AI & ChatGPT searches , social queries for CARTESIAN CLOSED-CATEGORY
Search references for CARTESIAN CLOSED-CATEGORY. Phrases containing CARTESIAN CLOSED-CATEGORY
See searches and references containing CARTESIAN CLOSED-CATEGORY!
Search references for CARTESIAN CLOSED-CATEGORY. Phrases containing CARTESIAN CLOSED-CATEGORY
See searches and references containing CARTESIAN CLOSED-CATEGORY!CARTESIAN CLOSED-CATEGORY
Type of category in category theory
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified
Cartesian_closed_category
Category whose hom objects correspond (di-)naturally to objects in itself
conditions. Cartesian closed categories are closed categories. In particular, any elementary topos is closed. The canonical example is the category of sets
Closed_category
Type of category in mathematics
in category theory, a closed monoidal category (or a monoidal closed category) is a category that is both a monoidal category and a closed category in
Closed_monoidal_category
Type of category in category theory
Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the
Cartesian_monoidal_category
Collection of objects and morphisms
, the category of complete partial orders with Scott-continuous functions. An elementary topos is a certain type of cartesian closed category in which
Category_(mathematics)
modelled in a locally Cartesian closed category (i.e., a category C {\displaystyle {\mathcal {C}}} such that all the slice categories C / A {\displaystyle
Semantics_of_type_theory
Topics referred to by the same term
a closed category in category theory Cartesian coordinate system, modern rectangular coordinate system Cartesian diagram, a construction in category theory
Cartesian
General theory of mathematical structures
theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language
Category_theory
Generalized object in category theory
value Inverse limit – Construction in category theory Cartesian closed category – Type of category in category theory Categorical pullback – Most general
Product_(category_theory)
Category whose objects are small categories and whose morphisms are functors
to the corresponding free categories: F : Quiv → Cat Cat has all small limits and colimits. Cat is a Cartesian closed category, with exponential D C {\displaystyle
Category_of_small_categories
Categorical generalization of a function space in set theory
all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products
Exponential_object
Category where each homset contains at most one morphism
cocomplete *-autonomous category. Conversely, categories, distributive categories, finitely cocomplete cartesian closed categories, and finitely cocomplete
Thin_category
Special kind of category with "dual objects"
is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. A symmetric monoidal category ( C , ⊗ , I )
Compact_closed_category
Mathematical set formed from two given sets
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is an
Cartesian_product
Relationship between programs and proofs
typed lambda calculus and cartesian closed categories. Under this correspondence, objects of a cartesian-closed category can be interpreted as propositions
Curry–Howard_correspondence
Special dagger category that is compact
In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio
Dagger_compact_category
Category admitting tensor products
Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as FdVect, the category of finite-dimensional
Monoidal_category
Theorem in category theory
William Lawvere in 1969. Lawvere's theorem states that, for any Cartesian closed category C {\displaystyle \mathbf {C} } and given an object B {\displaystyle
Lawvere's_fixed-point_theorem
Evaluation of a function on its argument
language of Cartesian closed categories is simply typed lambda calculus. The most general possible setting for Apply are the closed monoidal categories, of which
Function_application
Mathematical concept
bifunctor. Cartesian closed category – Type of category in category theory Limits and colimits in an ∞-category Mac Lane, Saunders (1998). Categories for the
Limit_(category_theory)
Property of topological spaces
shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesian closed category while still containing
Compactly_generated_space
kind of theory of computation for programmers, represented by Cartesian closed category and embedded into the combinatory logic. CAM is a transparent
Categorical_abstract_machine
Transforming a function in such a way that it only takes a single argument
objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesian closed categories is sufficient
Currying
Category whose objects are topological spaces and whose morphisms are continuous maps
{Top} } , and in particular the category is not preadditive. T o p {\displaystyle \mathbf {Top} } is not cartesian closed (and therefore also not a topos)
Category of topological spaces
Category_of_topological_spaces
Abstract mathematics relationship
equivalence F is an exact functor. C is a cartesian closed category (or a topos) if and only if D is cartesian closed (or a topos). Dualities "turn all concepts
