[{"data":1,"prerenderedAt":1190},["ShallowReactive",2],{"subject:category-theory":3,"course-wordcounts":75,"nav:category-theory":987},{"id":4,"title":5,"blurb":6,"body":7,"brief":18,"category":56,"description":57,"draft":58,"extension":59,"meta":60,"module":15,"navigation":25,"path":61,"practice":62,"rawbody":63,"readingTime":64,"seo":69,"sources":70,"status":71,"stem":72,"summary":15,"topics":73,"__hash__":74},"course\u002F06.category-theory\u002Findex.md","Category Theory","Mathematics organized by arrows — universal properties, functors, the\nYoneda lemma, limits, adjunctions, and the monads and closed categories\nthat connect it all to programming.\n",{"type":8,"value":9,"toc":14},"minimark",[10],[11,12,13],"p",{},"Category theory is the mathematics of structure-preserving maps: it describes\nobjects entirely by their arrows and by how those arrows compose. The sequence\nstarts with the definition itself — categories, the special morphisms\n(isomorphisms, monos, epis), functors, and natural transformations — then turns\nto universal properties, the technique of characterizing a construction by a\nsingle best map rather than by its internal makeup. From there it builds the\ncore theory: representable functors and the Yoneda lemma, limits and colimits as\nthe shared pattern behind products, equalizers, pullbacks, and their duals, and\nadjunctions as the deepest way two theories can correspond. The final arc carries\nthe subject into computing — monads and their algebras, the Kleisli category\nbehind effectful programs, and cartesian closed categories with their exact\nmatch to the typed lambda calculus. Leinster is the spine, Simmons supplies\nworked examples, and Barr & Wells draws out the connections to programming. Each\nlayer reuses the arrows of the one before it.",{"title":15,"searchDepth":16,"depth":16,"links":17},"",2,[],[19,21,26,30,32,34,38,40,44,46,50,52,54],{"p":20},"Category theory studies mathematical structure by how objects\n\u003Cstrong>map\u003C\u002Fstrong> to one another, not by what they contain. An object\nis opaque; everything you can say about it is said with arrows.\n",{"fig":22,"n":23,"caption":24,"large":25},"ct-square","001","A commuting square: two paths from A to D agree, so the diagram commutes.\n",true,{"fig":27,"n":28,"caption":29},"ct-composition","002","Arrows compose head to tail, and the composite is associative.\n",{"p":31},"A \u003Cstrong>category\u003C\u002Fstrong> is objects, arrows between them, and a way to\ncompose arrows that is associative and has identities. That is the whole\ndefinition — and almost every structure in mathematics is an instance of\nit.\n",{"p":33},"Instead of peering inside an object, you characterize it by a\n\u003Cem>universal property\u003C\u002Fem>: the maps into or out of it that make it the\nbest solution to a problem, unique up to a single isomorphism.\n",{"fig":35,"n":36,"caption":37},"ct-universal","003","A product: every test object factors through it by one mediating arrow.\n",{"p":39},"\u003Cstrong>Functors\u003C\u002Fstrong> carry one category into another, preserving\ncomposition and identities, so a whole theory can be transported and\ncompared against another intact.\n",{"fig":41,"n":42,"caption":43},"ct-functor","004","A functor F sends objects to objects and arrows to arrows, same direction.\n",{"p":45},"\u003Cstrong>Natural transformations\u003C\u002Fstrong> compare two functors arrow by\narrow. When their components make every square commute, the comparison is\ncanonical — independent of any arbitrary choice.\n",{"fig":47,"n":48,"caption":49},"ct-natural","005","A natural transformation links two functors so each naturality square commutes.\n",{"p":51},"The core results build on these three levels. The \u003Cem>Yoneda lemma\u003C\u002Fem>\nsays an object is fully known by the arrows into it; \u003Cstrong>limits\u003C\u002Fstrong>\nand colimits unify products, pullbacks, and quotients; and\n\u003Cstrong>adjunctions\u003C\u002Fstrong> pair functors that are optimal inverses of\none another.\n",{"p":53},"From adjunctions come \u003Cstrong>monads\u003C\u002Fstrong> and cartesian closed\ncategories — the structures that carry the subject into computer science,\nmodelling effects, the typed lambda calculus, and the semantics of\nprograms.\n",{"p":55},"The reward is leverage: prove something once about arrows and it holds in\nevery category at once, from sets and groups to spaces, logics, and types.\n","math","Category theory describes mathematical structure by how objects map to one\nanother, not by what they contain. Universal properties characterize\nconstructions up to unique isomorphism; functors and natural\ntransformations compare whole theories; Yoneda, limits, and adjunctions\nform the core results; and monads and cartesian closed categories carry the\nsubject into computer science. Leinster is the spine, with Simmons for\nworked examples and Barr & Wells for the computing connections.