Definition of monoids in slice category

Question

Can someone please tell me what would be an appropriate definition of an internal monoid in the slice category?

Or better yet, suppose you have an object $p : X \rightarrow A \in \mathcal{C}/{A}$ with pullbacks of $n$ copies (in $\mathcal{C}$), does that imply that the $\mathcal{C}/{A}$ is monoidal?

Answer 1

Internal monoids are usually defined in monoidal categories, I assume in an abstract slice category $\mathcal C/A$ the monoidal operation is the categorial direct product, now it is indeed the pullback in $\mathcal C$. If $f,g\in Ob \mathcal C/A$, let's denote their pullback accordingly by $f\times_A g$.

Then an internal monoid is an arrow $m:M\to A$ in $\mathcal C$, together with mappings in $\mathcal C/A$ $$\mu:m\times_A m\to m,\ \eta:1_A\to m$$ such that associativity and unitality holds. So, these are basically $\mu:M\times_A M\to M$ and $\eta:A\to M$ arrows in $\mathcal C$ where $M\times_AM$ is the upper left object of the pullback square of $m,m$. They have to make the arising triangles commute. Associativity is about the equality of the two compositions $M\times_A M\times_A M\to M$ and unitality can be written similarly.