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. These categories are particularly important in mathematical logic and the theory of programming, in that their internal language is the simply typed lambda calculus. They are generalized by closed monoidal categories, whose internal language, linear type systems, are suitable for both quantum and classical computation.[1]
^Baez, John C.; Stay, Mike (2011). "Physics, Topology, Logic and Computation: A Rosetta Stone" (PDF). In Coecke, Bob (ed.). New Structures for Physics. Lecture Notes in Physics. Vol. 813. Springer. pp. 95–174. arXiv:0903.0340. CiteSeerX 10.1.1.296.1044. doi:10.1007/978-3-642-12821-9_2. ISBN 978-3-642-12821-9. S2CID 115169297.
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In category theory, a category is Cartesianclosed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified...
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product; thus any category with a Cartesian product (and a final object) is a Cartesianclosedcategory. In graph theory, the Cartesian product of two graphs...
Examples include cartesianclosedcategories such as Set, the category of sets, and compact closedcategories such as FdVect, the category of finite-dimensional...
theory, where a cartesianclosedcategory is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language...
being a Cartesianclosedcategory while still containing all of the typical spaces of interest. This makes CGHaus a particularly convenient category of topological...
all finite products and exponential objects are called cartesianclosedcategories. Categories (such as subcategories of Top) without adjoined products...
category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesianclosed category...
in category theory, where it is right adjoint to currying in closed monoidal categories. A special case of this are the Cartesianclosedcategories, whose...
to the corresponding free categories: F : Quiv → Cat Cat has all small limits and colimits. Cat is a Cartesianclosedcategory, with exponential D C {\displaystyle...
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 cartesianclosedcategory. Mathematics...
objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesianclosedcategories is sufficient...
equivalence F is an exact functor. C is a cartesianclosedcategory (or a topos) if and only if D is cartesianclosed (or a topos). Dualities "turn all concepts...
shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesianclosedcategory while still containing...
automata homomorphisms defining the arrows between automata is a Cartesianclosedcategory, it has both categorical limits and colimits. An automata homomorphism...
functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesianclosedcategory. Here, eval (or, properly speaking, apply)...
the indiscrete category on that set. Exponential object. In a cartesianclosedcategory the endofunctor C → C given by –×A has a right adjoint –A. This...
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