Give an example of an additive functor H: 𝐀 𝐛 →𝐀 𝐛 that has neither a left nor a right adjoint.

Name: Problem 29, Chapter 5: An Introduction to Homological Algebra
Uploaded: 2021-08-04T18:26:12-08:00
Duration: 2 min 41 s
Channel: Monica Miller
Description: Give an example of an additive functor H: 𝐀 𝐛 →𝐀 𝐛 that has neither a left nor a right adjoint.

step 1 Okay, we are going to show that the derivative of a cross product is not communitive, meaning we

Give an example of an additive functor H: 𝐀 𝐛 →𝐀 𝐛 that has...

Key Concepts

Additive Functor

An additive functor between abelian categories is a mapping that preserves the additive structure of the hom?sets (i.e. it respects sums and the zero morphisms) and commutes with finite direct sums. In the context of the category of abelian groups, this means that the functor carries the abelian group operations in a way that makes its action compatible with the linear structure inherent in the groups.

Adjoint Functors

Adjoint functors occur in pairs (left and right) and are characterized by natural isomorphisms between certain Hom-sets. A functor F having a left adjoint means there is another functor G with Hom(F(A), B) ? Hom(A, G(B)) for all objects, while having a right adjoint means there is a functor H such that Hom(K(B), A) ? Hom(B, H(A)). The existence of these adjoints often relies on the functor preserving (co)limits, and when these conditions fail, the functor may lack adjoints.

Tensoring Functor

A common type of additive functor on the category of abelian groups is given by tensoring with a fixed abelian group. For example, the functor that sends an abelian group A to A ? Q is additive. However, because Q is not finitely presented as a Z-module, this tensoring functor typically fails to preserve the exactness properties required for the existence of a left or a right adjoint.

Abelian Category

The concept of an abelian category, exemplified by the category of abelian groups, provides a setting in which kernels, cokernels, and finite direct sums exist and behave in a well?controlled manner. These structural features are crucial when discussing properties of additive functors and the existence (or nonexistence) of adjoints.

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