\n",false,"md",{},"\u002Fcategory-theory",[],"---\ntitle: Category Theory\nstatus: available\ncategory: math\nblurb: |\n  Mathematics organized by arrows — universal properties, functors, the\n  Yoneda lemma, limits, adjunctions, and the monads and closed categories\n  that connect it all to programming.\ndescription: |\n  Category theory describes mathematical structure by how objects map to one\n  another, not by what they contain. Universal properties characterize\n  constructions up to unique isomorphism; functors and natural\n  transformations compare whole theories; Yoneda, limits, and adjunctions\n  form the core results; and monads and cartesian closed categories carry the\n  subject into computer science. Leinster is the spine, with Simmons for\n  worked examples and Barr & Wells for the computing connections.\nbrief:\n  - p: |\n      Category theory studies mathematical structure by how objects\n      \u003Cstrong>map\u003C\u002Fstrong> to one another, not by what they contain. An object\n      is opaque; everything you can say about it is said with arrows.\n  - fig: ct-square\n    n: \"001\"\n    caption: |\n      A commuting square: two paths from A to D agree, so the diagram commutes.\n    large: true\n  - fig: ct-composition\n    n: \"002\"\n    caption: |\n      Arrows compose head to tail, and the composite is associative.\n  - p: |\n      A \u003Cstrong>category\u003C\u002Fstrong> is objects, arrows between them, and a way to\n      compose arrows that is associative and has identities. That is the whole\n      definition — and almost every structure in mathematics is an instance of\n      it.\n  - p: |\n      Instead of peering inside an object, you characterize it by a\n      \u003Cem>universal property\u003C\u002Fem>: the maps into or out of it that make it the\n      best solution to a problem, unique up to a single isomorphism.\n  - fig: ct-universal\n    n: \"003\"\n    caption: |\n      A product: every test object factors through it by one mediating arrow.\n  - p: |\n      \u003Cstrong>Functors\u003C\u002Fstrong> carry one category into another, preserving\n      composition and identities, so a whole theory can be transported and\n      compared against another intact.\n  - fig: ct-functor\n    n: \"004\"\n    caption: |\n      A functor F sends objects to objects and arrows to arrows, same direction.\n  - p: |\n      \u003Cstrong>Natural transformations\u003C\u002Fstrong> compare two functors arrow by\n      arrow. When their components make every square commute, the comparison is\n      canonical — independent of any arbitrary choice.\n  - fig: ct-natural\n    n: \"005\"\n    caption: |\n      A natural transformation links two functors so each naturality square commutes.\n  - p: |\n      The core results build on these three levels. The \u003Cem>Yoneda lemma\u003C\u002Fem>\n      says an object is fully known by the arrows into it; \u003Cstrong>limits\u003C\u002Fstrong>\n      and colimits unify products, pullbacks, and quotients; and\n      \u003Cstrong>adjunctions\u003C\u002Fstrong> pair functors that are optimal inverses of\n      one another.\n  - p: |\n      From adjunctions come \u003Cstrong>monads\u003C\u002Fstrong> and cartesian closed\n      categories — the structures that carry the subject into computer science,\n      modelling effects, the typed lambda calculus, and the semantics of\n      programs.\n  - p: |\n      The reward is leverage: prove something once about arrows and it holds in\n      every category at once, from sets and groups to spaces, logics, and types.\n---\n\nCategory theory is the mathematics of structure-preserving maps: it describes\nobjects entirely by their arrows and by how those arrows compose. The sequence\nstarts with the definition itself — categories, the special morphisms\n(isomorphisms, monos, epis), functors, and natural transformations — then turns\nto universal properties, the technique of characterizing a construction by a\nsingle best map rather than by its internal makeup. From there it builds the\ncore theory: representable functors and the Yoneda lemma, limits and colimits as\nthe shared pattern behind products, equalizers, pullbacks, and their duals, and\nadjunctions as the deepest way two theories can correspond. The final arc carries\nthe subject into computing — monads and their algebras, the Kleisli category\nbehind effectful programs, and cartesian closed categories with their exact\nmatch to the typed lambda calculus. Leinster is the spine, Simmons supplies\nworked examples, and Barr & Wells draws out the connections to programming. Each\nlayer reuses the arrows of the one before it.