The category of monoids on $\smash {\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )}$ is isomorphic to the category of “monoids with right zero”1 and morphisms between them.


1A monoid with right zero is defined similarly as the monoids with zero of . Succinctly, they are monoids $\webleft (A,\mu _{A},\eta _{A}\webright )$ with a special element $0_{A}$ satisfying

\[ 0_{A}a=0_{A} \]

for each $a\in A$.

Monoids on $\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )$
A monoid on $\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )$ consists of:
  • The Underlying Object. A pointed set $\webleft (A,0_{A}\webright )$.
  • The Multiplication Morphism. A morphism of pointed sets

    \[ \mu _{A}\colon A\rhd A\to A, \]

    determining a right 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:

  1. Associativity. The diagram
  2. Left Unitality. The diagram

    commutes.

  3. Right Unitality. The diagram

    commutes.

Being a right-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:

  1. Associativity. The associativity condition acts as
    This gives
    \[ \webleft (ab\webright )c=a\webleft (bc\webright ) \]

    for each $a,b,c\in A$.

  2. Left Unitality. The left unitality condition acts as
    This gives
    \[ 1_{A}a=a \]

    for each $a\in A$.

  3. Right Unitality. The right unitality condition acts:
    1. On $1\rhd 0$ as
    2. On $a\rhd 1$ as

    This gives

    \begin{align*} a1_{A} & = a,\\ a0_{A} & = 0_{A} \end{align*}

    for each $a\in A$.

Thus we see that monoids with respect to $\rhd $ are exactly monoids with right zero.

Morphisms of Monoids on $\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )$
A morphism of monoids on $\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )$ 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:

  1. Compatibility With the Multiplication Morphisms. The diagram

    commutes.

  2. 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 right zero.

Identities and Composition
Similarly, the identities and composition of $\mathsf{Mon}\webleft (\mathsf{Sets}_{*},\rhd ,S^{0}\webright )$ can be easily seen to agree with those of monoids with right zero, which finishes the proof.


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