This is the key property of being a Cartesian closed category, and more generally, a closed monoidal category.

* Cartesian closed category, a closed category in category theory

Hyperbolas arise in practice in many ways: as the curve representing the function in the Cartesian plane, as the appearance of a circle viewed from within it, as the path followed by the shadow of the tip of a sundial, as the shape of an open orbit ( as distinct from a closed and hence elliptical orbit ), such as the orbit of a spacecraft during a gravity assisted swing-by of a planet or more generally any spacecraft exceeding the escape velocity of the nearest planet, as the path of a single-apparition comet ( one travelling too fast to ever return to the solar system ), as the scattering trajectory of a subatomic particle ( acted on by repulsive instead of attractive forces but the principle is the same ), and so on.

* Cartesian closed category

The category C is called Cartesian closed if and only if it satisfies the following three properties:

* A Heyting algebra is a Cartesian closed ( bounded ) lattice.

* Cartesian closed category

Citing Maturana and Varela, he defines an autopoietic system as " a closed topological space that ' continuously generates and specifies its own organization through its operation as a system of production of its own components, and does this in an endless turnover of components '", concluding that " Autopoietic systems are thus distinguished from allopoietic systems, which are Cartesian and which ' have as the product of their functioning something different from themselves '".

In a Cartesian coordinate system with coordinates ( x, y ) the unit square is defined as the square consisting of the points where both x and y lie in a closed unit interval from 0 to 1 on their respective axes.

That is, the unit square is the Cartesian product ×, where denotes the closed unit interval.

Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry – Howard isomorphism and they can be considered as the internal language of classes of categories, e. g. the simply typed lambda calculus is the language of Cartesian closed categories ( CCCs ).

A prototypical example of the use of type rules is in defining type inference in the simply typed lambda calculus, which is the internal language of Cartesian closed categories.

# REDIRECT Cartesian closed category

However, the modified definition is crucial if one wants the convenient category of compactly-generated weak Hausdorff spaces to be Cartesian closed, among other useful properties.

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of

# REDIRECT Cartesian closed category

More specifically, a closed orthant in R < sup > n </ sup > is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive.

This example is a Cartesian closed category.

* More generally, every Cartesian closed category is a symmetric monoidal closed category, when the monoidal structure is the cartesian product structure.

* Cartesian closed categories are closed categories.

The difference is that the Cartesian product can be interpreted simply as a pair of items ( or a list ), whereas the tensor product, used to define a monoidal category, is suitable for describing entangled quantum states.

* Cartesian diagram, a construction in category theory

* Cartesian morphism, formalisation of pull-back operation in category theory

Some philosophers, such as Dennett, reject both epiphenomenalism and the existence of qualia with the same charge that Gilbert Ryle leveled against a Cartesian " ghost in the machine ", that they too are category mistakes.

In the category of sets, for instance, the products are given by Cartesian products and the projections are just the natural projections onto the various factors.

For example, in the category of sets the categorical product is the usual Cartesian product, and the terminal object is a singleton set.

In the category of groups the categorical product is the Cartesian product of groups, and the terminal object is a trivial group with one element.

In fact, usual monoids are exactly the monoid objects in the monoidal category of sets with Cartesian product.

** Set, the category of sets with the Cartesian product, one-element sets serving as the unit.

This can easily be expressed in terms of a commutative diagram, and thus leads to the notion of a medial magma object in a category with a Cartesian product.

# REDIRECT Fibred category # Cartesian morphisms and functors

In category theory, a branch of mathematics, a pullback ( also called a fiber product, fibre product, fibered product or Cartesian square ) is the limit of a diagram consisting of two morphisms f: X → Z and g: Y → Z with a common codomain ; it is the limit of the cospan.

Though lying in the same isomorphism class in the category of rings, the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane.

One can generalize the concept of Cartesian coordinates to allow axes that are not perpendicular to each other, and / or different units along each axis.

In his Phenomenology of Perception ( first published in French in 1945 ), Merleau-Ponty developed the concept of the body-subject as an alternative to the Cartesian " cogito.

The evil demon, sometimes referred to as the evil genius, is a concept in Cartesian philosophy.

Baars argues that this is distinct from the concept of the Cartesian theater, since it is not based on the implicit dualistic assumption of " someone " viewing the theater, and is not located in a single place in the mind ( in Blackmore, 2005 ).

Ryle concludes that both Cartesian theory and behaviorist theory may be too rigid and mechanistic to provide us with an adequate understanding of the concept of mind.

Lewis Call has attempted to develop post-anarchist theory through the work of Friedrich Nietzsche, rejecting the Cartesian concept of the " subject.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

According to this Cartesian theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them.

From a Cartesian point of view, therefore, this was a faulty theory.

Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke – Platek set theory.

The Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be ' shallow ' for the purposes of the operation.

In set theory, the Cartesian product is a set of 2-tuples.

Free products of algebras of random variables play the same role in defining " freeness " in the theory of free probability that Cartesian products play in defining statistical independence in classical probability theory.

One may explain ( human ) vision by noting that light from the outside world forms an image on the retinas in the eyes and something ( or someone ) in the brain looks at these images as if they are images on a movie screen ( this theory of vision is sometimes termed the theory of the Cartesian Theater: it is most associated, nowadays, with the psychologist David Marr, see also: solipsism ).

The theory of interaction between World 1 and World 2 is an alternative theory to Cartesian dualism, which is based on the theory that the universe is composed of two essential substances: Res Cogitans and Res Extensa.

In the theory of functions of several complex variables, a branch of mathematics, a polydisc is a Cartesian product of discs.

By analogy, the term " hyperrectangle " or " box " refers to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers.

According to Ryle, the classical theory of mind, as represented by Cartesian rationalism, asserts that there is a basic distinction between mind and matter.

Cartesian theory holds that mental acts determine physical acts and that volitional acts of the body must be caused by volitional acts of the mind.

At the same time, however, he criticizes both Cartesian theory and behaviorist theory for being overly mechanistic.