\n\n",{"text":65,"minutes":66,"time":67,"words":68},"1 min 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Objects, and Arrows","\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category",[989],"A category is objects, arrows between them, a rule for composing arrows, and an identity arrow on every object, subject to associativity and the unit laws. The axioms mention no elements: arrows need not be functions, and an object is known only through the arrows into and out of it. Isomorphism, commutative diagrams, duality, and the terminal object are the first consequences.\n",{"title":999,"path":1000,"lessonNumber":16,"topics":1001,"summary":1002},"A Zoo of Categories","\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories",[989],"The axioms admit two very different kinds of model: large categories of structured sets and their structure-preserving maps (Set, Mon, Grp, Top, Vect), and small categories that are themselves single algebraic objects — a monoid as a one-object category, a poset as a thin category. The awkward cases Rel and Pfn have sets as objects but relations and partial functions as arrows, and a typed programming language presents its types and programs as a category.\n",{"title":1004,"path":1005,"lessonNumber":1006,"topics":1007,"summary":1008},"Isomorphisms, Monos, and Epis","\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms",3,[989],"Injectivity and surjectivity mention elements, so a general category re-expresses them by cancellation: monomorphisms cancel on the left, epimorphisms on the right. Sections and retractions are the split versions with an explicit one-sided inverse. Mono plus epi does not force an isomorphism, and subobjects are equivalence classes of monos into a fixed object.\n",{"title":1010,"path":1011,"lessonNumber":1012,"topics":1013,"summary":1014},"Functors: Maps Between Categories","\u002Fcategory-theory\u002Ffoundations\u002Ffunctors",4,[989],"A functor sends objects to objects and arrows to arrows while preserving composition and identities. Covariant and contravariant functors, the standard stock (forgetful, free, hom, and powerset), and the classification by faithfulness, fullness, and essential surjectivity all follow. Functors compose, so categories and functors form a category themselves.\n",{"title":1016,"path":1017,"lessonNumber":1018,"topics":1019,"summary":1020},"Natural Transformations and Functor Categories","\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations",5,[989],"A natural transformation is a map between two parallel functors: one component arrow per object, subject to a commuting square for every arrow of the source. Naturality is verified for the determinant, the double dual, and list operations; functors and natural transformations form the functor category [C, D]; and vertical and horizontal composition satisfy the Godement interchange law.\n",{"title":1022,"path":1023,"lessonNumber":1024,"topics":1025,"summary":1026},"Size: Small, Large, Locally Small","\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory",6,[989],"The objects of Set do not form a set, and pretending otherwise reproduces the classical paradoxes. Classes make the small\u002Flarge distinction precise, with locally small and essentially small as the intermediate notions. Cantor's theorem shows Set and its algebraic relatives are large, and the function-based axiomatization of sets is the one category theory prefers to ZFC.\n",{"module":1028,"moduleNumber":16,"slug":1029,"lessons":1030},"Universal Properties and Basic Constructions","universal-properties",[1031,1036,1041],{"title":1032,"path":1033,"lessonNumber":990,"topics":1034,"summary":1035},"Universal Properties, Initial and Terminal Objects","\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties",[1028],"A universal property characterizes an object by a for-all\u002Fexists-unique condition on the arrows into or out of it, and any two objects satisfying the same property are isomorphic by a unique isomorphism. Initial and terminal objects are the simplest cases; the free vector space, the discrete topology, and the ring of integers show the pattern at work.\n",{"title":1037,"path":1038,"lessonNumber":16,"topics":1039,"summary":1040},"Products and Coproducts","\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts",[1028],"The product of two objects is a wedge of projections through which every other wedge factors uniquely; the coproduct is the dual, built from injections. In Set these are the cartesian product and the disjoint union, in a poset the meet and join, and in abelian groups the two coincide. The mediating-arrow discipline established here is the template for all limits.