Equivalence_of_categories
Study of abstract machines and automata
automata homomorphisms defining the arrows between automata is a Cartesian closed category, it has both categorical limits and colimits. An automata homomorphism
Automata_theory
Object in category theory
an isomorphism. In a Cartesian closed category, every initial object is strict. Also, if C is a distributive or extensive category, then the initial object
Strict_initial_object
Mathematical-logic system
objects in the style of the lambda calculus Cartesian closed category – A setting for lambda calculus in category theory Categorical abstract machine – A
Lambda_calculus
Category where every morphism is invertible; generalization of a group
groupoids, and is denoted by Grpd. The category Grpd is, like the category of small categories, Cartesian closed: for any groupoids H , K {\displaystyle
Groupoid
Mathematical structures in category theory
. The category Cat {\displaystyle {\textbf {Cat}}} of all small categories with functors as morphisms is therefore a cartesian closed category. Mathematics
Functor_category
adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts
Distributive_category
Cartesian plane Cartesian tensor Cartesian monoid Cartesian monoidal category Cartesian closed category Cartesian oval Cartesian product Cartesian product of
List of things named after René Descartes
List_of_things_named_after_René_Descartes
Topological space characterized by sequences
topological spaces closed under sums and quotients and containing the metrizable spaces). The subcategory Seq is a Cartesian closed category with respect to
Sequential_space
Topics referred to by the same term
UK Royal Navy Color Cell Compression, an algorithm Cartesian closed category, a concept in category theory CCC, Roman numeral for 300 Countable chain condition
CCC
Relationship between two functors abstracting many common constructions
the indiscrete category on that set. Exponential object. In a cartesian closed category the endofunctor C → C given by –×A has a right adjoint –A. This
Adjoint_functors
Operation in graph theory
In graph theory, the Cartesian product G □ H of graphs G and H is a graph such that: the vertex set of G □ H is the Cartesian product V(G) × V(H); and
Cartesian_product_of_graphs
Mathematical form
(of a given type) that have Cartesian products is called a Cartesian category. Many of these are Cartesian closed categories. Sets are an example of such
Product_(mathematics)
How a type system assigns a type to a syntactic construction
simply typed lambda calculus, which is the internal language of Cartesian closed categories. Typing rules specify the structure of a typing relation that
Typing_rule
Function in a programming language, which evaluates a string
functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply)
Eval
Alternative foundation of mathematics
operators. Using the language of category theory, R. A. G. Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of
Intuitionistic_type_theory
Foundational theorem of quantum information processing
theory (in exact analogy to classical logic being founded on Cartesian closed categories). Suppose that there are two copies of an unknown quantum state
No-deleting_theorem
Structure-preserving correspondence between node-link graphs
Since these two operations are always defined, the category of graphs is a cartesian closed category. For the same reason, the lattice of equivalence classes
Graph_homomorphism
Category whose hom sets have algebraic structure
analogy, are categories enriched over (FinSet, ×), the category of finite sets with Cartesian product as the monoidal operation. If C is a closed monoidal
Enriched_category
Formal system in mathematical logic
the category of Cartesian closed categories, and the category of simply typed lambda theories. Part of this correspondence can be extended to closed symmetric
Simply_typed_lambda_calculus
Arithmetic operation
functor Y → T × Y . {\displaystyle Y\to T\times Y.} A category is called a Cartesian closed category, if direct products exist, and the functor Y → X × Y
Exponentiation
Set of functions between two fixed sets
orders that can model lambda calculus, by creating a well-behaved Cartesian closed category. In the representation theory of finite groups, given two finite-dimensional
Function_space
Concept in mathematical category theory
non-examples of symmetric monoidal categories: The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be
Symmetric_monoidal_category
American logician (born 1932)
theory; among its many advantages, the category of equilogical spaces is a cartesian closed category, whereas the category of domains is not. In 1994, he was
Dana_Scott
Generalized topological space
the case V = Set, that is, when the monoidal category V is specialized to the cartesian closed category Set of sets and their functions, but were not
Chu_space
Category whose objects are sets and whose morphisms are binary relations
by the cartesian product of sets. It is also a monoidal category if one defines the monoidal product by the disjoint union of sets. The category Rel was
