Subsection 5.3.1 (00DB): Foundations

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5.3.1 Foundations

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

Definition 5.3.1.1.1 ▶ The Left Tensor Product of Pointed Sets

The left tensor product of pointed sets is the functor¹

\[ \lhd \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

defined as the composition

\[ \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\overset {\mathsf{id}\times {\text{忘}}}{\to }\mathsf{Sets}_{*}\times \mathsf{Sets}\overset {\mathbf{\beta }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}}}{\to }\mathsf{Sets}\times \mathsf{Sets}_{*}\overset {\odot }{\to }\mathsf{Sets}_{*}, \]

where:

${\text{忘}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets}$ is the forgetful functor from pointed sets to sets.
${\mathbf{\beta }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}\times \mathsf{Sets}_{*}$ is the braiding of $\mathsf{Cats}_{\mathsf{2}}$, i.e. the functor witnessing the isomorphism

\[ \mathsf{Sets}_{*}\times \mathsf{Sets}\cong \mathsf{Sets}\times \mathsf{Sets}_{*}. \]
$\odot \colon \mathsf{Sets}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*}$ is the tensor functor of Item 1 of Proposition 5.2.1.1.6.

¹Further Notation: Also written $\lhd _{\mathsf{Sets}_{*}}$.

Remark 5.3.1.1.2 ▶ Unwinding Definition 5.3.1.1.1: Universal Property I

The left tensor product of pointed sets satisfies the following natural bijection:

\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ). \]

That is to say, the following data are in natural bijection:

Pointed maps $f\colon X\lhd Y\to Z$.
Maps of sets $f\colon X\times Y\to Z$ satisfying $f\webleft (x_{0},y\webright )=z_{0}$ for each $y\in Y$.

Remark 5.3.1.1.3 ▶ Unwinding Definition 5.3.1.1.1: Universal Property II

The left tensor product of pointed sets may be described as follows:

The left tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pair $\webleft (\webleft (X\lhd Y,x_{0}\lhd y_{0}\webright ),\iota \webright )$ consisting of
- A pointed set $\webleft (X\lhd Y,x_{0}\lhd y_{0}\webright )$;
- A left bilinear morphism of pointed sets $\iota \colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to X\lhd Y$;
satisfying the following universal property:
- Given another such pair $\webleft (\webleft (Z,z_{0}\webright ),f\webright )$ consisting of
  - A pointed set $\webleft (Z,z_{0}\webright )$;
  - A left bilinear morphism of pointed sets $f\colon \webleft (X\times Y,\webleft (x_{0},y_{0}\webright )\webright )\to X\lhd Y$;
  there exists a unique morphism of pointed sets $X\lhd Y\overset {\exists !}{\to }Z$ making the diagram
  
  commute.

Construction 5.3.1.1.4 ▶ The Left Tensor Product of Pointed Sets

In detail, the left tensor product of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pointed set $\webleft (X\lhd Y,\webleft [x_{0}\webright ]\webright )$ consisting of:

The Underlying Set. The set $X\lhd Y$ defined by

\begin{align*} X\lhd Y & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\lvert Y\right\rvert \odot X\\ & \cong \bigvee _{y\in Y}\webleft (X,x_{0}\webright ), \end{align*}

where $\left\lvert Y\right\rvert $ denotes the underlying set of $\webleft (Y,y_{0}\webright )$.
The Underlying Basepoint. The point $\webleft [\webleft (y_{0},x_{0}\webright )\webright ]$ of $\bigvee _{y\in Y}\webleft (X,x_{0}\webright )$, which is equal to $\webleft [\webleft (y,x_{0}\webright )\webright ]$ for any $y\in Y$.

Proof of Construction 5.3.1.1.4.

Since $\bigvee _{y\in Y}\webleft (X,x_{0}\webright )$ is defined as the quotient of $\coprod _{y\in Y}X$ by the equivalence relation $R$ generated by declaring $\webleft (y,x\webright )\sim \webleft (y',x'\webright )$ if $x=x'=x_{0}$, we have, by , , a natural bijection

\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright ), \]

where $\textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ is the set

\[ \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \textup{Hom}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )\ \middle |\ \begin{aligned} & \text{for each $x,y\in X$, if}\\ & \text{$\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$, then}\\ & \text{$f\webleft (y,x\webright )=f\webleft (y',x'\webright )$}\end{aligned} \webright\} . \]

However, the condition $\webleft (y,x\webright )\sim _{R}\webleft (y',x'\webright )$ only holds when:

We have $x=x'$ and $y=y'$.
We have $x=x'=x_{0}$.

