When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.

A related but different notion is a free abelian group.

In the case g = 1, the notion of abelian variety is the same as that of elliptic curve, and every complex torus gives rise to such a curve ; for g > 1 it has been known since Riemann that the algebraic variety condition imposes extra constraints on a complex torus.

Modules also generalize the notion of abelian groups, which are modules over the ring of integers.

Thus weights are primarily of interest for abelian Lie algebras, where they reduce to the simple notion of a generalized eigenvalue for space of commuting linear transformations.

There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.

Note: In SGA4, tome I, section 1, the notion of left ( right ) exact functors have been defined for general categories, and not just abelian ones.

The abelian case was the original framework for the notion of injectivity.

This generalizes to the notion of abelian scheme ; a group scheme G over a base S is abelian if the structural morphism from G to S is proper and smooth with geometrically connected fibers They are automatically projective, and they have many applications, e. g., in geometric class field theory and throughout algebraic geometry.

proceeded by the means of an explicit chain complex related to the Hochschild homology complex of A. Connes later found a more categorical approach to cyclic homology using a notion of cyclic object in an abelian category, which is analogous to the notion of simplicial object.

In the history of the subject they were introduced before the 1957 ' Tohoku ' paper of Alexander Grothendieck, which showed that the abelian category notion of injective object sufficed to found the theory.

One of the deepest discoveries of Gauss was the existence of a natural composition law on the set of classes of binary quadratic forms of given discriminant, which makes this set into a finite abelian group called the class group of discriminant D. Gauss also considered a coarser notion of equivalence, under which the set of binary quadratic forms of a fixed discriminant splits into several genera of forms and each genus consists of finitely many classes of forms.

The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.

The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.

This leads to the notion of an internal groupoid in a category.

In mathematics, especially in category theory and homotopy theory, a groupoid ( less often Brandt groupoid or virtual group ) generalises the notion of group in several equivalent ways.

Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

These categories, he argued, are " of the highest order of abstraction ", but he defended them as those necessary to " make possible a coherent account of what grammar is and of its place in language " In articulating the category unit, Halliday proposed the notion of a rank scale.

Such products are generically called internal products, as they can be described by the generic notion of a monoidal category.

The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.

* Descent ( category theory ), an idea extending the notion of " gluing " in topology

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits.

The notion of a natural transformation is categorical, and states ( informally ) that a particular map between functors can be done consistently over an entire category.

A factorization system for a category also gives rise to a notion of embedding.

In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.

Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions.

More generally, the notion of an exact sequence makes sense in any category with kernels and cokernels.

What is new here is that the above expresses the requirement for associativity without any explicit reference to individual morphisms in the enriched category C — again, these diagrams are for morphisms in hom-category M, and not in C — thus making the concept of associativity of composition meaningful in the general case where the hom-objects C ( a, b ) are abstract, and C itself need not even have any notion of individual morphism.

The notion that an ordinary category must have identity morphisms is replaced by the second and third diagrams, which express identity in terms of left and right unitors:

These should be distinguished from the morphisms that define the notion of identity for objects in the enriched category C, whether or not C can be considered to have individual morphisms of its own.

In this situation, F is called the lower adjoint of G and G is called the upper adjoint of F. Mnemonically, the upper / lower terminology refers to where the function application appears relative to ≤; the term adjoint relates the Galois connections to the notion with the same name from category theory as discussed further below.

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.

The partially ordered groups, together with this notion of morphism, form a category.

There is a haunting resemblance between the notion of cause in Copernicus and in Freud.

Within this notion clarity is possible, but for us who are neither Greek nor Jansenist there is not such clarity.

Hence, the only defensible procedure is to repress any and every notion, unless it gives evidence that it is perfectly safe.

This is the principal point made in this final section of Englishman No. 57, and it caps Steele's efforts in his other writing of these months to counteract the notion of the Tories as a `` Church Party '' supported by the body of the clergy.

The notion of `` inspiration '' is somehow cognate to this feeling.

The Aristotelian notion of catharsis, the purging of emotion, is a persistent and viable one.

The idea here is one of discharge but this must stand in opposition to a second view, Plato's notion of the arousal of emotion.

This is given some expression in Beardsley's notion of harmony and the resolution of indecision.

And there is one other point in the Poetics that invites moral evaluation: Aristotle's notion that the distinctive function of tragedy is to purge one's emotions by arousing pity and fear.

I refer to the notion that the structure of society is a microcosm of the cosmic design and that history conforms to patterns of justice and chastisement as if it were a morality play set in motion by the gods for our instruction.

That notion is fantastically wrong-headed from several points of view.

The notion of philosophy as Queen Bee may fit well with authoritarian modes of political ideology, but it has been noted that the price of such an imperial notion of philosophy is the frustration and flagellation of the social sciences.

The threadbare notion that belief, unlike behaviour, is not subject to objective analysis, has placed intuitive metaphysics squarely against the sociology of knowledge, since it is precisely the job of the sociology of knowledge to treat beliefs as social facts no less viable than social behaviour.

These cases, for all their rarity, are so dramatic that friends and relations repeat the story until the general population may get an entirely false notion of how often the hymen is a serious problem to newly-weds.

It is a notion which contains a gratuitous insult, implying, as it does, that Negroes can make no move unless they are manipulated.

They would like to convey the notion something is being done, even though it is something they know to be ineffectual.

( Pp. 228-229 ) in any event, it is obvious that the anti-trust laws did not prevent the formation of some of the greatest financial empires the world has ever known, held together by some of the most fantastic ideas, all based on the fundamental notion that a corporation is an individual who can trade and exchange goods without control by the government ''.

Still, the notion of altruism is modified in such a world-view, since the belief is that such a practice promotes our own happiness: " The more we care for the happiness of others, the greater our own sense of well-being becomes " ( Dalai Lama ).