How we can extract a vector space structure from a category with one object?

How can we associate a vector room framework to a group with one object? Exists an approved means of doing this?

2022-06-07 14:35:07
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Answers: 2

You have to have the ability to add vectors, yet usually there is no other way to "add" morphisms in a basic group with one object. Additionally, what should the base area be?

Allow $C$ be your group. What you can do, is to deal with an area $k$, and also take into consideration the openly created vector room $k[Hom(C)]$ (amusing symbols ). That is, the vector room with components official $k$ - straight mixes of the morphisms in $C$.

2022-06-07 14:55:31

Let is represent $*$ the (one-of-a-kind) object of your group. The (one-of-a-kind) set of maps of this group is the set of endomorphisms $\mathrm{Hom}(*,*)$. Necessarily of group, this is set is, well, a set. Yet there are conditions where it can have extra framework.

As an example, if your group is an abelian one, after that $\mathrm{Hom}(*,*)$ is an abelian team. This holds true, as an example, for the group of abelian teams, where the set of morphisms in between 2 abelian teams is additionally an abelian team.

When the collections of morphisms have the framework of $\mathbf{k}$ - vector rooms, the group is called $\mathbf{k}$ - linear ($\mathbf{k}$ an area). As an example, in the group of $\mathbf{k}$ - vector rooms, hom - collections are additionally $\mathbf{k}$ - vector rooms.

Therefore, a $\mathbf{k}$ - straight group with simply one object is basically the like a $\mathbf{k}$ - vector room. Particularly, $\mathrm{Hom}(*,*)$.

2022-06-07 14:55:08