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 categories are closed categories. In particular, any topos is closed. The canonical example is the category of sets. Compact closed categories...

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...

Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the...

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...

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...

value Inverse limit – Construction in category theory Cartesian closed category – Type of category in category theory Categorical pullback – Most general...

being a Cartesian closed category 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 cartesian closed categories. Categories (such as subcategories of Top) without adjoined products...

Dual (category theory) Groupoid Image (category theory) Coimage Commutative diagram Cartesian morphism Slice category Isomorphism of categories Natural...

a closed category in category theory Cartesian coordinate system, modern rectangular coordinate system Cartesian diagram, a construction in category theory...

theory, where a cartesian closed category is taken as a non-syntactic description of a lambda calculus. At the very least, category theoretic language...

bifunctor. Cartesian closed category – Type of category in category theory Limits and colimits in an ∞-category Mac Lane, Saunders (1998). Categories for the...

is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure. A symmetric monoidal category ( C , ⊗ , I )...

category, a distributive lattice as a small posetal distributive category, a Heyting algebra as a small posetal finitely cocomplete cartesian closed category...

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...

Examples include cartesian closed categories such as Set, the category of sets, and compact closed categories such as FdVect, the category of finite-dimensional...

typed lambda calculus and cartesian closed categories. Under this correspondence, objects of a cartesian-closed category can be interpreted as propositions...

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...

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...

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...

shortcomings of the category of topological spaces. In particular, under some of the definitions, they form a cartesian closed category while still containing...

objects in the style of the lambda calculus Cartesian closed category – A setting for lambda calculus in category theory Categorical abstract machine – A...

functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply)...

simply typed lambda calculus, which is the internal language of Cartesian closed categories. Typing rules specify the structure of a typing relation that...

. The category Cat {\displaystyle {\textbf {Cat}}} of all small categories with functors as morphisms is therefore a cartesian closed category. Mathematics...

William Lawvere in 1969. Lawvere's theorem states that, for any Cartesian closed category C {\displaystyle \mathbf {C} } and given an object B {\displaystyle...

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...

In category theory, a branch of mathematics, dagger compact categories (or dagger compact closed categories) first appeared in 1989 in the work of Sergio...

objects). Categories that do have both products and internal homs are exactly the closed monoidal categories. The setting of cartesian closed categories is sufficient...