I struggle real had trying to understand the definition of monoid in Category theory. At the first glance, the definition of monoid seems nothing but the definition in abstract algebra, but when I try to understand it further by looking at some examples, nothing really make sense to me. I've looked some related post on the site (like this and this and related post inside) and on wiki, nLab etc. None of these really help. The following is the definition given in wikipedia:
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) $(M, \mu, \eta)$ in a monoidal category $({\bf C}, \otimes, I)$ is an object $M$ together with two morphisms
$\bullet$ $\mu: M \otimes M \rightarrow M$ called multiplication,
$\bullet$ $\eta: I \rightarrow M$ called unit,
such that the pentagon diagram....
follow with diagrams saying the associativity law and identity law hold.
The definition itself looks very straight forward and intuitive as the categorical version of monoid.
As the first example, we have the monoid in $({\bf Set}, \times, I)$, category of set, just as the monoid with underlying set $C$. The multiplication is the binary operation of the monoid and the unit gives the identity of the monoid.
The second example, the one that confusion starts, is monoid in $({\bf Ab}, \otimes_\mathbb{Z}, \mathbb{Z})$, category of Abelian group, which recover (non-commutative) rings. The underlying object is just the abelian group. The multiplication is a bilinear map from the tensor product of abelian group with itself to itself associatively. Finally the unit map $\eta$ from $\mathbb{Z}$ to the object is suppose to give the identity of multiplication, but neither does ring always have the unity element, nor can I see how should it be a function from $\mathbb{Z}$. I feel like I'm missing something really obvious but I cannot get around it. It's very possible for my lack of understanding in tensor product though. I also had a look at the tensor product of abelian group in nLab, but that didn't help either.
Next example I tried to look at is monoid in ${\bf Top}$, the category of topological spaces, which gives the Topological monoid. It sounds like something induced from the categorical monoid and I have no idea what this is as well.
Other examples on the wiki are just totally out of my reach.
Thanks in advance!
Edit:
To make my confusion clearer, let me put down details of what I think it's happening in these examples.
The monoid in $({\bf Set}, \times, \{*\})$:
$\bullet$ An object $C$ is a set.
$\bullet$ The (associative) multiplication morphism $\mu: C\times C \rightarrow C$ defines the distributive, associative binary operation on the set $C$.
$\bullet$ The unit morphism $\eta : \{*\} \rightarrow C$ gives the identity element of the binary operation.
Together we get the monoid in abstract algebra.
The monoid in $({\bf Ab}, \otimes_\mathbb{Z}, \mathbb{Z})$:
$\bullet$ An object $A$ is an abelian group.
$\bullet$ The multiplication morphism $\mu : A\otimes A \rightarrow A$ again gives an associative, distributive binary operation on $A$, which is not the same as addition in $A$. Call it "multiplication".
$\bullet$ The unit morphism $\eta : \mathbb{Z} \rightarrow A$ gives the identity element of "multiplication"
Together we get a unital ring, except we didn't? If we only need "the identity" of multiplication, why do we need $\mathbb{Z}$ as the domain? In PrudiiArca's answer they says the case $I=\{*\}=0$ the trivial group has nothing interesting to discuss, but I can't really understand why.
Monoid in $({\bf Top}, \times, \{*\})$:
Thank to Qi Zhu's comment I think I now understand this example. Somehow in my head this wasn't a topological space with monoid structure but something unimaginable.
$\bullet$ An object $(X, \tau)$ is a topological space.
$\bullet$ I think multiplication and unit morphism are defined similarly to the morphisms in ${\bf Set}$.
and we get a topological space with monoid operation, the topological monoid.