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Tag Archives: functor
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
Species definition clarification and exercises
[This is the fourth in a series of posts about combinatorial species. Previous posts: And now, back to your regularly scheduled combinatorial species; Decomposing data structures; Combinatorial species definition.] In my previous post I neglected to mention something quite crucial, … Continue reading
Combinatorial species definition
Continuing from my previous post, recall that the goal of species is to have a unified theory of containers with labeled1 locations. So, how do we actually specify such things (leaving aside for the moment the question of how we … Continue reading
Themes on Streams, Part II
In a previous post I claimed that comonad structures on R > a are in onetoone correspondence with monoid structures on R. In this post and the next I’ll justify that claim. Characterizing comonads on R > a Suppose we … Continue reading
Posted in haskell, math
Tagged comonad, free theorem, functor, parametricity, representable
2 Comments
Themes on Streams
> {# LANGUAGE DeriveFunctor, FlexibleInstances #} Recall that a stream is a countably infinite sequence of values: > data Stream a = a :> Stream a > deriving (Functor, Show) > > sHead (a :> _) = a > sTail … Continue reading