The first possibility results in a closed interval of tangent points in the f-plane, the end points of which fall into category ( B ) or ( C ).

The automorphism group of an object X in a category C is denoted Aut < sub > C </ sub >( X ), or simply Aut ( X ) if the category is clear from context.

A category C consists of the following three mathematical entities:

A ( covariant ) functor F from a category C to a category D, written, consists of:

More specifically, every morphism in C must be assigned to a morphism in D. In other words, a contravariant functor acts as a covariant functor from the opposite category C < sup > op </ sup > to D.

* The functor category D < sup > C </ sup > has as objects the functors from C to D and as morphisms the natural transformations of such functors.

If one statement is true in a category C then its dual will be true in the dual category C < sup > op </ sup >.

It follows that the set of all endomorphisms of X forms a monoid, denoted End ( X ) ( or End < sub > C </ sub >( X ) to emphasize the category C ).

Identity functor in category C, written 1 < sub > C </ sub > or id < sub > C </ sub >, maps an object to itself and a morphism to itself.

Diagonal functor: The diagonal functor is defined as the functor from D to the functor category D < sup > C </ sup > which sends each object in D to the constant functor at that object.

Limit functor: For a fixed index category J, if every functor J → C has a limit ( for instance if C is complete ), then the limit functor C < sup > J </ sup >→ C assigns to each functor its limit.

Falling somewhere in a category between Einstein's theory and sand fleas -- difficult to see but undeniably there, nevertheless -- is the tropical green `` city '' of Islandia, a string of offshore islands that has almost no residents, limited access and an unlimited future.

It is natural from the marksman's viewpoint to call a bull's-eye a success, but in the mice example it is arbitrary which category corresponds to straight hair in a mouse.

In the third category the function is double-valued in this interval.

When alienation is used as an objective and diagnostic category, for example, it becomes clear that Fromm would have to say that awareness of alienation goes far toward conquering it.

There is a difference between evidence and illustration, and Fromm's citation of the other diagnosticians fits the latter category.

It is the only county in the state so far this month reporting a possible shortage in GA category, for which emergency allotment can be given by the state if necessary.

Because there is no canonical well-ordering of all sets, a construction that relies on a well-ordering may not produce a canonical result, even if a canonical result is desired ( as is often the case in category theory ).

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms ( usually called a small category ), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.

Abjads differ from abugidas, another category invented by Daniels, in that in abjads, the vowel sound is implied by phonology, and where vowel marks exist for the system, such as nikkud for Hebrew and harakāt for Arabic, their use is optional and not the dominant ( or literate ) form.

Abandoned land resulting from shifting cultivation is not included in this category.

The most general setting in which these words have meaning is an abstract branch of mathematics called category theory.

In category theory, an automorphism is an endomorphism ( i. e. a morphism from an object to itself ) which is also an isomorphism ( in the categorical sense of the word ).

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.

" Speaking of Milt Brown and himself working with songs done by Jimmie Davis, the Skillet Lickers, Jimmie Rodgers, and others, and songs he'd learned from his father, he said that " We'd pull these tunes down an set ' em in a dance category.

Being a form of visual art, it might also be said that many graffiti artists still fall in the category of the introverted archetypal artist.

ECPA President Mark Kuyper said, " In many ways this series established Christian fiction as a significant category in publishing in general.

Throughout most of the 1940s, Meher Baba worked with a category of spiritual aspirant called masts, whom he said are entranced or spellbound by internal spiritual experiences.

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, " I didn't invent categories to study functors ; I invented them to study natural transformations.

Regarding his belief system, he doesn't " want to provide a category for people to apply to " because he " would not want convictions reduced to a word " and states, " I have never said, although readers have freely informed me I am an atheist, an agnostic, or at the very least a secular humanist — which I am ".

In this case, the category is said to be well powered with respect to the class of embeddings.

However, there is an argument that nuclear and biological weapons do not belong in the same category as chemical and " dirty bomb " radiological weapons, which have limited destructive potential ( and close to none, as far as property is concerned ), whereas nuclear and biological weapons have the unique ability to kill large numbers of people with very small amounts of material, and thus could be said to belong in a class by themselves.

