[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-657835-105":59,"doc-detail-657835-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","category-theory-for-computing-science","Category Theory for Computing Science","","Category Theory for Computing Science presents an organized introduction to category theory with emphasis on concepts relevant to computation and programming language semantics. It covers foundational material on sets, functions, graphs, and homomorphisms; then builds the categorical framework through categories, functors, naturality, diagrams, and the Yoneda lemma. Subsequent chapters develop products and sums, cartesian closed categories and typed lambda calculus, sketches and their models, limits and colimits, fibrations, adjoints, algebras for endofunctors, and toposes, culminating in categories with monoidal structure.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/category-theory-for-computing-science/657835/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/category-theory-for-computing-science/657835.png","ImageObject",300,407,{"name":92,"@type":93},"Helloiii","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What prerequisite topics are covered before categories are introduced?","Question",{"text":112,"@type":113},"The book starts with preliminaries on sets, functions, graphs, and homomorphisms of graphs, preparing the reader to reason about mathematical structures.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How does the book relate category theory to computation?",{"text":117,"@type":113},"It develops categories and functors that mirror programming language structures, including chapters on functional programming languages as categories, typed lambda calculus, and deduction systems as categories.",{"name":119,"@type":110,"acceptedAnswer":120},"Which categorical constructions are emphasized in the middle chapters?",{"text":121,"@type":113},"Key constructions include products and sums, cartesian closed categories, finite product sketches, finite discrete sketches, and limits and colimits, with attention to how these ideas shape models.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},657835,1793074071,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145},5909897196768,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Category Theory for  \nComputing Science  \nMichael Barr Charles Wells  \n°c Michael Barr and Charles Wells, 1998  \nCategory Theory for Computing Science  \nMichael Barr  \nDepartment of Mathematics and  \nStatistics McGill University  \nCharles Wells Department of Mathematics Case Western Reserve University  \nFor  \nBecky, Adam and Joe and  \nMatt and Peter  \nContents  \nPreface xi  \n1 Preliminaries 1  \n1.1 Sets 1  \n1.2 Functions 3  \n1.3 Graphs 8  \n1.4 Homomorphisms of graphs 11  \n2 Categories 15  \n2.1 Basic de¯nitions 15  \n2.2 Functional programming languages as categories 20  \n2.3 Mathematical structures as categories 23  \n2.4 Categories of sets with structure 27  \n2.5 Categories of algebraic structures 32  \n2.6 Constructions on categories 35  \n2.7 Properties of objects and arrows in a category 40  \n2.8 Monomorphisms and subobjects 47  \n2.9 Other types of arrow 53  \n2.10 Factorization systems 58  \n3 Functors 65  \n3.1 Functors 65  \n3.2 Actions 74  \n3.3 Types of functors 80  \n3.4 Equivalences 84  \n3.5 Quotient categories 88  \n4 Diagrams, naturality and sketches 93  \n4.1 Diagrams 93  \n4.2 Natural transformations 101  \n4.3 Natural transformations between functors 109  \n4.4 The Godement calculus of natural transformations 117  \n4.5 The Yoneda Lemma and universal elements 121  \n4.6 Linear sketches (graphs with diagrams) 127  \n4.7 Linear sketches with constants: initial term models 133  \n4.8 2-categories 140  \nviii Contents  \n5 Products and sums 153  \n5.1 The product of two objects in a category 153  \n5.2 Notation for and properties of products 157  \n5.3 Finite products 168  \n5.4 Sums 178  \n5.5 Natural numbers objects 182  \n5.6 Deduction systems as categories 186  \n5.7 Distributive categories 188  \n6 Cartesian closed categories 195  \n6.1 Cartesian closed categories 195  \n6.2 Properties of cartesian closed categories 202  \n6.3 Typed ¸-calculus 208  \n6.4 ¸-calculus to category and back 210  \n6.5 Arrows vs. terms 212  \n6.6 Fixed points in cartesian closed categories 215  \n7 Finite product sketches 219  \n7.1 Finite product sketches 220  \n7.2 The sketch for semigroups 225  \n7.3 Notation for FP sketches 231  \n7.4 Arrows between models of FP sketches 234  \n7.5 The theory of an FP sketch 237  \n7.6 Initial term models for FP sketches 239  \n7.7 Signatures and FP sketches 245  \n8 Finite discrete sketches 251  \n8.1 Sketches with sums 251  \n8.2 The sketch for ¯elds 254  \n8.3 Term algebras for FD sketches 257  \n9 Limits and colimits 265  \n9.1 Equalizers 265  \n9.2 The general concept of limit 268  \n9.3 Pullbacks 273  \n9.4 Coequalizers 277  \n9.5 Cocones 280  \n9.6 More about sums 285  \n9.7 Uni¯cation as coequalizer 289  \n9.8 Properties of factorization systems 294  \nContents ix  \n10 More about sketches 299  \n10.1 Finite limit sketches 299  \n10.2 Initial term models of FL sketches 304  \n10.3 The theory of an FL sketch 307  \n10.4 General de¯nition of sketch 309  \n11 The category of sketches 313  \n11.1 Homomorphisms of sketches 313  \n11.2 Parametrized data types as pushouts 315  \n11.3 The model category functor 320  \n12 Fibrations 327  \n12.1 Fibrations 327  \n12.2 The Grothendieck construction 332  \n12.3 An equivalence of categories 338  \n12.4 Wreath products 341  \n13 Adjoints 347  \n13.1 Free monoids 347  \n13.2 Adjoints 350  \n13.3 Further topics on adjoints 356  \n13.4 Locally cartesian closed categories 360  \n14 Algebras for endofunctors 363  \n14.1 Fixed points for a functor 363  \n14.2 Recursive categories 368  \n14.3 Triples 372  \n14.4 Factorizations of a triple 374  \n14.5 Scott domains 376  \n15 Toposes 383  \n15.1 De¯nition of topos 384  \n15.2 Properties of toposes 387  \n15.3 Is a two-element poset complete? 391  \n15.4 Presheaves 393  \n15.5 Sheaves 395  \n15.6 Fuzzy sets 400  \n15.7 External functors 403  \n15.8 The realizability topos 408  \nx Contents  \n16 Categories with monoidal structure 413  \n16.1 Closed monoidal categories 413  \n16.2 Properties of A ¡± C 417  \n16.3 ¤-autonomous categories 422  \n16.4 The Chu construction 424  \nSolutions to the exercises 431  ","cbCaipzbtRMjSI0m","https://ap.wps.com/l/cbCaipzbtRMjSI0m","pdf",2097033,556,"English","# Preface\n# 1 Preliminaries\n## 1.1 Sets\n## 1.2 Functions\n## 1.3 Graphs\n## 1.4 Homomorphisms of graphs\n# 2 Categories\n## 2.1 Basic de¯nitions\n## 2.2 Functional programming languages as categories\n## 2.3 Mathematical structures as categories\n## 2.4 Categories of sets with structure\n## 2.5 Categories of algebraic structures\n## 2.6 Constructions on categories\n## 2.7 Properties of objects and arrows in a category\n## 2.8 Monomorphisms and subobjects\n## 2.9 Other types of arrow\n## 2.10 Factorization systems\n# 3 Functors\n## 3.1 Functors\n## 3.2 Actions\n## 3.3 Types of functors\n## 3.4 Equivalences\n## 3.5 Quotient categories\n# 4 Diagrams, naturality and sketches\n## 4.1 Diagrams\n## 4.2 Natural transformations\n## 4.3 Natural transformations between functors\n## 4.4 The Godement calculus of natural transformations\n## 4.5 The Yoneda Lemma and universal elements\n## 4.6 Linear sketches (graphs with diagrams)\n## 4.7 Linear sketches with constants: initial term models\n## 4.8 2-categories\n# 5 Products and sums\n## 5.1 The product of two objects in a category\n## 5.2 Notation for and properties of products\n## 5.3 Finite products\n## 5.4 Sums\n## 5.5 Natural numbers objects\n## 5.6 Deduction systems as categories\n## 5.7 Distributive categories","[{\"question\":\"What prerequisite topics are covered before categories are introduced?\",\"answer\":\"The book starts with preliminaries on sets, functions, graphs, and homomorphisms of graphs, preparing the reader to reason about mathematical structures.\"},{\"question\":\"How does the book relate category theory to computation?\",\"answer\":\"It develops categories and functors that mirror programming language structures, including chapters on functional programming languages as categories, typed lambda calculus, and deduction systems as categories.\"},{\"question\":\"Which categorical constructions are emphasized in the middle chapters?\",\"answer\":\"Key constructions include products and sums, cartesian closed categories, finite product sketches, finite discrete sketches, and limits and colimits, with attention to how these ideas shape models.\"}]","Category Theory for Computing Science | PDF",1792991660,1401]