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 ( 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.

A category intuitively consists of a collection of objects, and defined relationships between them.

Since plug-ins and extensions both increase the utility of the original application, Mozilla uses the term " add-on " as an inclusive category of augmentation modules that consists of plug-ins, themes, and search engines.

Grand theft generally consists of the theft of something of value over $ 1000 ( it can be money, labor or property but is lower with respect to various specified property ), while petty theft is the default category for all other thefts.

An initial morphism from X to U is an initial object in the category of morphisms from X to U. In other words, it consists of a pair ( A, φ ) where A is an object of D and φ: X → U ( A ) is a morphism in C, such that the following initial property is satisfied:

A category consists of at least two further elements: the pattern and template elements.

If ( E, M ) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense.

A category C consists of

The category Rel consists of all sets, with binary relations as morphisms.

In this sense, each monoid is a category that consists of one object and a collection of specific morphisms to itself.

Then an enriched category C ( alternatively, in situations where the choice of monoidal category needs to be explicit, a category enriched over M, or M-category ), consists of

The predrum category consists of simple drum-like percussion instruments.

A concrete category, when defined without reference to the notion of a category, consists of a class of objects, each equipped with an underlying set ; and for any two objects A and B a set of functions, called morphisms, from the underlying set of A to the underlying set of B.

In the study of comparative religion, the category of Abrahamic religions consists of the three monotheistic religions, Christianity, Islam and Judaism, which claim Abraham ( Hebrew Avraham א ַ ב ְ ר ָ ה ָ ם ; Arabic Ibrahim إبراهيم ) as a part of their sacred history.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

The ecrs module identifier consists of either chk, sks, ksk or loc followed by a slash and a category specific value.

Since plug-ins and extensions both increase the utility of the original application, Mozilla uses the term " add-on " as an inclusive category of augmentation modules that consists of plug-ins, themes, and search engines.

* The center of a monoidal category C consists of pairs ( A, u ) where A is an object of C, and a natural isomorphism satisfying certain axioms.

If is a category, a monad on consists of a functor together with two natural transformations: ( where denotes the identity functor on ) and ( where is the functor from to ).

The Jew's harp, jaw harp, mouth harp, Ozark harp, trump or juice harp, is a lamellophone instrument, which is in the category of plucked idiophones: it consists of a flexible metal or bamboo tongue or reed attached to a frame.

In category theory, currying can be found in the universal property of an exponential object, which gives rise to the following adjunction in cartesian closed categories: There is a natural isomorphism between the morphisms from a binary product and the morphisms to an exponential object.

Note that, following the property of opposite category,.

In the following decades, fear of communism and the Great Depression of 1929 – 1933 led to the rise of extreme nationalist governmentssometimes loosely grouped under the category of fascismin Italy ( 1922 ), Germany ( 1933 ), Spain ( after a civil war ending in 1939 ) and other countries such as Hungary ( 1944 ), Romania ( 1940 ) and Slovakia ( 1939 ).

Since homomorphisms are morphisms, the following specific kinds of morphisms defined in any category are defined for homomorphisms as well.

Taking a more positive position, Mettinger argues in his recent book that the category of rise and return to life is significant for the following deities: Ugaritic Baal, Melqart, Adonis, Eshmun, Osiris and Dumuzi.

Locke then elaborates on the nature of this third category, naming it Σημειωτικη ( Semeiotike ) and explaining it as " the doctrine of signs " in the following terms:

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual ( opposite ) notions:

Having played diverse roles in many well-received films, Ryder won a Golden Globe Award for Best Supporting Actress and an Academy Award nomination in the same category for her role in The Age of Innocence in 1993, as well as another Academy Award nomination for Little Women the following year for Best Actress.

The following year, she played the title role in the smash hit There's Something About Mary ( 1998 ), for which she was nominated for a Golden Globe Award for the category of Best Actress – Musical or Comedy.

Other linguists, following Huddleston, dispute whether such a category exists.

In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense:

The film was nominated for the following Academy Awards, winning in one category:

At the 9th Academy Awards ceremony ( 1936 ), however, the Best Supporting Actress category was specifically introduced as a distinct award following complaints that the single Best Actress category necessarily favored leading performers with the most screen time.

The following year " Best Special Effects " became a recognized category, although on occasion the Academy has chosen to honor a single film outright rather than nominate two or more films.

For a category C, the following are all equivalent:

Kitniyot, qit ' niyyoth () ( literally legumes ) is a category of foods that may not be eaten during Passover by Jews following traditional Ashkenazi laws and customs.

( i ) Secondary materials: cite in the following order: uniform codes, model codes, and restatements ( in reverse chronological order by category ), books and pamphlets, works in journals, books reviews not written by students, student-written law review materials, annotations, magazine and newspaper articles, working papers, unpublished materials not forthcoming, electronic sources, including Internet sources.

It states that for the following commutative diagram ( in any abelian category, or in the category of groups ), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well.

Consider the following commutative diagram in any abelian category ( such as the category of abelian groups or the category of vector spaces over a given field ) or in the category of groups.