Is Hom an exact functor?

Is Hom an exact functor?

The most basic examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups.

What does Hom mean in math?

Hom(M,N) refers to the set of A-module homomorphisms from M to N. These form an abelian group under pointwise addition (define f+g by (f+g)(x)=f(x)+g(x)), and if A is commutative they in fact form an A-module by pointwise scalar multiplication (define a⋅f by (af)(x)=a⋅f(x)).

Is Hom left or right exact?

Hom Functor is Left Exact Since gf = 0, we also have. Thus. Conversely, suppose. so h : N → M is a map such that gh : N → M” is the zero map.

What is the difference between Morphism and homomorphism?

is that morphism is (mathematics|formally) an arrow in a category while homomorphism is (algebra) a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces.

What are the types of Morphism?

Here are 5 of the morphisms that I currently have.

  • Isomorphism.
  • Homomorphism.
  • Homeomorphism.
  • Monomorphism.
  • Epimorphism.

What are the properties of the hom functor?

Properties. The internal hom functor preserves limits; that is, sends limits to limits, while sends limits in , that is colimits , into limits. In a certain sense, this can be taken as the definition of a limit or colimit.

How are Hom functors used in category theory?

These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics. Let C be a locally small category (i.e. a category for which hom-classes are actually sets and not proper classes ). For all objects A and B in C we define two functors to the category of sets as follows:

Is the hom functor a contravariant functor?

The functor Hom (–, B) is also called the functor of points of the object B . Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor.

How is the hom functor used to preserve limits?

Preservation of limits. The hom-functor preserves limits in both arguments separately. This means: for fixed object c∈Cc in C the functor hom(c,−):C→Sethom(c,-) : C to Set sends limit diagrams in CC to limit diagrams in SetSet;

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