Category of abelian groups information

mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every...

prototypical example of an abelian category is the category of abelian groups, Ab. Abelian categories are very stable categories; for example they are...

group. Subgroups, quotients, and direct sums of abelian groups are again abelian. The finite simple abelian groups are exactly the cyclic groups of prime...

this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from groups to sets...

abelian groups, together with the group homomorphisms, form an abelian category which is a Serre subcategory of the category of abelian groups. Note that...

theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic...

in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab...

branch of mathematics. It is defined differently, but analogously, for different kinds of structures. As an example, the direct sum of two abelian groups A...

For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar...

functor from the category of commutative monoids to the category of abelian groups which sends the commutative monoid M to its Grothendieck group K. This functor...

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. The five...

in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product (including the empty product...

specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence...

locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped...

groups Category of abelian groups Category of rings Category of magmas Initial object Terminal object Zero object Subobject Group object Magma object...

the category of groups is conormal (since it is the cokernel of its own kernel), so this category is conormal. In an abelian category, every monomorphism...

Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups. An abelian group (...

functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups. The existence of the abelianization functor...

Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological...

In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used...

commutative Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms Metabelian group, a group where the commutator...

is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every...

{\displaystyle V} . The category of presheaves of sets (abelian groups, rings) on X {\displaystyle X} is then the same as the category of contravariant functors...

One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically...

the fact that the first Ext group Ext1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold...

Quillen and is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels...