Mathematics and Computation | Hask is not a category
Hask is not a category 06 August 2016 Andrej Bauer Computation, Programming This post is going to draw an angry Haskell mob, but I just have to say it out loud: I have never seen a definition of the so-called category Hask and I do not actually believe there is one until someone does some serious work. Let us look at the matter a bit closer. The Haskell wiki page on Hask says: The objects of Hask
Hask - HaskellWiki
Hask is the category of Haskell types and functions. Informally, the objects of Hask are Haskell types, and the morphisms from objects A to B are Haskell functions of type A -> B. The identity morphism for object A is id :: A -> A, and the composition of morphisms f and g is f . g = \x -> f (g x). However, subtleties arise from questions such as the following. When are two morphisms equal? People
Category Theory for Programmers: The Preface
Category Theory for Programmers: The Preface Posted by Bartosz Milewski under C++, Category Theory, Functional Programming, Haskell, Programming [184] Comments Table of Contents Part One Category: The Essence of Composition Types and Functions Categories Great and Small Kleisli Categories Products and Coproducts Simple Algebraic Data Types Functors Functoriality Function Types Natural Transformati
Haskell/åœè«– - Wikibooks
ã“ã®é …ç›®ã§ã¯ Haskell ã«é–¢é€£ã™ã‚‹å†…容ã«é™ã£ã¦åœè«–ã®æ¦‚観を与ãˆã‚‹ã“ã¨ã‚’試ã¿ã‚‹ã€‚ãã®ãŸã‚ã«ã€æ•°å¦çš„ãªå®šç¾©ã«ä½µã›ã¦ Haskell コードも示ã™ã€‚絶対的ãªåŽ³å¯†ã•ã¯æ±‚ã‚ãªã„。ãã®ã‹ã‚ã‚Šã€åœè«–ã®æ¦‚念ã¨ã¯ã©ã‚“ãªã‚‚ã®ã‹ã€ã©ã®ã‚ˆã†ã« Haskell ã«é–¢é€£ã™ã‚‹ã‹ã®ç›´æ„Ÿçš„ãªç†è§£ã‚’èªè€…ã«ä¸Žãˆã‚‹ã“ã¨ã‚’追求ã™ã‚‹ã€‚ ３ã¤ã®å¯¾è±¡A, B, Cã€ï¼“ã¤ã®æ’ç‰å°„, , ã¨ã€ã•ã‚‰ã«åˆ¥ã®å°„, ã‹ã‚‰ãªã‚‹å˜ç´”ãªåœã€‚３ã¤ã‚ã®è¦ç´ (ã©ã®ã‚ˆã†ã«å°„ã‚’åˆæˆã™ã‚‹ã‹ã®å®šç¾©)ã¯ç¤ºã—ã¦ã„ãªã„。 本質的ã«ã€åœã¨ã¯å˜ç´”ãªé›†ã¾ã‚Šã§ã‚る。ã“ã‚Œã¯æ¬¡ã®ï¼“ã¤ã®è¦ç´ ã‹ã‚‰ãªã‚‹ã€‚ 対象(Object)ã®é›†ã¾ã‚Šã€‚ ãµãŸã¤ã®å¯¾è±¡(source objectã¨target object)ã‚’ã²ã¨ã¤ã«çµã³ã¤ã‘ã‚‹å°„ã®é›†ã¾ã‚Šã€‚(ã“れらã¯arrowã¨å‘¼ã°ã‚Œã‚‹ã“ã¨ã‚‚ã‚ã‚‹ãŒã€Haskellã§ã¯ã“ã‚Œã¯åˆ¥ã®æ„味をæŒã¤ç”¨èªžãªã®ã§ã€ã“ã“ã§ã¯ã“ã®ç”¨èªžã‚’é¿ã‘ã‚‹ã“ã¨ã«ã™ã‚‹ã€‚) f ãŒã‚½ãƒ¼ã‚¹ã‚ªãƒ–
