# Kernel and Image definition in general category theory.

Let $$A \xrightarrow{f} B$$ be a morphism of objects $$A,B$$ of a category $$\mathcal{C}$$. Then

• We say that a morphism $$K \xrightarrow{\iota_K} A$$ of objects of $$\mathcal{C}$$ is a $$\mathit{kernel}$$ of $$f$$ if $$f\iota_K = 0$$ and whenever $$K' \xrightarrow{\iota_{K'}} A$$ is another morphism satisfying $$f\iota_{K'} = 0$$ there exists a unique morphism $$K' \xrightarrow{\mathscr{k}} K$$ such that $$\iota_{K'} = \iota_K\mathscr{k}$$.
• We say that a morphism $$A \xrightarrow{e_I} I$$ of objects of $$\mathcal{C}$$ is an $$\mathit{image}$$ of $$f$$ if there exists a monomorphism $$I \xrightarrow{\varepsilon_I} B$$ with $$f = \varepsilon_I e_I$$ such that whenever $$A \xrightarrow{e_{I'}} I’$$ is a morphism such there exists a monomorphism $$I' \xrightarrow{\varepsilon_{I'}} B$$ with $$f = \varepsilon_{I'}e_{I'}$$ there exists a unique morphism $$I \xrightarrow{\mathscr{i}} I'$$ such that $$\varepsilon_I = \varepsilon_{I'}\mathscr{i}$$.

My question concerns the directions of the arrows $$\mathscr{k}, \mathscr{i}$$. In the case of the kernel, $$K$$ is the codomain of $$\mathscr{k}$$ whilst in the case of the image, $$I$$ is the domain of $$\mathscr{i}$$. Is there a reason it is defined this way? And if so is there a good way to remember which is which? Note that the definition of cockerel/coimage reverses these too.

• Your map $e_I$ in the definition of the image should be a map $A\to I$, not $I\to A$. – Andrew Hubery May 9 at 21:28
• @AndrewHubery too right! Fixing it – Adam Higgins May 9 at 21:29

You can easily tell why kernel has to be a limit: it's the largest subobject of domain of $$f$$ on which $$f$$ vanishes. So, if $$f$$ vanishes on something, there's an appropriate morphism going into kernel.
Similarly, image is colimit since it's the smallest subobject of codomain of $$f$$ through which $$f$$ factors. If there is any other such subobject, image is contained in it, i.e. there is an appropriate morphism from image.