Monoid Action

An left action of a Monoid $M$ on a set $X$ is a function $\alpha:M\times X\to X$ such that

• $\alpha(\varepsilon,x)=x$
• $\alpha(m_{1}\star m_{2},x)=\alpha(m_{1},\alpha(m_{2},x))$

Dually, a right monoid action is a function $\alpha:X\times M\to X$ such that

• $\alpha(x,\varepsilon)=x$
• $\alpha(m_{1}\star m_{2},x)=\alpha(m_{2},\alpha(m_{1},x))$

As Decategorifications

Left and right monoid actions are decategorifications of Copresheaves and Presheaves. More explicitly, a left monoid action is a copresheaf on the Delooping of $M$, and a right monoid action is a presheaf on the Delooping of $M$.