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Example 12.3.13. Let $k$ be a field. Consider the category of filtered vector spaces over $k$. (See Definition 12.19.1.) Consider the filtered vector spaces $(V, F)$ and $(W, F)$ with $V = W = k$ and

\[ F^ iV = \left\{ \begin{matrix} V & \text{if} & i < 0 \\ 0 & \text{if} & i \geq 0 \end{matrix} \right. \text{ and } F^ iW = \left\{ \begin{matrix} W & \text{if} & i \leq 0 \\ 0 & \text{if} & i > 0 \end{matrix} \right. \]

The map $f : V \to W$ corresponding to $\text{id}_ k$ on the underlying vector spaces has trivial kernel and cokernel but is not an isomorphism. Note also that $\mathop{\mathrm{Coim}}(f) = V$ and $\mathop{\mathrm{Im}}(f) = W$. This means that the category of filtered vector spaces over $k$ is not abelian.

Comments (2)

Comment #7840 by Zhenhua Wu on

The category of even dimension vector space over a field is an easier example that is additive but not abelian, since the kernel of the linear map is one dimensional. The word 'filtered vector spaces' may scare people.

Comment #8064 by on

The category of filtered vector spaces has kernels and cokernels but is not abelian. So it is interesting in that it shows one needs the Im Coim axiom.

There are also:

  • 10 comment(s) on Section 12.3: Preadditive and additive categories

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