\n",{"title":1042,"path":1043,"lessonNumber":1006,"topics":1044,"summary":1045},"Opposite, Product, Slice, and Comma Categories","\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories",[1028],"Categories are themselves mathematical structures, and the standard algebraic constructions apply: opposites, products, subcategories, slices, and the comma category that subsumes them. The opposite category yields the duality principle, halving the subject's proofs; slice and comma categories repackage every universal property as an initial or terminal object.\n",{"module":1047,"moduleNumber":1006,"slug":1048,"lessons":1049},"Representables and the Yoneda Lemma","representables-yoneda",[1050,1055,1060],{"title":1051,"path":1052,"lessonNumber":990,"topics":1053,"summary":1054},"Hom-Functors and Representables","\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors",[1047],"Fixing an object A of a locally small category produces a set-valued functor, the hom-functor A(A,-), that records every map out of A. A functor is representable when it is naturally isomorphic to such a hom-functor. We define the covariant and contravariant hom-functors, collect the standard representables (identity, forgetful, powerset), and read maps as generalized elements of varying shape.\n",{"title":1056,"path":1057,"lessonNumber":16,"topics":1058,"summary":1059},"The Yoneda Lemma","\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma",[1047],"The Yoneda lemma computes the natural transformations out of a representable presheaf: they form a set in natural bijection with X(A). The proof fixes a single degree of freedom, the image of the identity arrow, and shows naturality forces everything else. We prove the bijection, verify naturality in both variables, and read off that a natural transformation out of a representable is just one element.\n",{"title":1061,"path":1062,"lessonNumber":1006,"topics":1063,"summary":1064},"The Yoneda Embedding and Its Uses","\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences",[1047],"Three corollaries turn the Yoneda lemma into working machinery. A representation of a presheaf is the same thing as a universal element; the Yoneda embedding of a category into its presheaf category is full and faithful; and two objects are isomorphic exactly when their representables are. Together they justify constructing arrows by constructing natural transformations between hom-functors, and they contain Cayley's theorem as the one-object case.\n",{"module":1066,"moduleNumber":1012,"slug":1067,"lessons":1068},"Limits and Colimits","limits-colimits",[1069,1074,1079,1084,1089],{"title":1070,"path":1071,"lessonNumber":990,"topics":1072,"summary":1073},"Cones and Limits","\u002Fcategory-theory\u002Flimits-colimits\u002Flimits",[1066],"A diagram is a functor from a small shape category; a cone over it is an object with compatible legs to every node; and a limit is the terminal cone, the one every other cone factors through uniquely. Products and terminal objects reappear as limits over particular shapes, and the whole construction is unique up to a single isomorphism.\n",{"title":1075,"path":1076,"lessonNumber":16,"topics":1077,"summary":1078},"Equalizers and Pullbacks","\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks",[1066],"The equalizer of a parallel pair is the universal arrow that makes the two composites agree; the pullback of a cospan is the universal commutative square. In Set they are solution sets and fibered products, every equalizer is monic, monics are stable under pullback, and products plus equalizers together generate all limits.\n",{"title":1080,"path":1081,"lessonNumber":1006,"topics":1082,"summary":1083},"Colimits: Coproducts, Coequalizers, Pushouts","\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits",[1066],"Colimits are limits in the opposite category: cocones replace cones, and the universal cocone is initial rather than terminal. Coproducts glue objects side by side, coequalizers impose relations and produce quotients, pushouts glue along a shared part, and in Set every colimit is a quotient of a disjoint union. Directed colimits admit a clean elementwise description.\n",{"title":1085,"path":1086,"lessonNumber":1012,"topics":1087,"summary":1088},"Computing Limits in Concrete Categories","\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits",[1066],"In Set the limit of any diagram is the set of threads: choice functions through the nodes that commute with every edge. In Pos, Mon, and Top the recipe is the same limit downstairs plus the unique structure that makes the projections structure-preserving — pointwise order, componentwise operations, the topology generated by the projections. The pattern is what \"the forgetful functor creates limits\" means concretely.