Subsection 7.3.9 (00EG): Monoids With Respect to the Left Tensor Product of Pointed Sets

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.

¹A monoid with left 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 $\webleft (\mathsf{Sets}_{*},\lhd ,S^{0}\webright )$

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:

1.
Associativity. The diagram
2.
Left Unitality. The diagram
commutes.
3.
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:

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:
1. (a)
  On $0\lhd a$ as
2. (b)
  On $1\lhd a$ as
This gives

\begin{align*} 1_{A}a & = a,\\ 0_{A}a & = 0_{A} \end{align*}

for each $a\in A$.
3.
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:

1.
Compatibility With the Multiplication Morphisms. The diagram
commutes.
2.
Compatibility With the Unit Morphisms. The diagram
commutes.

These act on elements as

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.

The Clowder Project

7.3.9 Monoids With Respect to the Left Tensor Product of Pointed Sets