Definition 7.4.5.1.1 (00FE): The Right Skew Left Unitor of $\rhd $

Table of Contents
Part 2: Sets
Chapter 7: Tensor Products of Pointed Sets
Section 7.4: The Right Tensor Product of Pointed Sets
Subsection 7.4.5: The Right Skew Left Unitor
Definition 7.4.5.1.1: The Right Skew Left Unitor of $\rhd $ (cite)

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7.4.5 The Right Skew Left Unitor

Definition 7.4.5.1.1 ▶ The Right Skew Left Unitor of $\rhd $

The skew left unitor of the right tensor product of pointed sets is the natural transformation

whose component

\[ \lambda ^{\mathsf{Sets}_{*},\rhd }_{X} \colon X \to S^{0}\rhd X \]

at $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ is given by the composition

\begin{align*} X & \rightarrow X\vee X\\ & \cong |S^{0}|\odot X\\ & \cong S^{0}\rhd X, \end{align*}

where $X\to X\vee X$ is the map sending $X$ to the second factor of $X$ in $X\vee X$.

Proof of Definition 7.4.5.1.1.

(Proven below in a bit.)

Remark 7.4.5.1.2 ▶ Unwinding Definition 7.4.5.1.1

In other words, $\smash {\lambda ^{\mathsf{Sets}_{*},\rhd }_{X}}$ acts on elements as

\[ \lambda ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (1,x\webright )\webright ] \]

i.e. by

\[ \lambda ^{\mathsf{Sets}_{*},\rhd }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}1\rhd x \]

for each $x\in X$.

Remark 7.4.5.1.3 ▶ Non-Invertibility of the Skew Left Unitor of $\rhd $

The morphism $\smash {\lambda ^{\mathsf{Sets}_{*},\rhd }_{X}}$ is non-invertible, as it is non-surjective when viewed as a map of sets, since the elements $0\rhd x$ of $S^{0}\rhd X$ with $x\neq x_{0}$ are outside the image of $\lambda ^{\mathsf{Sets}_{*},\rhd }_{X}$, which sends $x$ to $1\rhd x$.

Proof of Definition 7.4.5.1.1.