Category_of_relations
Mathematics construct
A category is said to be locally cartesian closed if every slice of it is cartesian closed (see above for the notion of slice). Locally cartesian closed
Comma_category
System of resource-aware logic
interpretation of intuitionistic logic by replacing cartesian (closed) categories by symmetric monoidal (closed) categories, or the interpretation of classical logic
Linear_logic
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Functor mapping hom objects to an underlying category
language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear
Hom_functor
Formalism in computer science
of certain classes of categories. For example, the simply typed lambda calculus is the language of Cartesian closed categories (CCCs). Various typed lambda
Typed_lambda_calculus
theory Directed complete partial order Knaster–Tarski theorem Cartesian closed category Yoneda lemma Graph reduction Combinator graph reduction Strict
List of functional programming topics
List_of_functional_programming_topics
Design pattern in functional programming to build generic types
product functor and the exponential functor, which exists in any cartesian closed category by definition. A continuation monad with return type R maps type
Monad (functional programming)
Monad_(functional_programming)
Computer programming paradigm
value-free style of FP is closely related to the equational logic of a cartesian-closed category. The canonical function-level programming language is FP. Others
Function-level_programming
Mathematical construction of a set with an equivalence relation
"The Interpretation of Intuitionistic Type Theory in Locally Cartesian Closed Categories—an Intuitionistic Perspective", Electronic Notes in Theoretical
Setoid
Object in category theory
uniqueness is not required, then N is called a weak NNO. NNOs in cartesian closed categories (CCCs) are sometimes defined in the following equivalent way
Natural_numbers_object
Category whose objects are sets and whose morphisms are functions
in particular cartesian closed and exact in the sense of Barr). It in fact the prime example of a Grothendieck topos, being the category of sheaved on
Category_of_sets
Mathematical phrase
the complete partial orders with Scott-continuous maps form a cartesian closed category. Every order-preserving self-map f of a pointed dcpo (P, ⊥) has
Complete_partial_order
French polymath (1596–1650)
ISBN 978-88-452-8071-9 Bucket argument Cartesian circle Cartesian plane Cartesian product Cartesian product of graphs Cartesian theater Cartesian tree Descartes number
René_Descartes
Definition of continuity for functions between posets
of X is open with respect to the Scott topology. For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous
Scott_continuity
Overview of and topical guide to category theory
Dual (category theory) Groupoid Image (category theory) Coimage Commutative diagram Cartesian morphism Slice category Isomorphism of categories Natural
Outline_of_category_theory
Family of type systems based on substructural logic
language of closed symmetric monoidal categories, much in the same way that simply typed lambda calculus is the language of Cartesian closed categories. More
Substructural_type_system
a sub-object classifier, and some limit conditions (to make a cartesian-closed category, at least). For a while this notion of topos was called 'elementary
History_of_topos_theory
Characterizing property of mathematical constructions
Natural transformation Adjoint functor Monad (category theory) Variety of algebras Cartesian closed category Jacobson (2009), Proposition 1.6, p. 44. See
Universal_property
Topics referred to by the same term
of the SETcc instructions in the x86 instruction set SetC; see Cartesian closed category Southeastern Theatre Conference, a non-profit organization; see
SETC
programming languages. [clarification needed] Cartesian closed category Currying Exponential object, category-theoretic equivalent First-class function Function
Function_type
Category whose objects are measurable spaces and whose morphisms are measurable maps
retracts. Meas is not cartesian closed (and therefore also not a topos) since it does not have exponential objects for all spaces. Category of topological spaces –
Category_of_measurable_spaces
polynomial functors, and the theory can be developed in any locally cartesian closed category with W-types in the categorical sense of Moerdijk and Palmgren
Container_(type_theory)
Mathematical theory of data types
a category may be thought of as a type theory shorn of its syntax." A number of significant results follow in this way: cartesian closed categories correspond
Type_theory
Branch of logic using category theory to study mathematical structures
of category. A classic example is the correspondence between theories of βη-equational logic over simply typed lambda calculus and Cartesian closed categories
Categorical_logic
Class of geometric plane curves
In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These
Cartesian_oval
Topics referred to by the same term
