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.
Monoids on $\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$
A monoid on $\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$ consists of: - 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 $A$.
satisfying the following conditions:
-
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
\[ 0_{A}a=0_{A} \]
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.
Morphisms of Monoids on $\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$
A morphism of monoids on $\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$ from $\webleft (A,\mu _{A},\eta _{A},0_{A}\webright )$ to $\webleft (B,\mu _{B},\eta _{B},0_{B}\webright )$ is a morphism of pointed sets
\[ f\colon \webleft (A,0_{A}\webright )\to \webleft (B,0_{B}\webright ) \]
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
\begin{gather*} f\webleft (ab\webright ) = f\webleft (a\webright )f\webleft (b\webright ),\\ \begin{aligned} f\webleft (0_{A}\webright ) & = 0_{B},\\ f\webleft (1_{A}\webright ) & = 1_{B}, \end{aligned}\end{gather*}
for each $a,b\in A$, which is exactly a morphism of monoids with left zero.
Identities and Composition
Similarly, the identities and composition of $\mathsf{Mon}\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$ can be easily seen to agree with those of monoids with left zero, which finishes the proof.
. Succinctly, they are monoids $\webleft (A,\mu _{A},\eta _{A}\webright )$ with a special element $0_{A}$ satisfying