So, given $f\in \textup{Hom}_{\mathsf{Sets}}\webleft (\coprod _{y\in Y}X,Z\webright )$ with a corresponding $\overline{f}\colon X\lhd Y\to Z$, the latter case above implies

\begin{align*} f\webleft (\webleft [\webleft (y,x_{0}\webright )\webright ]\webright ) & = f\webleft (\webleft [\webleft (y',x_{0}\webright )\webright ]\webright )\\ & = f\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright ), \end{align*}

and since $\overline{f}\colon X\lhd Y\to Z$ is a pointed map, we have

\begin{align*} f\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright ) & = \overline{f}\webleft (\webleft [\webleft (y_{0},x_{0}\webright )\webright ]\webright )\\ & = z_{0}. \end{align*}

Thus the elements $f$ in $\textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )$ are precisely those functions $f\colon X\times Y\to Z$ satisfying the equality

\[ f\webleft (x_{0},y\webright )=z_{0} \]

for each $y\in Y$, giving an equality

\[ \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright )=\textup{Hom}^{\otimes ,\mathrm{L}}_{\mathsf{Sets}_{*}}\webleft (X\times Y,Z\webright ) \]

of sets, which when composed with our earlier isomorphism

\[ \mathsf{Sets}_{*}\webleft (X\lhd Y,Z\webright ) \cong \textup{Hom}^{R}_{\mathsf{Sets}}\webleft (X\times Y,Z\webright ), \]

gives our desired natural bijection, finishing the proof.

Notation 5.3.1.1.5 ▶ Elements of Left Tensor Products of Pointed Sets

We write¹ $x\lhd y$ for the element $\webleft [\webleft (y,x\webright )\webright ]$ of

\[ X\lhd Y\cong \left\lvert Y\right\rvert \odot X. \]

¹Further Notation: Also written $x\lhd _{\mathsf{Sets}_{*}}y$.

Remark 5.3.1.1.6 ▶ Basepoints of Left Tensor Products of Pointed Sets

Employing the notation introduced in Notation 5.3.1.1.5, we have

\[ x_{0}\lhd y_{0}=x_{0}\lhd y \]

for each $y\in Y$, and

\[ x_{0}\lhd y=x_{0}\lhd y' \]

for each $y,y'\in Y$.

Proposition 5.3.1.1.7 ▶ Properties of Left Tensor Products of Pointed Sets

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto X\lhd Y$ define functors
\[ \begin{array}{ccc} X\lhd -\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -\lhd Y\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ -_{1}\lhd -_{2}\colon \mkern -15mu & \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]

In particular, given pointed maps

\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (A,a_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (B,b_{0}\webright ), \end{align*}

the induced map

\[ f\lhd g\colon X\lhd Y\to A\lhd B \]

is given by

\[ \webleft [f\lhd g\webright ]\webleft (x\lhd y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (x\webright )\lhd g\webleft (y\webright ) \]

for each $x\lhd y\in X\lhd Y$.
Adjointness I. We have an adjunction
witnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\webleft [Y,Z\webright ]^{\lhd }_{\mathsf{Sets}_{*}}\webright ) \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, where $\webleft [X,Y\webright ]^{\lhd }_{\mathsf{Sets}_{*}}$ is the pointed set of Definition 5.3.2.1.1.
Adjointness II. The functor
\[ X\lhd -\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

does not admit a right adjoint.
Adjointness III. We have a ${\text{忘}}$-relative adjunction
witnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}}\webleft (|Y|,\mathsf{Sets}_{*}\webleft (X,Z\webright )\webright ) \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

Proof of Proposition 5.3.1.1.7.

Item 1: Functoriality

This follows from the definition of $\lhd $ as a composition of functors (Definition 5.3.1.1.1).

Item 2: Adjointness I

This follows from Item 3 of Proposition 5.2.1.1.6.

Item 3: Adjointness II

For $X\lhd -$ to admit a right adjoint would require it to preserve colimits by

. However, we have

\begin{align*} X\lhd \text{pt}& \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|\text{pt}|\odot X\\ & \cong X\\ & \ncong \text{pt}, \end{align*}

and thus we see that $X\lhd -$ does not have a right adjoint.

Item 4: Adjointness III

This follows from Item 2 of Proposition 5.2.1.1.6.

Remark 5.3.1.1.8 ▶ On the Failure of $X\lhd -$ To Be a Left Adjoint

Here is some intuition on why $X\lhd -$ fails to be a left adjoint. Item 4 of Proposition 5.3.1.1.7 states that we have a natural bijection

\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}}\webleft (|Y|,\mathsf{Sets}_{*}\webleft (X,Z\webright )\webright ), \]

so it would be reasonable to wonder whether a natural bijection of the form

\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\lhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (Y,\textbf{Sets}_{*}\webleft (X,Z\webright )\webright ), \]

also holds, which would give $X\lhd -\dashv \textbf{Sets}_{*}\webleft (X,-\webright )$. However, such a bijection would require every map

\[ f\colon X\lhd Y\to Z \]

to satisfy

\[ f\webleft (x\lhd y_{0}\webright )=z_{0} \]

for each $x\in X$, whereas we are imposing such a basepoint preservation condition only for elements of the form $x_{0}\lhd y$. Thus $\textbf{Sets}_{*}\webleft (X,-\webright )$ can’t be a right adjoint for $X\lhd -$, and as shown byItem 3 of Proposition 5.3.1.1.7, no functor can.¹

¹The functor $\textbf{Sets}_{*}\webleft (X,-\webright )$ is instead right adjoint to $X\wedge -$, the smash product of pointed sets of Definition 5.5.1.1.1. See Item 2 of Proposition 5.5.1.1.10.

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