory <HomotopyT...@googlegroups.com>
Subject: A syntax for higher inductive-inductive types
Date: Thu, 8 Feb 2018 03:23:32 -0800 (PST) [thread overview]
Message-ID: <3214e47f-39f7-49d7-aa4a-ac6a8eac45d8@googlegroups.com> (raw)
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Dear all,
We have a draft titled "A Syntax for Higher Inductive-Inductive Types"
<https://akaposi.github.io/hiit.pdf> which may be of interest to you.
There's also an Agda formalization <https://bitbucket.org/akaposi/elims>,
and a Haskell implementation <https://bitbucket.org/akaposi/elims> of a
HIIT type/positivity checker and eliminator-generator which you can play
around with. Here
<https://bitbucket.org/akaposi/elims/src/0fd2d3e8226e08757e2ab460a5b5966e6a6fb4c4/haskell/elims-demo/examples/?at=master>
are a number of examples.
The main idea is to describe HIIT-s as contexts in a "domain specific" type
theory. This type theory has a universe closed under identity types, and a
function space restricted to small (i. e. contained in the universe) domain
types, which enforces strict positivity. Then, one can compute types of
algebras, induction methods and eliminators/beta rules by taking various
models of this type theory. Non-closed HIITs and infinitary constructors
can be also represented by additional function spaces, with domain types
built from an "external" non-inductive context. We do not say anything
about existence or semantics of specified types yet.
András Kovács
Ambrus Kaposi
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