モナドã¸ã®è¿‘é“・Haskell ã‹ã‚‰ã®å¯„é“
モナドã¸ã®è¿‘é“・Haskell ã‹ã‚‰ã®å¯„é“ ä¸æ‘ç¿”å¾âˆ— 2007 å¹´ 1 月 26 æ—¥ 本稿ã®è¶£æ—¨ åœè«–ã®ãƒ¢ãƒŠãƒ‰ã¨ãƒ—ãƒã‚°ãƒ©ãƒŸãƒ³ã‚°è¨€èªž (Haskell) ã®é–“ã«ã¯ã©ã®æ§˜ãªé–¢ä¿‚ãŒã‚ã‚‹ã®ã‹ï¼Žã“ã®å•ã„ã«ç”ãˆã‚‹ãŸ ã‚ã«ã¯ï¼Œå½“然ã ãŒï¼Œå°‘ãªãã¨ã‚‚モナドãŒç†è§£ã§ãる程度ã«åœè«–を知らãªã‘ã‚Œã°ãªã‚‰ãªã„．本稿ã¯ï¼Œåœè«–ã®çŸ¥ è˜ãŒå…¨ããªã„ã¨ã“ã‚ã‹ã‚‰ï¼Œãƒ¢ãƒŠãƒ‰ã‚’ç†è§£ã™ã‚‹ã¾ã§ã®ï¼Œæœ€çŸã‚³ãƒ¼ã‚¹ã®é“ã®ã‚Šã‚’ãŸã©ã‚Š (第 1,2 ç« )，ã“ã‚Œã‚’å‰ æã«ã—㦠Haskell ã¨åœè«–ã¨ã®é–¢ä¿‚を考察ã™ã‚‹ (第 3 ç« )．èªè€…ã«ã¯åœè«–ã®çŸ¥è˜ã‚’è¦æ±‚ã—ãªã„ãŒï¼ŒHaskell 㮠文法ã®åŸºæœ¬çš„ãªç†è§£ã¯å‰æã¨ã—ã¦ã„る．ã¾ãŸï¼Œåœè«–ã«é–¢ã—ã¦ã¯ï¼Œã‚ãã¾ã§ã‚‚「モナドã¸ã®æœ€çŸã‚³ãƒ¼ã‚¹ã€ãªã® ã§ï¼Œç±³ç”°ã®è£œé¡Œï¼Œæ™®é性，コンマåœï¼Œæ¥µé™ãªã©ã®é‡è¦ãªæ¦‚念ã¯è§£èª¬ã—ã¦ã„ãªã„． 1 基礎知è˜ã®æº–å‚™ 定義 1 (メタグラフ metagraph) メタグラフã¯å°„ (arrow
Boost.勉強会 #6 æœå¹Œã§å–‹ã£ã¦ããŸè©±ã¨ã„ã†ã‹é–¢æ•°ãŒç´”粋ã§å¼·ã型付ã‘られã¦ã„ã‚‹ã“ã¨ã®é‡è¦æ€§ã¨ã„ã†ã‹ - 梶本裕介ã®æ—¥è¨˜
https://public.me.com/yusuke1984/ja/発表資料ã¯ã“ã“ã«ãŠã„ã¦ã¾ã™ãŒã€ustã¯é€”ä¸ã§é€”切れã¡ã‚ƒã£ãŸã¿ãŸã„ãªã®ã§è¨€ã„ãŸã‹ã£ãŸã“ã¨çºã‚ã¨ãã¾ã™ã¨ã€ä»Šå›žã¯çµå±€ãƒ—ãƒã‚°ãƒ©ãƒ 「をã€è¨ˆç®—ã™ã‚‹è©±ãŒã—ãŸã‹ã£ãŸã®ã§ã—ãŸã€‚関数プãƒã‚°ãƒ©ãƒŸãƒ³ã‚°ã§ã¯ã€å®£è¨€çš„ã«è¨˜è¿°ã•ã‚ŒãŸé–¢æ•°ã®å®šç¾©å¼ãŒå®Ÿè¡Œå¯èƒ½ãªä»•æ§˜ã¨å‘¼ã¹ã‚‹ã»ã©åˆ†ã‹ã‚Šã‚„ã™ããªã‚‹ã“ã¨ãŒã‚ã‚Šã¾ã™ã€‚ã—ã‹ã—ãªãŒã‚‰ãã®ãƒ—ãƒã‚°ãƒ©ãƒ ã¯ç¾å®Ÿã®å•é¡Œè§£æ±ºã§ç”¨ã„ã‚‹ã®ã«å分ãªã»ã©é«˜é€Ÿã§ã¯ãªã„ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“。ã¨ã“ã‚ã§å‚ç…§é€éŽæ€§ãŒä¿ãŸã‚Œã¦ã„ã‚‹ã¨ã€ãƒ—ãƒã‚°ãƒ©ãƒ ã‚’å½¢æˆã™ã‚‹å¼ã‚’ã€ãã‚Œã¨ç‰ã—ã„値をæŒã¤å¼ã¨ç½®ãæ›ãˆã¦è¡Œãã“ã¨ãŒã§ãã¾ã™ã€‚ãã“ã§ã€ã¾ãšé–¢æ•°åž‹è¨€èªžã‚’用ã„ã¦ç°¡æ½”ã§æ˜Žæ¾„ãªå•é¡Œã®è¨˜è¿°ã‚’è¡Œã„ã€ãã“ã‹ã‚‰ã€ãã®é–¢æ•°åž‹è¨€èªžä¸Šã§æ—¢ã«è¨¼æ˜Žã•ã‚Œã¦ã„る定ç†ã‚’用ã„ã¦ã€å¼å¤‰å½¢ã«ã‚ˆã£ã¦åŠ¹çŽ‡çš„ãªå®Ÿè£…ã‚’å°Žã出ã™ã¨è¨€ã†æ‰‹æ³•ãŒè€ƒãˆã‚‰ã‚Œã¾ã™ã€‚Pearls of Functional Algorithm D
Category theory for Haskell programmers | Geoff Hulette
