Monoid Object

Let \((\cat{C}, \otimes, I)\) be a monoidal category.

Then a Monoid Object consists of an object \(M\in\cat{C}\) equipped with

• a multiplication map \(\mu\colon M\otimes M\to M\)
• a unit \(\eta\colon I\to M\)

such that

• the pentagon diagram holds (i.e., the multiplication map is associative)

\begin{equation} \require{AMScd} \begin{CD} (M\otimes M)\otimes M @>e>> M\otimes(M\otimes M) \\ @V{\mu\otimes1}VV @VV{1\otimes\mu}V \\ M\otimes M @. M\otimes M \\ @V{\mu}VV @V{\mu}VV \\ M @= M \end{CD} \end{equation}

• the left and right unit

\begin{equation} \begin{CD} I\otimes M @>{\eta\otimes1}>> M\otimes M @<{1\otimes\eta}<< M\otimes I\\ @V{\lambda}VV @V{\mu}VV @V{\rho}VV\\ M @= M @= M \end{CD} \end{equation}

Given two monoids \((M,\mu,\eta)\) and \((M', \mu', \eta')\) in a monoidal category \(\cat{C}\), a morphism \(f\colon M\to M'\) is a Morphism of Monoids when

• \(f\circ\mu = \mu'\circ(f\otimes f)\)
• \(f\circ\eta = \eta'\)

• In Set, a monoid object is just a familiar Monoid we all know and love
• In Ab, a monoid object is a ring
• If \(\cat{C}\) is a monoidal category, then we can form a category of monoid objects internal to \(\cat{C}\) and this is denoted \(\mathrm{Mon}(\cat{C})\).