I'm currently reading an abstract algebra textbook, written in a category-theoretical way. The contents are similar to any other algebra textbook. It covers groups, rings, $R$-modules, $R$-algebras, fields, etc. One special thing (in my opinion) about this textbook is that the author states definitions, theorems and exercises using "$\square\square$", which is an interface for algebraic structures mentioned above. For example,
Definition. Let $X$, $X'$ be a $\square\square$. For each binary operation $\ast$ equipped in $X$, if there exists a bijection $\varphi: X\rightarrow X'$ that satisfies $$\varphi(x\ast y) = \varphi(x)\ast\varphi(y)$$ then we say that "$X$ and $X'$ are isomorphic as $\square\square$". Moreover, $\varphi$ is called an $\square\square$-isomorphism.
I have no problem understanding this - I can plug in groups, rings in $\square\square$ - and I really like the way the author introduces isomorphism in this way, since it gives the general idea of an isomorphism and isomorphic structures.
I was doing fine, until I started a chapter on subobjects...
The author chose not to define the (category theoretical) term object, and introduced subobjects as the following.
Definition. Let $X$ be a $\square\square$, and let $Y\subseteq X$. Using the binary operations of $X$, if $Y$ itself forms a $\square\square$, then $Y$ is a sub-$\square\square$ of $X$, and we write $Y\leq X$.
I also had no problem understanding the definition, (I don't know if this is actually correct, I have no reason not to believe the definition in the book) since I have already studied subgroups and (vector) subspaces. But as I read on, I felt that there must be something I don't know yet.
Let $\varphi:X\rightarrow Y$ be a $\square\square$-homomorphism. Show that $\ker \varphi \leq X$.
The statement itself looks really simple. But I got stuck when I tried to prove this. If $\square\square$ were some concrete algebraic structure such as groups or rings, I could prove this, using the subgroup/subring criterion.
One thing I realized was that such criterion is different for different algebraic structures.
My Question (TL;DR)
- What is a subobject?
- Is the statement in the problem true for general algebraic structure $\square\square$?
- If the statement is true, how would I prove this general statement? (Other than proving for each algebraic structure)
The question became too long, but thanks for reading and any references, hints are always welcome. Thanks in advance.
Note. I keep referring to category theory, but I have not studied category theory. It's just that I got curious and did some googling online for objects and subobjects (since the author won't be defining it) and found many results being related to category theory.