In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified...
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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...
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a closed category in category theory Cartesian coordinate system, modern rectangular coordinate system Cartesian diagram, a construction in category theory...
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Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the...
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Cartesian closed categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets. Compact closed categories...
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Dual (category theory) Groupoid Image (category theory) Coimage Commutative diagram Cartesian morphism Slice category Isomorphism of categories Natural...
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product; thus any category with a Cartesian product (and a final object) is a Cartesian closed category. In graph theory, the Cartesian product of two graphs...
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Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as FdVect, the category of finite-dimensional...
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and CPO, the category of complete partial orders with Scott-continuous functions. A topos is a certain type of cartesian closed category in which all...
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value Inverse limit – Construction in category theory Cartesian closed category – Type of category in category theory Categorical pullback – Most general...
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theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language...
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being a Cartesian closed category while still containing all of the typical spaces of interest. This makes CGHaus a particularly convenient category of topological...
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Exponential object (redirect from Exponential (category theory))
all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products...
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in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesian closed categories, whose...
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{\displaystyle \operatorname {Hom} } bifunctor. Cartesian closed category – Type of category in category theory Equaliser (mathematics) – Set of arguments...
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category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category...
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Lawvere's fixed-point theorem (category 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...
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normalizing" cartesian closed category. To clarify this distinction, the underlying syntactic structure of cartesian closed categories is rephrased below...
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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...
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non-examples of symmetric monoidal categories: The category of sets. The tensor product is the set theoretic cartesian product, and any singleton can be...
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. The category Cat {\displaystyle {\textbf {Cat}}} of all small categories with functors as morphisms is therefore a cartesian closed category. Mathematics...
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functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply)...
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theory Directed complete partial order Knaster–Tarski theorem Cartesian closed category Yoneda lemma Graph reduction Combinator graph reduction Strict...
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Compactly generated space (redirect from K-closed set)
shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesian closed category while still containing...
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kind of theory of computation for programmers, represented by Cartesian closed category and embedded into the combinatory logic. CAM is a transparent...
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Currying (section Category theory)
objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesian closed categories is sufficient...
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analogy, are categories enriched over (FinSet, ×), the category of finite sets with Cartesian product as the monoidal operation. If C is a closed monoidal...
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Adjoint functors (redirect from Unit (category theory))
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...
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is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. A symmetric monoidal category ( C , ⊗ , I )...
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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...
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