Abstract
We present a construction of W-types in the setoid model of extensional Martin-L\"of type theory using dependent W-types in the underlying intensional theory. More precisely, we prove that the internal category of setoids has initial algebras for polynomial endofunctors. In particular, we characterise the setoid of algebra morphisms from the initial algebra to a given algebra as a setoid on a dependent W-type. We conclude by discussing the case of free setoids. We work in a fully intensional theory and, in fact, we assume identity types only when discussing free setoids. By using dependent W-types we can also avoid elimination into a type universe. The results have been verified in Coq and a formalisation is available on the author's GitHub page.
Highlights
The present paper is a contribution to the study of models of extensional properties in intensional type theories and is in particular concerned with W-types
We present a construction of W-types in the setoid model of extensional Martin-Lof type theory using dependent W-types in the underlying intensional theory
We prove that the internal category of setoids has initial algebras for polynomial endofunctors
Summary
The present paper is a contribution to the study of models of extensional properties in intensional type theories and is in particular concerned with W-types. The characterisation in Theorem 4.18 allows us to reduce the problem of finding a unique algebra morphism, to the problem of finding a unique telescopic function To this aim, and thanks to the inductive nature of telescopic functions, we directly use the elimination principle of dependent W-types. Thanks to the inductive nature of telescopic functions, we directly use the elimination principle of dependent W-types In this sense, we believe that Theorem 4.18 makes explicit, in the case of setoids, the connection between the commutativity condition for an algebra morphism out of the initial algebra, and the inductive definition of the morphism itself. A common aspect of arguments that construct W-types is the use of the set, or setoid, of all subtrees of a tree, usually obtained as the transitive closure of the immediate subtree relation. The Coq formalisation of the present paper is available on the author’s GitHub page [Emm18] and it includes Definitions 2.2 and 2.3 and all numbered definitions and results in Sections 2.5 and 3 to 5, except for Proposition 2.9, a Coq proof of which can be found at [Pal12a]
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