\n",{"title":1090,"path":1091,"lessonNumber":1018,"topics":1092,"summary":1093},"Preservation, Reflection, and Creation of Limits","\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors",[1066],"A functor preserves limits if it sends limit cones to limit cones, reflects them if it recognizes them, and creates them if limits downstairs lift uniquely upstairs. Representable functors preserve all limits, forgetful functors from algebra create them, and limits in functor categories are computed pointwise, one evaluation at a time.\n",{"module":1095,"moduleNumber":1018,"slug":1096,"lessons":1097},"Adjunctions","adjunctions",[1098,1103,1108,1113],{"title":1099,"path":1100,"lessonNumber":990,"topics":1101,"summary":1102},"Adjoint Functors via Hom-Set Bijections","\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions",[1095],"An adjunction is a natural bijection between two hom-sets: maps out of $F(A)$ in one category correspond to maps into $G(B)$ in the other. We give the definition, spell out the naturality axioms that make the correspondence compatible with composition, and work the flagship examples — free vector spaces, free groups, discrete and indiscrete topologies, and currying.\n",{"title":1104,"path":1105,"lessonNumber":16,"topics":1106,"summary":1107},"Units, Counits, and the Triangle Identities","\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits",[1095],"The whole hom-set bijection of an adjunction is generated by two natural transformations: the unit, obtained by transposing identity maps on one side, and the counit, by transposing them on the other. Two triangle identities are all they must satisfy, and any pair satisfying them determines a unique adjunction. The same correspondence specializes to order-preserving maps between posets and to free constructions.\n",{"title":1109,"path":1110,"lessonNumber":1006,"topics":1111,"summary":1112},"Adjunctions from Universal Arrows","\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows",[1095],"The unit component at a single object is an initial object of a comma category, and this universal property alone rebuilds the whole adjunction. A functor has a left adjoint exactly when every object admits such a universal arrow, and the left adjoint is assembled from them one object at a time. We prove the equivalence of all three formulations of adjointness.\n",{"title":1114,"path":1115,"lessonNumber":1012,"topics":1116,"summary":1117},"Free Constructions and Free–Forgetful Adjunctions","\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions",[1095],"Free monoids, free groups, and free vector spaces are left adjoints to forgetful functors, and the universal mapping property is all one needs to prove it. Some forgetful functors also have right adjoints (co-free constructions like the indiscrete topology), producing three-functor chains. Contravariant adjunctions, symmetric in their two functors, close the lesson with the pattern behind duality and representation theorems.\n",{"module":1119,"moduleNumber":1024,"slug":1120,"lessons":1121},"Adjoints, Representables, and Limits Together","adjoints-limits",[1122,1130,1135,1140],{"title":1123,"path":1124,"lessonNumber":990,"topics":1125,"summary":1129},"Limits as Adjoints and as Representables","\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints",[1126,1127,1128],"Adjoints","Representables","and Limits Together","A cone on a diagram is a natural transformation from a constant diagram, so a limit is a representation of the cone functor and, equivalently, a value of the right adjoint to the diagonal functor. We prove both rephrasings, derive uniqueness and functoriality of limits from them, and record the dual statement that a colimit is the left adjoint to the diagonal.\n",{"title":1131,"path":1132,"lessonNumber":16,"topics":1133,"summary":1134},"Limits and Colimits of Presheaves","\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits",[1126,1127,1128],"Representables preserve limits, and limits in a functor category are computed one object at a time, so a presheaf category is complete and cocomplete with all its structure inherited pointwise from Set. The Yoneda embedding then preserves limits but not colimits, and the density theorem repairs the colimit side: every presheaf is a canonical colimit of representables.\n",{"title":1136,"path":1137,"lessonNumber":1006,"topics":1138,"summary":1139},"Right Adjoints Preserve Limits (RAPL)","\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits",[1126,1127,1128],"A functor with a left adjoint preserves every limit that exists, and dually a functor with a right adjoint preserves colimits. The proof is a four-line chain of natural isomorphisms through the adjunction and the continuity of representables. The theorem yields product-and-exponential arithmetic in Set, another proof that limits commute with limits, and a standard test for proving that a functor has no adjoint.