Questions about the definition of normal category and wording of an exercise.

Question

The following is taken from 'Theory of Categories" by Barry Mitchell

$\color{Green}{Background:}$

$\textbf{(1) Definition:}$

If $\alpha;A'\to A$ is a monomorphism, we shall call $A'$ a $\textbf{subobject}$ of $A,$ and we shall refer to $\alpha$ as the $\textbf{inclusion}$ of $A'$ in $A.$ Sometimes we shall write $\alpha;A'\subset A,$ or simply $A'\subset A$ when we want to indicate that $A'$ is a subobject of $A,$ and we shall say that $A'$ is $\textbf{contained}$ in $A,$ or that $A$ $\textbf{contains}$ $A'.$

$\textbf{(2) Definition:}$

If $A'\to A$ is the kernel of some morphism then we call $A'$ a $\textbf{normal}$ subobject of $A.$ If every monomorphism in a category is normal then we say that the category is $\textbf{normal}.$

$\textbf{(3) Definition:}$ Let $A$ be a category with a zero object, and let $\alpha:A\to B.$ We will call a morphism $u:K\to A$ the $\textbf{kernel}$ of $\alpha$ if $\alpha =0,$ and if for every morphism $u':K\to A$ such that $\alpha u'=0$ we have a unique morphism $\gamma:K'\to K$ such that $u\gamma =u'.$ Equivaluently, the kernel of $\alpha$ is given by the pullback diagram

$$\begin{array}{ccccccccc} K & \xrightarrow{} & 0\\ \small {u}\big\downarrow & & \big\downarrow\small {} & \\ A & \xrightarrow{} & B \end{array}$$

In other words $K=\alpha^{-1}(0),$ so that in particular $u$ must be a monomorphism, and any two kernels must be isomorphic subobjects of $A.$ The object $K$ is frequently denoted by $\mathrm{Ker}(\alpha).$ If $\alpha$ is a monomorphism then $\mathrm{Ker}(\alpha)=0,$ but the converse is not true in general (cf $\textbf{(4) Exercise}$ below). If $\mathrm{Ker}{(\beta\alpha)}$ and $\mathrm{Ker}{(\alpha)}$ are defined, then $\mathrm{Ker}{(\alpha})\subset \mathrm{Ker}{(\beta\alpha)}.$ If $\beta$ is a monomorphism, then

$$\mathrm{Ker}{(\alpha)}=\mathrm{Ker}{(\beta\alpha)}$$

in the sense that if either side is defined then so is the other and they are equal. Also if either of $\mathrm{Equ}(\alpha,0)$ and $\mathrm{Ker}(\alpha)$ are defined then so is the other and they are equal, so that in particular if $A$ has equalizers then $A$ has kernels. Frequently we will want to know if a morphism $u$ is the kernel of some morphism $\alpha,$ knowing in advance that $\alpha u=0$ and that $u$ is a monomorphism. In such a case it suffices to test for the existence of the morphism $\gamma.$ Uniqueness will be automatic since $u$ is a monomorphism.

$\textbf{(4) Exercise:}$

In a normal category with equalizers, a morphism is an epimorphism if and only if its cokernel is $0.$

$\color{Red}{Questions:}$

I am having trouble doing the (4) Exercise above due to difficulty translating it into math notation. Specifically I am having trouble understanding three things,

1. what does it mean to say that $A'\to A$ is the kernel of some morphism?

2. What does it mean for it to say that the morphism of a cokernel is 0

3. By the referral of morphism in the exercise, which morphism is it referring to, the pair of maps that are the equalizers or the map that come before that precompose with the pair of maps which form equalisers?

Thank you in advance

Answer 1

1. $f : A' \to A$ is the kernel of some morphism iff there is some morphism $g : A \to B$ such that $f = \text{ker}(g)$.

2. I don't have the text here but I would only define kernels and cokernels in a category with a zero object $0$, meaning an object which is both initial and terminal (like the zero abelian group). In that context, to say that the cokernel of a morphism $f : A \to B$ is zero means that the zero map $B \to 0$ (which exists and is unique because $0$ is the terminal object) satisfies the universal property of the cokernel.

3. Neither. The exercise says that a morphism $f : A \to B$ is an epimorphism iff $\text{coker}(f) = 0$ (in the sense above). There's no explicit mention of an equalizer in the statement of the problem, it's just indicating that you'll probably have to use equalizers to solve it.

I have to say I really don't like the texts you seem to be working from. This concept of a normal subobject or normal category is not common or useful. I suppose it's a language for describing the difference between how $\text{Grp}$ and $\text{Ab}$ behave with respect to their kernels and cokernels (normal subobjects here generalize normal subgroups) but really you should be acquiring experience with how groups and abelian groups behave concretely before dealing with abstract stuff like this, especially abstract stuff that isn't even particularly common or useful.