For example, in category theory, a category is said to be connected if each pair of objects in it is joined by a sequence of morphisms.

An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category, with the last terminology being used in some influential texts, such as MacLane's.

The 710-page work followed the pattern of Coke's Institutes of the Lawes of England, but was far more methodical ; James Fitzjames Stephen said that Hale's work " was not only of the highest authority but shows a depth of thought which puts it in quite a different category from Coke's Institute ... is far more of a treatise and far less of an index or mere work of practice ".

Thus, physical disorders can be said to produce delirium as a mental side-effect or symptom, although primary mental disorders which produce the symptom cannot be put into this category once identified.

A third category of mitzvot are called Chukim whose rationale and purpose are said to be no longer obvious in post-Biblical contexts.

On 21 July 2011: Former MP, Evan Harris, " said that every scientific advisor had agreed it was a good idea to declassify cannabis from category B to C. But because of a story in the Mail on Sunday, the Labour government broke the law.

He said that when needs in a higher category are not met then individuals redouble the efforts invested in a lower category need.

Germaine Gioia, vice president of licensing at THQ, said " The Simpsons and Buffy properties have enjoyed tremendous consumer appeal in nearly every product category, including video game successes.

It has been obvious to the assessors, particularly those in shore communities, that boats comprise the largest category of tangible personal property which they have been unable to reach.

Notable in this category are the Jupiter and Thor intermediate range ballistic missiles, which have been successfully developed, produced, and deployed, but the relative importance of which has diminished with the increasing availability of the Atlas intercontinental ballistic missile.

The Bible can fall into both the category of ecclesiastical sources and also this category, as the Beowulf poet would have relied on Old English translations.

Jefferson is known to have done an unusual amount of traveling for the time in the American South, which is reflected in the difficulty of pigeonholing his music into one regional category.

Within the transfer education category, comprehensive schools typically have articulation agreements in place that provide prearranged acceptance into specific four-year institutions.

In fact, what we have done is define a category of categories and functors – the objects are categories, and the morphisms ( between categories ) are functors.

These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.

At the very least, category theoretic language clarifies what exactly these related areas have in common ( in some abstract sense ).

These categories surely have some objects that are " special " in a certain way, such as the empty set or the product of two topologies, yet in the definition of a category, objects are considered to be atomic, i. e., we do not know whether an object A is a set, a topology, or any other abstract concept – hence, the challenge is to define special objects without referring to the internal structure of those objects.

The United States Environmental Protection Agency ( EPA ) includes discarded CRT monitors in its category of " hazardous household waste " but considers CRTs that have been set aside for testing to be commodities if they are not discarded, speculatively accumulated, or left unprotected from weather and other damage.

But they are all in the CISC category because they have " load-operate " instructions that load and / or store memory contents within the same instructions that perform the actual calculations.

The lone exception to this pattern has been the very smallest category of bookseller, shops with between one and four employees, which appear to have withstood the trend.

But in addition, statistical analysis indicates that lower-mass stars ( red dwarfs, of spectral category M ) are less likely to have planets massive enough to detect.

Nevertheless, overlap between the diagnoses and neurological abnormalities ( when found ) have continued, and in fact a diagnostic category of schizoaffective disorder would be brought in to cover the intermediate cases.

A number of writings, belonging in this category, have been entirely lost.

People with seizures that occur only after specific precipitants (" provoked seizures "), have " epilepsies " that fall into this category.

In other words, we have a functor from the category of topological spaces with base point to the category of groups.

According to the WHO, some practices regarded as legal in countries that have outlawed FGM do fall under the category of Type IV ( see below ), but the organization decided to maintain a broad definition to avoid loopholes that could allow FGM to continue.

The inflectional categories of the Greek verb have likewise remained largely the same over the course of the language's history, though with significant changes in the number of distinctions within each category and their morphological expression.

This origin gave rise to a wide variety of dishes called gazpacho and to some others that don't have this name but belong to the same category.

Often depending on which constitutional category ( above ) a head of state belongs to, they may have some or all of the roles listed below, and various other ones.