Here is some advice, gained through my own trial and error, on how to teach yourself basic category theory. Â This advice will probably be useful only if you are like me, i.e. a computer scientist and Haskell programmer without a strong background in advanced math. The development of category theory was motivated by problems in the mathematical field of abstract algebra, and most of the literature
apfelmus - Monoids and Finger Trees
In this article, we explain how monoids enable one and the same data structure, the finger tree, to implement virtually any other data structure; they arise by different choices of monoids. The technique is discussed using the examples of a list with random access and a priority queue. This post grew out of the big monoid discussion on the haskell-cafe mailing list. Introduction A very powerful ap
A Neighborhood of Infinity: Haskell Monoids and their Uses
Haskell is a great language for constructing code modularly from small but orthogonal building blocks. One of these small blocks is the monoid. Although monoids come from mathematics (algebra in particular) they are found everywhere in computing. You probably use one or two monoids implicitly with every line of code you write, whatever the language, but you might not know it yet. By making them ex
Haskel モナドã®è§£èª¬ from 2ch - From a Far East Island
735 ：デフォルトã®åç„¡ã—ã•ã‚“ ：2007/02/15(木) 00:11:39 >>733 3è¡Œã§èª¬æ˜Žã™ã‚‹ã®ã¯ä¿ºã«ã¯ç„¡ç†ï¼Ž ã¾ãšå®šç¾©5．ã“ã‚Œã¯å®Ÿéš›ã«æ‰‹ã‚’å‹•ã‹ã™ã¨è¦‹ãˆã¦ãã‚‹ã®ã§ï¼Œ 具体例ã§ã‚„ã‚‹ã®ãŒè‰¯ã„ã¨æ€ã†ï¼Žä»¥ä¸‹ã¯ãã® pdf ã«ã‚‚ã‚る例． 1. リスト函手 ・型 A ã«å¯¾ã— T A 㯠A ã®ãƒªã‚¹ãƒˆã‚’作る： T A = [A] ・関数 f :: A -> B ã«å¯¾ã— T f :: [A] -> [B] ã¯æ¬¡ã®é–¢æ•°ã‚’作る： T f = map f ・μ_X 㯠X åž‹ã®ãƒªã‚¹ãƒˆã®ãƒªã‚¹ãƒˆã‚’ãªã‚‰ã—㦠X åž‹ã®ãƒªã‚¹ãƒˆã‚’作る： μ_X = concat ・η_X 㯠X åž‹ã®å€¤ x ã‹ã‚‰ãã‚Œã ã‘ã‹ã‚‰ãªã‚‹ [X} åž‹ã®ãƒªã‚¹ãƒˆ [x] を作る： η_X = singleton = (:[]) ã“れらを図ã«ä»£å…¥ã—ã¦ï¼Œã¡ã‚ƒã‚“ã¨å¯æ›ã«ãªã‚‹ã“ã¨ã‚’確èªã™ã‚‹ã‚ˆã‚ã—． 736 ：デフォルトã®åç„¡ã—ã•ã‚“ ：2007/0
Category theory/Natural transformation - HaskellWiki
