The category of monoids on $\smash {\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )}$ is isomorphic to the category of “monoids with left zero”[1] and morphisms between them.
- The Underlying Object. A pointed set $\webleft (A,0_{A}\webright )$.
- The Multiplication Morphism. A morphism of pointed sets
\[ \mu _{A}\colon A\lhd A\to A, \]
determining a left bilinear morphism of pointed sets
- The Unit Morphism. A morphism of pointed sets
\[ \eta _{A}\colon S^{0}\to A \]
picking an element $1_{A}$ of $A$.
-
Associativity. The diagram
-
Left Unitality. The diagram
commutes.
-
Right Unitality. The diagram
commutes.
Being a left-bilinear morphism of pointed sets, the multiplication map satisfies
for each $a\in A$. Now, the associativity, left unitality, and right unitality conditions act on elements as follows:
-
Associativity. The associativity condition acts as This gives
\[ \webleft (ab\webright )c=a\webleft (bc\webright ) \]
for each $a,b,c\in A$.
-
Left Unitality. The left unitality condition acts:
- On $0\lhd a$ as
- On $1\lhd a$ as
This gives
\begin{align*} 1_{A}a & = a,\\ 0_{A}a & = 0_{A} \end{align*}for each $a\in A$.
-
Right Unitality. The right unitality condition acts as This gives
\[ a1_{A}=a \]
for each $a\in A$.
Thus we see that monoids with respect to $\lhd $ are exactly monoids with left zero.
satisfying the following conditions:
-
Compatibility With the Multiplication Morphisms. The diagram
commutes.
-
Compatibility With the Unit Morphisms. The diagram
commutes.
These act on elements as
and and giving
for each $a,b\in A$, which is exactly a morphism of monoids with left zero.