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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?

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why do you use striken text, instead of properly editing what you want to say? – magma Feb 14 '13 at 17:54
up vote 2 down vote accepted

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.

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