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searching for Category of abelian groups 32 found (98 total)

alternate case: category of abelian groups

AB5 category (113 words) [view diff] exact match in snippet view article find links to article

various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according

Splitting lemma (1,092 words) [view diff] no match in snippet view article find links to article

In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent

Bredon cohomology (96 words) [view diff] exact match in snippet view article find links to article

the category of G-complex with equivariant homotopy maps to the category of abelian groups together with the connecting homomorphism satisfying some conditions

Topological half-exact functor (172 words) [view diff] exact match in snippet view article find links to article

spaces) to an abelian category (most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following

Bivariant theory (433 words) [view diff] exact match in snippet view article find links to article

theory is a covariant functor from the category of spaces to the category of abelian groups, while a cohomology theory is a contravariant functor from the

Size functor (523 words) [view diff] case mismatch in snippet view article find links to article

real numbers, and A b {\displaystyle \mathrm {Ab} \ } is the category of Abelian groups, defined in the following way. For x ≤ y {\displaystyle x\leq

Exact category (1,382 words) [view diff] exact match in snippet view article find links to article

take A to be the category of left-exact functors from E into the category of abelian groups, which is itself abelian and in which E is a natural subcategory

Operator K-theory (526 words) [view diff] exact match in snippet view article find links to article

from the category of C*-algebras and *-homomorphisms, to the category of abelian groups and group homomorphisms. The higher K-functors are defined via

Presheaf with transfers (2,676 words) [view diff] exact match in snippet view article find links to article

the category of finite correspondences (defined below) to the category of abelian groups (in category theory, “presheaf” is another term for a contravariant

Normal morphism (280 words) [view diff] exact match in snippet view article find links to article

its kernel. Thus, abelian categories are always binormal. The category of abelian groups is the fundamental example of an abelian category, and accordingly

Homology (mathematics) (6,435 words) [view diff] exact match in snippet view article

covariant functor from the category of chain complexes to the category of abelian groups (or modules). If the chain complex depends on the object X in

Generator (category theory) (320 words) [view diff] exact match in snippet view article

The dual concept is called a cogenerator or coseparator. In the category of abelian groups, the group of integers Z {\displaystyle \mathbf {Z} } is a generator:

Injective cogenerator (513 words) [view diff] exact match in snippet view article find links to article

is injective. For example, the integers are a generator of the category of abelian groups (since every abelian group is a quotient of a free abelian group)

Eilenberg–Steenrod axioms (750 words) [view diff] exact match in snippet view article find links to article

( X , A ) {\displaystyle (X,A)} of topological spaces to the category of abelian groups, together with a natural transformation ∂ : H i ( X , A ) → H

Forgetful functor (1,161 words) [view diff] exact match in snippet view article find links to article

Deleting × {\displaystyle \times } and 1 yields a functor to the category of abelian groups, which assigns to each ring R {\displaystyle R} the underlying

Zig-zag lemma (656 words) [view diff] exact match in snippet view article find links to article

in every abelian category. In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let ( A

Five lemma (867 words) [view diff] exact match in snippet view article find links to article

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

Biproduct (1,027 words) [view diff] exact match in snippet view article find links to article

empty, or nullary, biproduct is always a zero object. In the category of abelian groups, biproducts always exist and are given by the direct sum. The

Unit (ring theory) (1,519 words) [view diff] exact match in snippet view article

forgetful functor from the category of commutative rings to the category of abelian groups). Suppose that R is commutative. Elements r and s of R are called

Injective object (1,031 words) [view diff] exact match in snippet view article find links to article

uniquely determined by X up to a non-canonical isomorphism. In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily

Snake lemma (1,434 words) [view diff] exact match in snippet view article find links to article

connecting homomorphisms. In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), consider

Cohomology (6,691 words) [view diff] exact match in snippet view article find links to article

CW-pairs (X, A) (so X is a CW complex and A is a subcomplex) to the category of abelian groups, together with a natural transformation ∂i: hi(X, A) → hi−1(A)

Monoidal functor (1,261 words) [view diff] exact match in snippet view article find links to article

{Z} )\rightarrow (\mathbf {Set} ,\times ,\{\ast \})} from the category of abelian groups to the category of sets. In this case, the map ϕ A , B : U ( A

Yoneda lemma (3,361 words) [view diff] exact match in snippet view article find links to article

additive contravariant functors from the original category into the category of abelian groups; these are functors which are compatible with the addition of

Direct sum (2,822 words) [view diff] exact match in snippet view article find links to article

of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category

Local system (2,671 words) [view diff] exact match in snippet view article find links to article

the category of local systems of abelian groups on X and the category of abelian groups endowed with an action of π 1 ( X , x ) {\displaystyle \pi _{1}(X

Dold–Thom theorem (1,909 words) [view diff] exact match in snippet view article find links to article

from the category of basepointed, connected CW complexes to the category of abelian groups a reduced homology theory if they satisfy If f ≃ g: X → Y, then

K-theory of a category (1,642 words) [view diff] exact match in snippet view article find links to article

group construction is a functor from the category of rings to the category of abelian groups. The higher K-theory should then be a functor from the category

Grothendieck topology (4,520 words) [view diff] exact match in snippet view article find links to article

require either that a presheaf F is a contravariant functor to the category of abelian groups (or rings, or modules, etc.), or that F be an abelian group (ring

Tensor (9,356 words) [view diff] exact match in snippet view article find links to article

ISBN 978-1-4612-9839-7. ...for example the monoid M ... in the category of abelian groups, × is replaced by the usual tensor product... Bamberg, Paul; Sternberg

Triangulated category (5,798 words) [view diff] exact match in snippet view article find links to article

\operatorname {Hom} ({\text{-}},B)} are cohomological, with values in the category of abelian groups. (To be precise, the latter is a contravariant functor, which

Glossary of algebraic topology (7,540 words) [view diff] exact match in snippet view article find links to article

contravariant functor from the category of pairs of spaces to the category of abelian groups that satisfies all of the Eilenberg–Steenrod axioms except the