Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. In broad terms, it is a theory about the transition from a type theory, understood to be within an intuitionistic logic or constructive mathematics setting, to a category, by means of a translation that respects both the syntax and the intended computational meaning of type-theoretic constructions. The subject has been recognisable in these terms since about 1970, when the needs of domain theory started to call on category theory. The earlier history is relatively complex, and contains some ironies. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ... Computer science (informally, CS or compsci) is, in its most general sense, the study of computation and information processing, both in hardware and in software. ... At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into collections called types. ... Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ... In the philosophy of mathematics, constructivism asserts that it is necessary to find (or construct) a mathematical object to prove that it exists. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... Syntax, originating from the Greek words συν (syn, meaning co- or together) and τάξις (táxis, meaning sequence, order, arrangement), can be described as the study of the rules, or patterned relations that govern the way the words in a sentence come together. ... 1970 (MCMLXX) was a common year starting on Thursday. ... Domain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. ...

Categorical logic originated within sheaf theory, as a suitable version of Kripke semantics one can say with hindsight, and emerged as a theory with a character of its own only in shedding the necessary connection with sheaves. This can be traced in a number of stages, from 1960 onwards: the formulation of the Grothendieck topos, and then of the elementary topos, giving rise first to topos theory. Topos theory, as would now be understood, is the intuitionistic replacement for set theory. In particular, where L. E. J. Brouwer laboured hard to construct a theory of 'species', which were to be the real numbers in the intuitionistic setting, there is a theory of real numbers in each topos (which contradicts the idea that there is a single, correct concept). Computationally speaking, real numbers are anyway an idealisation of what one carries out in an actual calculation. Set theory may be a universal container for mathematical concepts, but turns out to be a rather large, baggy syntax if one wants to express computation. Computer science, on the other hand, by intention rations and constrains computations. Less expressive theories, from the mathematical logic viewpoint, have lower-level category theory counterparts. For example the concept of an algebraic theory leads to Gabriel-Ulmer duality. In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... Kripke semantics (also known as possible world semantics, relational semantics, or frame semantics) is a formal semantics for modal logic systems, created in late 1950s and early 1960s by Saul Kripke. ... 1960 was a leap year starting on Friday (link will take you to calendar). ... In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. ... For discussion of topoi in literary theory, see literary topos. ... For discussion of topoi in literary theory, see literary topos. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

Actual computation needs recursion, something isolated in category theory as a natural number object (this corresponds to a minimal theory of recursion). On the other hand functional programming is the cleanest model of computation (computational paradigm) and was realised to match up with the cartesian closed category concept. In mathematics and computer science, recursion specifies (or constructs) a class of objects (or an object from a certain class) by defining a few very simple base cases (often just one), and then defining rules to break down complex cases into simpler cases. ... In category theory, a natural number object (nno) is an object endowed with a recursive structure similar to natural numbers. ... The Haskell programming language logo Functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions. ... In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ...

The founders of elementary topos theory were Lawvere and Tierney. Lawvere's writings, sometimes couched in a philosophical jargon, isolated some of the basic concepts as adjoint functors (which he explained as 'objective' in a Hegelian sense, not without some justification). A subobject classifier is a strong property to ask of a category, since with cartesian closure and finite limits it gives a topos (axiom bashing shows how strong the assumption is). Lawvere's further work in the 1960s gave a theory of attributes, which in a sense is a subobject theory more in sympathy with type theory. Major influences subsequently have been Martin-Löf type theory from the direction of logic, type polymorphism and the calculus of constructions from functional programming, linear logic from proof theory, game semantics and the projected synthetic domain theory. The abstract categorical idea of fibration has been much applied. Francis William Lawvere is a mathematician who is known for his work in category theory and the philosophy of mathematics. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... Georg Wilhelm Friedrich Hegel (August 27, 1770 - November 14, 1831) was a German philosopher born in Stuttgart, Württemberg, in present-day southwest Germany. ... In category theory, a subobject classifier is a special object Ω of a category; intuitively, the subobjects of an object X correspond to the morphisms from X to Ω. Introductory example As an example, the set Ω = {0,1} is a subobject classifier in the category of sets and functions... An attribute is the following: Generally, an attribute is an abstraction characteristic of an entity In database management, an attribute is a property inherent in an entity or associated with that entity for database purposes. ... In category theory, there is a general definition of subobject extending the idea of subset and subgroup. ... Intuitionistic Type Theory, or Constructive Type Theory, or Martin-Löf Type Theory or just Type Theory (with capital letters) is at the same time a functional programming language, a logic and a set theory based on the principles of mathematical constructivism. ... In computer science, polymorphism is the idea of allowing the same code to be used with different types, resulting in more general and abstract implementations. ... The calculus of constructions (CoC) is a higher-order typed lambda calculus where types are first-class values. ... In mathematical logic, linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. ... Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ... Game semantics is an approach to the semantics of logic that bases the concepts of truth or validity on game-theoretic concepts, such as the existence of a winning strategy for a player. ... This article may be too technical for most readers to understand. ...

To go back historically, the major irony here is that in large-scale terms, intuitionistic logic had reappeared in mathematics, in a central place in the Bourbaki-Grothendieck program, a generation after the messy Hilbert-Brouwer controversy had ended, with Hilbert the apparent winner. Bourbaki, or more accurately Jean Dieudonné, having laid claim to the legacy of Hilbert and the Göttingen school including Emmy Noether, had revived intuitionistic logic's credibility, as the logic of an arbitrary topos, where classical logic was that of 'the' topos of sets. This was one consequence, certainly unanticipated, of Grothendieck's relative point of view; and not lost on Pierre Cartier, one of the broadest of the core group of French mathematicians around Bourbaki and IHES. Cartier was to give a Séminaire Bourbaki exposition of intuitionistic logic. Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ... Alexander Grothendieck (born March 28, 1928, Berlin) is one of the greatest mathematicians of the 20th century, with major contributions to algebraic geometry, homological algebra, and functional analysis. ... David Hilbert David Hilbert (January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Brouwer is the last name of different people. ... Jean-Alexandre-Eugène Dieudonné (July 1, 1906 - November 29, 1992) was a French mathematician, known for research in abstract algebra and functional analysis, for close involvement with the Nicolas Bourbaki pseudonymous group and the Éléments de géométrie algébrique project of Alexander Grothendieck, and as a... Emmy Noether (born Nöther) (March 23, 1882 – April 14, 1935) was one of the most talented mathematicians of the early 20th century, with penetrating insights that she used to develop elegant abstractions which she formalized beautifully. ... Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ... Pierre Cartier (born in Sedan, France in 1932) is a mathematician - more specifically, a category theorist. ... IHÉS main building The Institut des Hautes Études Scientifiques (I.H.É.S.) is a French institute supporting advanced research in mathematics and theoretical physics. ... The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. ...

In an even broader perspective, one might take category theory to be to the mathematics of the second half of the twentieth century, what measure theory was to the first half. It was Kolmogorov who applied measure theory to probability theory, the first convincing (if not the only) axiomatic approach. Kolmogorov was also a pioneer writer in the early 1920s on the formulation of intuitionistic logic, in a style entirely supported by the later categorical logic approach (again, one of the formulations, not the only one; the realizability concept of Stephen Kleene is also a serious contender here). Another route to categorical logic would therefore have been through Kolmogorov, and this is one way to explain the protean Curry-Howard isomorphism. (19th century - 20th century - 21st century - more centuries) Decades: 1900s 1910s 1920s 1930s 1940s 1950s 1960s 1970s 1980s 1990s The 20th century lasted from 1901 to 2000 in the Gregorian calendar (often from (1900 to 1999 in common usage). ... In mathematics, a measure is a function that assigns a number, e. ... Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (Андре́й Никола́евич Колмого́ров) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Russian mathematician who made major advances in the fields of probability theory and topology. ... Probability theory is the mathematical study of probability. ... Sometimes referred to as the Jazz Age or primarily in North America and in Australia as the Roaring Twenties . In Europe it is sometimes refered to as the Golden Twenties. ... Stephen Cole Kleene (January 5, 1909 - January 25, 1994) was an American mathematician whose work at the University of Wisconsin-Madison helped lay the foundations for theoretical computer science. ... The Curry-Howard correspondence is the close relationship between computer programs and mathematical proofs; the correspondence is also known as the Curry-Howard isomorphism, or the formulae-as-types correspondence. ...