In the followings, this example will be used to illustrate the notion of natural transformation. If the examples are exaggerated and/or the definitions are incomprehensible, try #External links. Definition Let C, D denote categories. Let Φ,Ψ:C→D be functors. Let X,Y∈Ob(C). Let f∈HomC(X,Y). Let us define the η:Φ→Ψ natural transformation. It associates to each object of C a morphism of D in the foll
Category theory - HaskellWiki
Category theory can be helpful in understanding Haskell's type system. There exists a "Haskell category", of which the objects are Haskell types, and the morphisms from types a to b are Haskell functions of type a -> b. The Haskell wikibooks has an introduction to Category theory, written specifically with Haskell programmers in mind. Definition of a category A category consists of two collections
The Typeclassopediaを訳ã—ã¾ã—ãŸ, The Typeclassopedia - #3(2009-10-20)
â– [Haskell] The Typeclassopediaを訳ã—ã¾ã—㟠The Monad.Readerã®Issue 13ã«æŽ²è¼‰ã•ã‚ŒãŸThe Typeclassopediaã¨ã„ã†è¨˜äº‹ãŒã€Functor, Monad, Monoid, Applicative, Foldable, Traversable, Arrowã¨ã„ã£ãŸã‚ˆã†ãªåž‹ã‚¯ãƒ©ã‚¹ã«ã¤ã„ã¦è‰¯ãã¾ã¨ã¾ã£ã¦ã„ã¦ã€ãã®ã‚ãŸã‚Šã‚’知りãŸã„時ã®å–ã£æŽ›ã‹ã‚Šã«ãªã‚Šãã†ã ã£ãŸã®ã§ç¿»è¨³ã—ã¦ã¿ã¾ã—ãŸã€‚ 作者ã®Brent Yorgeyã•ã‚“ã‹ã‚‰ã‚‚許å¯ãŒã„ãŸã ã‘ãŸã®ã§å…¬é–‹ã—ã¾ã™ã€‚翻訳ã«æ…£ã‚Œã¦ã„ãªã„ã®ã§å¤‰ãªæ—¥æœ¬èªžï¼ˆç‰¹ã«å°‚門用語ã®æ—¥æœ¬èªžè¨³ã¯ã‹ãªã‚Šæ€ªã—ã„）ãŒã‚ã£ãŸã‚Šã€ãã‚‚ãã‚‚é–“é•ã£ã¦ã„ãŸã‚Šã™ã‚‹ã‹ã‚‚ã—ã‚Œã¾ã›ã‚“ã®ã§ã€ä½•ã‹è¦‹ã¤ã‘ãŸã‚‰ã‚³ãƒ¡ãƒ³ãƒˆã‚’é ‚ã‘ã‚‹ã¨åŠ©ã‹ã‚Šã¾ã™ã€‚ â– [Haskell] The Typeclassopedia by Brent Yorgey <first
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Where do values fit in Category of Hask?
Re: [Haskell-cafe] Basic question concerning the category Hask (was: concerning data constructors)
A Neighborhood of Infinity: "What Category do Haskell Types and Functions Live In?"
GitHub - ekmett/category-extras: category-theoretic goodness for Haskell
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