プãƒã‚°ãƒ©ãƒžãƒ¼ã®ãŸã‚ã®åœè«–（上） - bitterharvest’s diary
『プãƒã‚°ãƒ©ãƒžã®ãŸã‚ã®åœè«–ã€ã¯ã“ã‚Œã¾ã§ã®åˆ†ã‚’ã¾ã¨ã‚ã¦PDFファイルã«ã—ã¾ã—ãŸã€‚å‚考ã«ã—ã¦ãã ã•ã„。
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
Enriched Categories
This is part 28 of Categories for Programmers. Previously: Kan Extensions. See the Table of Contents. A category is small if its objects form a set. But we know that there are things larger than sets. Famously, a set of all sets cannot be formed within the standard set theory (the Zermelo-Fraenkel theory, optionally augmented with the Axiom of Choice). So a category of all sets must be large. Ther
Steve Awodey著『Category Theoryã€ã‚’èªã‚€ã€€2.3 Generalized elements - Qiita
ã¨ã„ã†æ•°å¼ãŒå‡ºã¦ãã¦ã€ã‚‚～ギブアップã£ã¦æ„Ÿã˜ã«ãªã‚Šã¾ã™ã€‚ $X$ã¨ã„ã†ã®ã¯$Sets$ã®ä¸ã®å¯¾è±¡ã§ã€ä½•ã‹ã®é›†åˆã‚‰ã—ã„ã§ã™ã€‚ $1$ã¨ã„ã†ã®ã¯$Sets$ã®ä¸ã®çµ‚対象ã§ã€ãªã‚“ã§ã‚‚ã„ã„ã‹ã‚‰è¦ç´ æ•°ãŒ1ã®é›†åˆã ãã†ã§ã™ã€‚ ãã—ã¦$Hom_{Sets}(1,X)$ã¯çµ‚対象ã‹ã‚‰$X$ã¸ã®å°„ã‚’å…¨ã¦é›†ã‚ãŸé›†åˆã ãã†ã§ã™ã€‚ ã¨ãªã‚‹ã¨ã€ã“ã®æ•°å¼ã¯ã€ã‚る集åˆã¨ã€$Sets$ã®ä¸ã®å°„ã®é›†åˆãŒåŒã˜ã‚‚ã®ã ã¨è¨€ã£ã¦ã„ã‚‹ã“ã¨ã«ãªã‚Šã¾ã™ã€‚ ãã‚“ãªã“ã¨è¨€ã‚ã‚Œã¦ã‚‚ã€ãœã‚“ãœã‚“åŒã˜ã«è¦‹ãˆã¾ã›ã‚“ï¼ ã•ã¦ã€ã“ã‚“ãªæ™‚ã«åŠ©ã‘ã«ãªã‚‹ã®ã¯ã€ã‚„ã£ã±ã‚Šå…·ä½“例ã§ã™ã€‚ ã¨ã«ã‹ã具体例を見ã¦ã€ã ã„ãŸã„ã®æ„Ÿã˜ã§ã„ã„ã‹ã‚‰ã¤ã‹ã¿ã¾ã—ょã†ã€‚ ãã®ãŸã‚ã«$Sets_{fin}$を使ã„ã¾ã™ã€‚ ã“ã®ã‚³ãƒ¼ãƒ‰ã‚’コピペã™ã‚‹ãªã‚Šãªã‚“ãªã‚Šã—ã¦ã€æ‰‹å…ƒã«SetsFin.hsを用æ„ã—ã¦ãã ã•ã„。 ã“れをghciã«èªã¿è¾¼ã¾ã›ã¾ã™ã€‚
Kleisli Categories
Kleisli Categories Posted by Bartosz Milewski under C++, Category Theory, Haskell [34] Comments In the previous installment of Categories for Programmers, Categories Great and Small, I gave a few examples of simple categories. In this installment we’ll work through a more advanced example. If you’re new to the series, here’s the Table of Contents. Composition of Logs You’ve seen how to model types
Categories for the Working Haskeller | SkillsCast
8th October 2014 in London There are 7 other SkillsCasts available from Haskell eXchange 2014 Please log in to watch this conference skillscast. In this talk, I hope to explain a little bit about how categories can help the working Haskeller. I'll focus on two aspects: categories as an organising principle, helping us to manage generic libraries; and categories as a reasoning principle, helping us
