Lie algebras: Assigning to every real ( complex ) Lie group its real ( complex ) Lie algebra defines a functor.

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments.

Another way to describe G-sets is the functor category, where is the groupoid ( category ) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X = F and for every g in G ( i. e. for every morphism in ) induces a bijection F < sub > g </ sub >: X → X.

The categorical structure of the functor F assures us that F defines a G-action on the set X.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

If X is an object of a locally small category C, then the assignment defines a covariant functor.

This is rather vague, though suggestive, and can be made precise in the language of category theory: a construction is most efficient if it satisfies a universal property, and is formulaic if it defines a functor.

The two facts that this method of turning rngs into rings is most efficient and formulaic can be expressed simultaneously by saying that it defines an adjoint functor.

The most general setting for a free object is in category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor.

In category theory, given any family P < sub > n </ sub > of invertible n-by-n matrices defining a similarity transformation for all rectangular matrices sending the m-by-n matrix A into P < sub > m </ sub >< sup >− 1 </ sup > AP < sub > n </ sub >, the family defines a functor that is an automorphism of the category of all matrices, having as objects the natural numbers and morphisms from n to m the m-by-n matrices composed via matrix multiplication.

Note that natural transformations can be composed: if μ ( X ): F ( X ) → G ( X ) is a natural transformation from the functor F: C → D to the functor G: C → D, and η ( X ): G ( X ) → H ( X ) is a natural transformation from the functor G to the functor H, then the collection η ( X ) μ ( X ): F ( X ) → H ( X ) defines a natural transformation from F to H. With this composition of natural transformations ( known as vertical composition, see natural transformation ), D < sup > C </ sup > satisfies the axioms of a category.

The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then F < sub > A </ sub >( X ) = Hom < sub > A </ sub >( A, X ) defines a covariant left-exact functor from A to the category Ab of abelian groups.

The correspondence defines an exact functor from the category of sheaves over to the category of sheaves of. The following two statements are the heart of Serre's GAGA theorem ( as extended by Grothendieck, Neeman et al.

The correspondence G → Bohr ( G ) defines a covariant functor on the category of topological groups and continuous homomorphisms.

If C has all inverse limits, then the limit defines a functor pro-C ' C.

It can be shown that this construction defines a functor.

The original ISO 10646 standard defines a 31-bit encoding form called UCS-4, in which each encoded character in the Universal Character Set ( UCS ) is represented by a 32-bit friendly code value in the code space of integers between 0 and hexadecimal 7FFFFFFF.

The ITU defines several " capability levels " for this protocol, starting with Capability Set 1 ( CS-1 ).

The XMP standard also defines particular namespaces for defined sets of core properties ( e. g. a namespace for the Dublin Core Metadata Element Set ).

His Ethics defines `` possessions as the property of the community, of which the individual is sovereign steward.

In discussing the process of communication, Loomis defines it as `` the process by which information, decisions, and directives are transmitted among actors and the ways in which knowledge, opinions, and attitudes are formed, or modified by interaction ''.

For every criterion which defines what something is, at the same time proclaims -- implicitly if not openly -- what that something is not.

Instead, the limbs of agnostids closely resemble those of stem group crustaceans, although they lack the proximal endite, which defines that group.

Another view defines anxiety as " a future-oriented mood state in which one is ready or prepared to attempt to cope with upcoming negative events ," suggesting that it is a distinction between future and present dangers which divides anxiety and fear.

In this work, Alan uses prose and verse to illustrate the way in which nature defines its own position as inferior to that of God.

The Dictionary of Politics: Selected American and Foreign Political and Legal Terms defines the term Alford plea as: " A plea under which a defendant may choose to plead guilty, not because of an admission to the crime, but because the prosecutor has sufficient evidence to place a charge and to obtain conviction in court.

" The book Plea Bargaining's Triumph: A History of Plea Bargaining in America published by Stanford University Press defines the plea as one in " which the defendant adheres to his / her claim of innocence even while allowing that the government has enough evidence to prove his / her guilt beyond a reasonable doubt ".

which defines the Abbe number with respect to the yellow Fraunhofer d ( or D < sub > 3 </ sub >) helium line at 587. 5618 nm wavelength.

Most traffic classes also introduce the concept of Cell Delay Variation Tolerance ( CDVT ), which defines the " clumping " of cells in time.

Queen Charlotte Sound defines its western side, while to the south lies Tory Channel, which is on the sea route from Wellington in the North Island to Picton.

It defines the administrative act, the most common form of action in which the public administration occurs against a citizen.

The American Institute of Certified Public Accountants ( AICPA ) defines accountancy as " the art of recording, classifying, and summarizing in a significant manner and in terms of money, transactions and events which are, in part at least, of financial character, and interpreting the results thereof.

72 ) defines affray as taking part in a fight in a public highway or taking part in a fight of such a nature as to alarm the public in any other place to which the public have access.

The formalisation defines algorithm complexity, problem complexity and reducibility which in turn allows equivalence classes to be defined.

On September 21, 1996, barely three years after the " Don't Ask, Don't Tell " imbroglio, and further straining relations with the LGBT community, Clinton signed into law the Defense of Marriage Act ( DOMA ), which defines marriage as the legal union of one man and one woman.

This defines a future horizon, which limits the events in the future that we will be able to influence.

; Independent countries: This category has independent countries, which the CIA defines as people " politically organized into a sovereign state with a definite territory ".

Well-known communications protocols are Ethernet, a hardware and Link Layer standard that is ubiquitous in local area networks, and the Internet Protocol Suite, which defines a set of protocols for internetworking, i. e. for data communication between multiple networks, as well as host-to-host data transfer, and application-specific data transmission formats.

Within states, whether sovereign or federated, a constitution defines the principles upon which the state is based, the procedure in which laws are made and by whom.

Although the Copenhagen interpretation is often confused with the idea that consciousness causes collapse, it defines an " observer " merely as that which collapses the wave function.

Constitutional law is the body of law which defines the relationship of different entities within a state, namely, the executive, the legislature, and the judiciary.