The Comonad.Reader » The State Comonad
Is State a Comonad? Not Costate or rather, Store as we tend to call it today, but actually State s itself? Let's see! Recently there was a post to reddit in which the author King_of_the_Homeless suggested that he might have a Monad for Store. Moreover, it is one that is compatible with the existing Applicative and ComonadApply instances. My knee-jerk reaction was to disbelieve the result, but I'm
The Comonad.Reader » Adjoint Triples
A common occurrence in category theory is the adjoint triple. This is a pair of adjunctions relating three functors: F ⊣ G ⊣ H F ⊣ G, G ⊣ H Perhaps part of the reason they are so common is that (co)limits form one: colim ⊣ Δ ⊣ lim where Δ : C -> C^J is the diagonal functor, which takes objects in C to the constant functor returning that object. A version of this shows up in Haskell (with some exte
The Comonad.Reader » Recursion Schemes: A Field Guide (Redux)
About a year back I posted a field guide of recursion schemes on this blog and then lost it a few months later when I lost a couple of months of blog entries to a crash. I recently recovered the table of recursion schemes from the original post thanks to Google Reader's long memory and the help of Jeff Cutsinger. The following recursion schemes can be found in category-extras, along with variation
The Comonad.Reader » On the unsafety of interleaved I/O
One area where I'm at odds with the prevailing winds in Haskell is lazy I/O. It's often said that lazy I/O is evil, scary and confusing, and it breaks things like referential transparency. Having a soft spot for it, and not liking most of the alternatives, I end up on the opposite side when the topic comes up, if I choose to pick the fight. I usually don't feel like I come away from such arguments
The Comonad.Reader » Categories of Structures in Haskell
In the last couple posts I've used some 'free' constructions, and not remarked too much on how they arise. In this post, I'd like to explore them more. This is going to be something of a departure from the previous posts, though, since I'm not going to worry about thinking precisely about bottom/domains. This is more an exercise in applying some category theory to Haskell, "fast and loose". (Advan
The Comonad.Reader » Domains, Sets, Traversals and Applicatives
Last time I looked at free monoids, and noticed that in Haskell lists don't really cut it. This is a consequence of laziness and general recursion. To model a language with those properties, one needs to use domains and monotone, continuous maps, rather than sets and total functions (a call-by-value language with general recursion would use domains and strict maps instead). This time I'd like to t
The Comonad.Reader » Free Monoids in Haskell
It is often stated that Foldable is effectively the toList class. However, this turns out to be wrong. The real fundamental member of Foldable is foldMap (which should look suspiciously like traverse, incidentally). To understand exactly why this is, it helps to understand another surprising fact: lists are not free monoids in Haskell. This latter fact can be seen relatively easily by considering
The Comonad.Reader » Letter to a Young Haskell Enthusiast
The following letter is not about what "old hands" know and newcomers do not. Instead, it is about lessons that we all need to learn more than once, and remind ourselves of. It is about tendencies that are common, and understandable, and come with the flush of excitement of learning any new thing that we understand is important, and about the difficulty, always, in trying to decide how best to con
The Comonad.Reader » Iteratees, Parsec and Monoids (Slides)
I was asked to give two talks at the Boston Area Haskell User Group for this past Tuesday. The first was pitched at a more introductory level and the second was to go deeper into what I have been using monoids for lately. The first talk covers an introduction to the mathematical notion of a monoid, introduces some of the features of my Haskell monoids library on hackage, and starts to motivate the
Monoids
All About All About Monoids Monoids Edward Kmett Overview Overview Monoids (definition, examples) Reducers Generators Benefits of Monoidal Parsing ◦ Incremental Parsing (FingerTrees) ◦ Parallel Parsing (Associativity) ◦ Composing Parsers (Products, Layering) ◦ Compressive Parsing (LZ78, Bentley-McIlroy) Going Deeper (Seminearrings) What is a What is a Monoid Monoid? ? A Monoid is any a
The Comonad.Reader » Representing Applicatives
In the previous two posts, we've built up a whole range of applicatives, out of Const, Identity, Reader, Compose, Product, Sum, and Fix (and some higher-order analogues). Sum has given us the most trouble, but in some sense has been the most powerful, letting us write things like possibly eventually terminating lists, or trees, or in fact any sort of structure with branching alternatives. In this
The Comonad.Reader » Japanese “ekmett” Workshop (Part 1 of 2)
A couple of weeks back one of my coworkers brought to my attention a several hour long workshop in Japan to go over and describe a number of my libraries, hosted by TANAKA Hideyuki — not the voice actor, I checked! I was incredibly honored and I figured that if that many people (they had 30 or so registered attendees and 10 presentations) were going to spend that much time going over software that
The Comonad.Reader » Deriving Strength from Laziness
No, this isn't some uplifting piece about deriving courage from sloth in the face of adversity. What I want to talk about is monadic strength. Transcribing the definition from category theory into Haskell we find that a strong monad is a functor such that there exists a morphism: with a couple of conditions on it that I'll get to later. Currying that to get something that feels more natural to a H
The Comonad.Reader » Algebras of Applicatives
While the previous post in this series was relatively immediately applicable, this one has constructions I definitely wouldn't recommend in production code. However, they do take us further in exploring the universe of applicative functors, and, more broadly, exploring which data types provide which properties by construcion. It's well known that if you have any Functor F a, you can take its "fixp
The Comonad.Reader » Abstracting with Applicatives
Consider the humble Applicative. More than a functor, less than a monad. It gives us such lovely syntax. Who among us still prefers to write liftM2 foo a b when we could instead write foo <$> a <*> b? But we seldom use the Applicative as such — when Functor is too little, Monad is too much, but a lax monoidal functor is just right. I noticed lately a spate of proper uses of Applicative —Formlets (
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The Comonad.Reader » Internalized Guarded Recursion for Equational Reasoning
The Comonad.Reader » Some Rough Notes on Univalent Foundations and B-Systems, Part I
The Comonad.Reader » Fast Circular Substitution
http://comonad.com/haskell/Lenses-Folds-and-Traversals-NYC.pdf
The Comonad.Reader » Unnatural Transformations and Quantifiers