Procrustean Mathematics - School of Haskell | School of Haskell
Interactive code snippets not yet available for SoH 2.0, see our Status of of School of Haskell 2.0 blog post This post is a bit of a divergence from my norm. I'm going to editorialize a bit about mathematics, type classes and the tension between different ways of fitting ideas together, rather than about any one algorithm or data structure. I apologize in advance for the fact that my examples are
Why Category Theory Matters · rs.io
Wed, Apr 30, 2014 I hope most mathematicians continue to fear and despise category theory, so I can continue to maintain a certain advantage over them. —John Baez The above is a graph of the number of times the phrase “category theory†has been used in books, from about 1950 through the present. It speaks for itself. But why? What’s the big deal? Why does category theory matter? I’m about a quarte
category-extras
category-extras: A meta-package documenting various packages inspired by category theory The obsolete category-extras package provided a monolithic set of modules designed for the use of category theory in Haskell. It was exploded into more focused, self-contained packages (listed in the dependencies below); this meta-package documents where the code has gone. In addition to the core definitions,
Isomorphism, retraction, section | tnomuraã®ãƒ–ãƒã‚°
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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–
IIJ Research Laboratory
ãƒãƒƒãƒˆãƒ¯ãƒ¼ã‚¯ã®è¨ˆæ¸¬ã¨è§£æž インターãƒãƒƒãƒˆã®ä½¿ã‚れ方やãƒãƒƒãƒˆãƒ¯ãƒ¼ã‚¯ã®æŒ™å‹•ã‚’把æ¡ã™ã‚‹äº‹ã¯ã€ãƒãƒƒãƒˆãƒ¯ãƒ¼ã‚¯ã‚’é‹ç”¨ã—ã€ãã®æŠ€è¡“開発を行ㆠãŸã‚ã«æ¬ ã‹ã›ã¾ã›ã‚“。ã—ã‹ã—ã€è¦³æ¸¬ã§å¾—られるデータé‡ã¯è†¨å¤§ã§ã™ãŒãƒŽã‚¤ã‚ºãŒå¤šãã€ã¾ãŸã€è¦³æ¸¬ã§ãã‚‹ã®ã¯æ¥µã‚ã¦é™ã‚‰ã‚ŒãŸéƒ¨åˆ†ã§ã—ã‹ã‚ã‚Šã¾ã›ã‚“。