\n",{"title":1141,"path":1142,"lessonNumber":1012,"topics":1143,"summary":1144},"The Adjoint Functor Theorem","\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem",[1126,1127,1128],"RAPL makes limit preservation necessary for having a left adjoint; the adjoint functor theorems identify when it is sufficient. For ordered sets no extra hypothesis is needed. In general the candidate adjoint is a limit over a comma category that may be large, and the general adjoint functor theorem tames it with a weakly initial set. We prove GAFT in full and apply it to free groups and, through the special adjoint functor theorem, the Stone–Čech compactification.\n",{"module":1146,"moduleNumber":1147,"slug":1148,"lessons":1149},"Monads and Algebras",7,"monads-algebras",[1150,1155,1160,1165],{"title":1151,"path":1152,"lessonNumber":990,"topics":1153,"summary":1154},"Monads from Adjunctions","\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads",[1146],"A monad on a category is an endofunctor equipped with a unit and a multiplication satisfying associativity and unit laws — the data of a monoid, written internally to the category of endofunctors. Every adjunction induces one, and the list, exception, and state constructions that model computational effects are all monads on Set.\n",{"title":1156,"path":1157,"lessonNumber":16,"topics":1158,"summary":1159},"Algebras for a Monad","\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore",[1146],"An algebra for a monad is an object with a structure map that interacts correctly with the unit and multiplication. The algebras form the Eilenberg–Moore category, whose free–forgetful adjunction induces the monad back; a comparison functor relates any other inducing adjunction to it, and for the list monad the algebras are exactly monoids.\n",{"title":1161,"path":1162,"lessonNumber":1006,"topics":1163,"summary":1164},"The Kleisli Category and Monads in Programming","\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming",[1146],"The Kleisli category of a monad has the same objects as the base but takes arrows A to TB, composed by mapping and flattening. These arrows are effectful programs, Kleisli composition is the bind of functional programming, and the Kleisli adjunction is the initial resolution of the monad, with Eilenberg–Moore at the terminal end.\n",{"title":1166,"path":1167,"lessonNumber":1012,"topics":1168,"summary":1169},"Algebras for an Endofunctor and Recursion","\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors",[1146],"Dropping the monad laws leaves algebras for a bare endofunctor, whose initial objects are the least fixed points of the functor by Lambek's lemma. The natural numbers, lists, and trees are initial algebras; the unique map out of an initial algebra is the fold of functional programming; and the Smyth–Plotkin fixed-point technique builds Scott domains the same way.\n",{"module":1171,"moduleNumber":1172,"slug":1173,"lessons":1174},"Cartesian Closed Categories and Typed Lambda Calculus",8,"cartesian-closed-lambda",[1175,1180,1185],{"title":1176,"path":1177,"lessonNumber":990,"topics":1178,"summary":1179},"Cartesian Closed Categories","\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories",[1171],"A cartesian closed category has a terminal object, binary products, and for every pair of objects an exponential object that internalizes the hom-set as an object of the category. The defining data is an evaluation arrow and a currying operation, packaged by the adjunction between product-with-A and exponential-by-A. Set, Boolean and Heyting algebras, functor categories, and Cat are all cartesian closed.\n",{"title":1181,"path":1182,"lessonNumber":16,"topics":1183,"summary":1184},"Typed Lambda Calculus and CCCs","\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence",[1171],"The typed lambda calculus and the cartesian closed category are two presentations of the same theory. Types become objects, terms with one free variable become arrows, product types become products, and function types become exponentials, with abstraction matching currying and application matching evaluation. Building the category of a lambda theory and the internal language of a category are mutually inverse up to equivalence.\n",{"title":1186,"path":1187,"lessonNumber":1006,"topics":1188,"summary":1189},"Fixed Points in Cartesian Closed Categories","\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion",[1171],"The untyped lambda calculus has a fixed-point combinator; the typed calculus cannot, and Lawvere's fixed-point theorem explains why: any point-surjection onto an exponential forces every endomap to have a fixed point, which is the abstract form of Cantor's diagonal argument. Recursion is recovered instead by restricting to omega-complete partially ordered objects, where every continuous endomap has a least fixed point built by iterating from bottom. This gives While loops a semantics.\n",1786059479193]