# 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?

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

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, $\mathrm{Hom}(*,*)$.

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