ãã“ã§ã€è†¨å¤§ãªãƒ‡ãƒ¼ã‚¿ã‹ã‚‰æ„味ã®ã‚ã‚‹æƒ…å ±ã‚’æŠ½å‡ºã—ãŸã‚Šã€éƒ¨åˆ†çš„ãªè¦³æ¸¬ã‹ã‚‰ã‚ˆã‚Šä¸€èˆ¬çš„ãªå‚¾å‘を推測ã™ã‚‹äº‹ãŒå¿…è¦ã¨ãªã‚Šã¾ã™ã€‚... インターãƒãƒƒãƒˆåŸºç›¤æŠ€è¡“ 速ãã¦ã€å®‰å…¨ã§ã€ä¿¡é ¼æ€§ãŒé«˜ãã€ä½¿ã„ã‚„ã™ãã€ãªã©ã€ã‚¤ãƒ³ã‚¿ãƒ¼ãƒãƒƒãƒˆã‚µãƒ¼ãƒ“スã¸ã®è¦æ±‚ã¯ã¾ã™ã¾ã™é«˜ã¾ã£ã¦ã„ã¾ã™ã€‚ã“れらã®è¦æ±‚ã«å¿œãˆã‚‹ãŸã‚ã«ã€ã‚¤ãƒ³ã‚¿ãƒ¼ãƒãƒƒãƒˆã® 基盤技術も日々進æ©ã—ã¦ã„ã¾ã™ã€‚ã„ã¾ã‚„インターãƒãƒƒãƒˆã¯ã¤ãªãŒã‚‹ã ã‘ã®ã‚µãƒ¼ãƒ“スã§ã¯ãªãã€é«˜åº¦ã§è¤‡é›‘ãªæ©Ÿèƒ½ã‚’å‚™ãˆãŸç¤¾ä¼šåŸºç›¤ã¨ãªã‚Šã¾ã—ãŸã€‚IIJæŠ€è¡“ç ”ç©¶æ‰€ã¯ã€ã‚¤ãƒ³ã‚¿ãƒ¼ãƒãƒƒãƒˆã®åŸºç›¤ã¨ã—ã¦å®Ÿç¾ãŒæœŸå¾…ã•ã‚Œã‚‹æ©Ÿèƒ½ã‚’æä¾›ã™ã‚‹ãŸã‚ã«ã€ã•ã¾ã–ã¾ãªæŠ€è¡“課題ã«å–り組んã§
åœè«–ã®è¨€è‘‰ã€€ãã®ï¼’ | tnomuraã®ãƒ–ãƒã‚°
endomap endomap 㯠domain 㨠codomain ãŒåŒã˜å°„ã 。ãã†çã—ã„ã‚‚ã®ã§ã¯ãªãã€æ•´æ•°ã®é›†åˆã‚’ domain ã¨ã—ã€æ•´æ•°ã®2å€ã®å€¤ã‚’è¿”ã™é–¢æ•°ï¼ˆå°„）ã¯ã€codomain ãŒæ•´æ•°ã«ãªã‚‹ã€‚ Prelude> :set +m Prelude> let Prelude| double :: Int -> Int Prelude| double x = 2 * x Prelude| Prelude> double 3 6 有é™å€‹ã®æ•°å€¤ã®é›†åˆã® endomap ã®ã†ã¡ã€å…¨å˜å°„ã«ãªã‚‹ã‚‚ã®ã¯ã€é †åˆ—ã 。 Prelude> let Prelude| f 1 = 4 Prelude| f 2 = 3 Prelude| f 3 = 2 Prelude| f 4 = 1 Prelude| Prelude> map f [1,2,3,4] [4,3,2,1] identity map endo
åœè«–ã®è¨€è‘‰ | tnomuraã®ãƒ–ãƒã‚°
Haskell を勉強ã—始ã‚㦠IO モナドãªã‚“ã‹ã«å‡ºãã‚ã•ãªã‘ã‚Œã°åœè«–ãªã©è¿‘ã¥ã“ã†ã¨ã‚‚ã—ãªã‹ã£ãŸã ã‚ã†ãŒã€ãƒ¢ãƒŠãƒ‰ã‚’æ£ç¢ºã«ç†è§£ã—ãŸããªã£ã¦ã€åœè«–ã®å…¥é–€æ›¸ Conceptual Mathematics: A First Introduction to Categories F. William Lawvere〠Stephen H. Schanuel ã‚’èªã¿å§‹ã‚ãŸã€‚é«˜æ ¡å’æ¥ç¨‹åº¦ã®äºˆå‚™çŸ¥è˜ã§ã‚‚èªã‚るよã†ã«æ‡‡åˆ‡ä¸å¯§ã«æ›¸ã‹ã‚Œã¦ã„ã‚‹ã®ãŒã†ã‚Œã—ã„。 ã—ã‹ã—ã€åœè«–ã¯åœè«–ãªã®ã§ã€ãã†ç°¡å˜ã«ç†è§£ã§ãるよã†ã«ã¯æ€ãˆãªã„。ãã“ã§ã€åˆ†ã‹ã£ãŸã¨æ€ãˆãŸã¨ã“ã‚ã‚’ã¾ã¨ã‚ã¦ã¿ã‚‹äº‹ã«ã—ãŸã€‚ åœè«–ã¯é©ç”¨ç¯„囲ãŒåºƒã„抽象的ãªè€ƒãˆæ–¹ãªã®ã§ã€å…·ä½“çš„ãªã‚¤ãƒ¡ãƒ¼ã‚¸ãŒä½œã‚Šã«ãã„。やã¯ã‚Šã€æ…£ã‚Œè¦ªã—ã‚“ã 集åˆã¨å†™åƒã‚’使ãˆã‚‹ã€Œé›†åˆã®åœã€ã‹ã‚‰å§‹ã‚ãŸæ–¹ãŒè‰¯ã„よã†ã 。集åˆã¨å†™åƒã‚’ã€å¯¾è±¡ã¨å°„ã¨ã—ã¦å–り扱ã†äº‹ã«ã‚ˆã£ã¦ã€ä»–ã®åœã«å¯¾ã—ã¦ã‚‚イメージを広ã’ã¦ã„ã事ãŒã§
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åœè«–勉強会 @ ワークスアプリケーションズ
Reddit - Dive into anything
Reddit - Dive into anything
Covariant and contravariant in categories and geometry
https://docs.sympy.org/0.7.2/modules/categories.html
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