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Tag Archives: types
Anafunctors
This is part four in a series of posts on avoiding the axiom of choice (part one, part two, part three). In my previous post, we considered the “Axiom of Protoequivalence”—that is, the statement that every fully faithful, essentially surjective … Continue reading
Posted in category theory, math, species
Tagged AC, anafunctor, axiom of choice, category, constructive, equivalence, functor, isomorphism, theory, types, unique
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AC and equivalence of categories
This is part three in a series of posts on avoiding the axiom of choice (part one, part two). In my previous post, I explained one place where the axiom of choice often shows up in category theory, namely, when … Continue reading
Posted in category theory, math, species
Tagged AC, axiom of choice, category, constructive, equivalence, functor, isomorphism, theory, types, unique
5 Comments
Unique isomorphism and generalized “the”
This is part two in a series of posts on avoiding the axiom of choice; you can read part one here. In category theory, one is typically interested in specifying objects only up to unique isomorphism. In fact, definitions which … Continue reading
Posted in category theory, math, species
Tagged AC, axiom of choice, category, constructive, functor, isomorphism, theory, types, unique
8 Comments
Avoiding the axiom of choice, part I
I’m hard at work on my dissertation, and plan to get back to doing a bit of blogging based on stuff I’m writing and thinking about, as a way of forcing myself to explain things clearly and to potentially get … Continue reading
Posted in category theory, math, species
Tagged AC, axiom of choice, category, constructive, theory, types
17 Comments
Unordered tuples and type algebra
At Hac Phi a few weekends ago (which, by the way, was awesome), Dan Doel told me about a certain curiosity in type algebra, and we ended up working out a bunch more details together with Gershom Bazerman, Scott Walck, … Continue reading
Typed typelevel programming: status report
A few people have been bugging me (you know who you are ;) about the current status of the project to allow automatic “lifting” of Haskell data constructors to the type level, to allow for typed typelevel programming. I’ve written … Continue reading
Posted in haskell, projects
Tagged extension, GHC, hacking, lifting, progress, refactoring, types
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On a Problem of sigfpe
> {# LANGUAGE TypeFamilies, EmptyDataDecls, TypeOperators, GADTs #} At the end of his most recent blog post, Divided Differences and the Tomography of Types, Dan Piponi left his readers with a challenge: In preparation for the next installment, here’s a … Continue reading
Typed typelevel programming in Haskell, part IV: collapsing types and kinds
In Part III, we saw how the current state of the art in Haskell typelevel programming leaves some things to be desired: it requires duplicating both data declarations and code, and even worse, it’s untyped. What to do? Currently, GHC’s … Continue reading
Typed typelevel programming in Haskell, part II: type families
In my previous post, we saw how multiparameter type classes with functional dependencies in Haskell allow us to do typelevel programming in a logic programming style. (If you’re not clear on why this corresponds to a logic programming style, see … Continue reading