Monoid object

In category theory, the monoid object is a generalization of the concept of the monoid .

definition

Let there be a monoidal category with the functor , the unit object , the natural transformation with the components , and the natural transformations and . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {-}\otimes {-}\colon {\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}}$ ${\displaystyle I\in |{\mathcal {C}}|}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\to A\otimes (B\otimes C)}$ ${\displaystyle \lambda \colon (I\otimes {-})\to \mathrm {Id} _{\mathcal {C}}}$ ${\displaystyle \rho \colon ({-}\otimes I)\to \mathrm {Id} _{\mathcal {C}}}$

A monoid object is now an object together with two arrows and , for which the equations ${\displaystyle M\in |{\mathcal {C}}|}$ ${\displaystyle \eta \colon I\to M}$ ${\displaystyle \mu \colon M\otimes M\to M}$

• ${\displaystyle \mu \circ (\mu \otimes M)=\mu \circ (M\otimes \mu )\circ \alpha _{M,M,M}\ \colon (M\otimes M)\otimes M\to M}$,
• ${\displaystyle \mu \circ (M\otimes \eta )=\rho _{M}\ \colon M\otimes I\to M}$ and
• ${\displaystyle \mu \circ (\eta \otimes M)=\lambda _{M}\ \colon I\otimes M\to M}$

be valid.

Examples

• Monoids are monoid objects in the category of sets, which is monoidal with the Cartesian product .
• Group objects are monoid objects.
• In the category of monoids (monoidal through direct products), monoid objects are commutative monoids.
• Is any category, so that is functor with the Funktorkomposition monoidal. Monoid objects in are monads . ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\mathcal {C}}}$ ${\displaystyle {\mathcal {C}}^{\mathcal {C}}}$

literature

• Saunders Mac Lane: Categories for the Working Mathematician . 2nd Edition. Springer-Verlag, 1997, p. 170 f .