a convention center in Lancaster, Pennsylvania, USA Locally cartesian closed category in mathematics Search for "lccc" on Wikipedia. All pages with
LCCC
Axiomatic set theory devised by W.V.O. Quine
image of A {\displaystyle A} . As a consequence, the category of sets in NF is not Cartesian closed. NF may seem to run afoul of problems similar to those
New_Foundations
}} if they map smooth curves to smooth curves. This leads to a Cartesian closed category of smooth mappings between c ∞ {\displaystyle c^{\infty }} -open
Convenient_vector_space
Canadian mathematician (1922–2014)
developing the connections between typed lambda calculus and cartesian closed categories (see Curry–Howard–Lambek correspondence). His last works were
Joachim_Lambek
Mathematical notion
Palmgren, Erik (2008). "Locally cartesian closed categories without chosen constructions". Theory and Applications of Categories. 20: 5–17. Roberts, David M
Anafunctor
Philosophical theory
John Foster, Stewart Goetz, Richard Swinburne and Charles Taliaferro. Cartesian dualism, most famously defended by René Descartes, argues that there are
Mind–body_dualism
Contravariant functor to Set
a small category, the functor category C ^ = S e t C o p {\displaystyle {\widehat {C}}=\mathbf {Set} ^{C^{\mathrm {op} }}} is cartesian closed. The poset
Presheaf_(category_theory)
Technical treatment of Boolean algebras
function Boolean-valued function Boolean-valued model Cartesian closed category Closed monoidal category Complete Boolean algebra Elementary topos Field of
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Programming language feature
typed lambda calculus corresponds to the internal language of Cartesian closed categories. Functional programming languages, such as Erlang, Scheme, ML
First-class_function
Function that takes two inputs
X\times Y\rightarrow Z} where X × Y {\displaystyle X\times Y} is the Cartesian product of X {\displaystyle X} and Y . {\displaystyle Y.} Set-theoretically
Binary_function
Category used in algebraic topology
{Map} (Y,Z))} that is natural in X, Y, and Z. In short, the category is Cartesian closed in an enriched sense. A finite product of CW complexes is a CW
Category of compactly generated weak Hausdorff spaces
Category_of_compactly_generated_weak_Hausdorff_spaces
"Small" subset of a topological space
Consequently, any closed subset with empty interior is meagre. Thus a closed subset of X {\displaystyle X} that is of the second category in X {\displaystyle
Meagre_set
Branch of logic
corresponding category-theoretic structure. Proofs in intuitionistic logic can be interpreted in cartesian closed categories, that is, categories with finite
Bunched_logic
Branch of mathematical logic
to a three way correspondence, the third leg of which are the cartesian closed categories. Other research topics in structural theory include analytic
Proof_theory
Topology on Cartesian products of topological spaces
In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology
Product_topology
Category whose objects are abelian groups and whose morphisms are group homomorphisms
isomorphism of categories. The product in A b {\displaystyle \mathbf {Ab} } is given by the product of groups, formed by taking the Cartesian product of the
Category_of_abelian_groups
Concept in category theory
Lawvere, Francis William (1969). "Diagonal arguments and Cartesian closed categories". Category Theory, Homology Theory and their Applications II (Lecture
Point-surjective_morphism
the category of assemblies. If C is an additive category, then Cex is an abelian category. If C is cartesian closed or locally cartesian closed, then
Exact_completion
Mathematical construction
i ∈ I {\displaystyle b_{\bullet }=\left(b_{i}\right)_{i\in I}} of the Cartesian product ∏ i ∈ I M i , {\textstyle {\textstyle \prod \limits _{i\in I}}M_{i}
Ultraproduct
Axiom of set theory
the Cartesian product of the sets in X {\displaystyle X} , and vice versa. Therefore an equivalent form of the axiom of choice is: The Cartesian product
Axiom_of_choice
Generalization in mathematics
denote the actions. Using that the category of small categories C a t {\displaystyle \mathbf {Cat} } is cartesian closed, the profunctor ϕ {\displaystyle
Profunctor
Mathematical set with an ordering
decreasing sets of pairs, three of the possible partial orders on the Cartesian product of two partially ordered sets are (see Fig. 4): the lexicographical
Partially_ordered_set
All-encompassing set or class
desired Cartesian products. Then the superstructure also contains functions and relations, since these may be represented as subsets of Cartesian products
Universe_(mathematics)
Axioms for the natural numbers
axioms can also be understood using category theory. Let C be a category with terminal object 1C, and define the category of pointed unary systems, US1(C)
Peano_axioms
monoidal category as a one-object 2-category, i.e. an enriched category over ( C a t , × ) {\displaystyle (\mathbf {Cat} ,\times )} with the Cartesian product
Premonoidal_category
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY
CARTESIAN CLOSED-CATEGORY