# Initial and terminal objects in $\textbf{FinVect}_R$

I am teaching myself category theory and I am having difficulties identifying the initial and terminal object of the category of $\textbf{FinVect}_R$.

I was thinking that because it is finite vectors then the initial and terminal should be the same object ( since they are finite, we can operate in those vectors until we reached that last one since it is finite).

Any help will be greatly appreciated.

• Is the category $\operatorname{FinVect}_R$ the category of real finite-dimensional vector spaces? If so, could you edit that into your question? I think it would make it a bit clearer and easier to answer. – Potato Jun 8 '13 at 5:13

I believe the terminal and initial object are both the zero-dimensional vector space $0$.
There is one map from $0$ to any other vector space $V$, since we must send $0$ to $0_V$. This follows from the definition of a linear map, and linear maps are the morphisms in this category. So $0$ is the initial object.
Similarly, there is exactly one morphism from any vector space $V$ to $0$: the map sending all elements to $0$. So $0$ satisfies the definition of terminal object.
Note that this argument is independent of the base field. So you could consider the category of vector spaces over $\mathbb C$, for example, and the answer would be the same.