Remark 7.3.1.1.3 (00DE) — The Clowder Project

7.3.1 Foundations

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

Definition 7.3.1.1.1 ▶ 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 7.2.1.1.6.

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

Remark 7.3.1.1.2 ▶ Unwinding Definition 7.3.1.1.1: Universal Property I

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

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

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

1.
Pointed maps $f\colon X\lhd Y\to Z$.
2.
Maps of sets $f\colon X\times Y\to Z$ satisfying $f(x_{0},y)=z_{0}$ for each $y\in Y$.

Remark 7.3.1.1.3 ▶ Unwinding Definition 7.3.1.1.1: Universal Property II

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

•
The left tensor product of $(X,x_{0})$ and $(Y,y_{0})$ is the pair $((X\lhd Y,x_{0}\lhd y_{0}),\iota )$ consisting of
- •
  A pointed set $(X\lhd Y,x_{0}\lhd y_{0})$;
- •
  A left bilinear morphism of pointed sets $\iota \colon (X\times Y,(x_{0},y_{0}))\to X\lhd Y$;
satisfying the following universal property:
- (★)

Construction 7.3.1.1.4 ▶ The Left Tensor Product of Pointed Sets

In detail, the left tensor product of $(X,x_{0})$ and $(Y,y_{0})$ is the pointed set $(X\lhd Y,[x_{0}])$ 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}(X,x_{0}), \end{align*}

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

Proof of Construction 7.3.1.1.4.

Since $\bigvee _{y\in Y}(X,x_{0})$ is defined as the quotient of $\coprod _{y\in Y}X$ by the equivalence relation $R$ generated by declaring $(y,x)\sim (y',x')$ if $x=x'=x_{0}$, we have, by Chapter 10: Conditions on Relations, , a natural bijection

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

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

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

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

1.
We have $x=x'$ and $y=y'$.
2.
We have $x=x'=x_{0}$.

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

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

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

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

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

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

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

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

of sets, which when composed with our earlier isomorphism

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

gives our desired natural bijection, finishing the proof.

Notation 7.3.1.1.5 ▶ Elements of Left Tensor Products of Pointed Sets

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

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

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

Remark 7.3.1.1.6 ▶ Basepoints of Left Tensor Products of Pointed Sets

Employing the notation introduced in Notation 7.3.1.1.5, 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 7.3.1.1.7 ▶ Properties of Left Tensor Products of Pointed Sets

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

1.
Functoriality. The assignments $X,Y,(X,Y)\mapsto X\lhd Y$ 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 (X,x_{0}) \to (A,a_{0}),\\ g & \colon (Y,y_{0}) \to (B,b_{0}), \end{align*}

the induced map

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

is given by

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

for each $x\lhd y\in X\lhd Y$.
2.
Adjointness I. We have an adjunction
witnessed by a bijection of sets

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\lhd Y,Z)\cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X,[Y,Z]^{\lhd }_{\mathsf{Sets}_{*}}) \]

natural in $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$, where $[X,Y]^{\lhd }_{\mathsf{Sets}_{*}}$ is the pointed set of Definition 7.3.2.1.1.
3.
Adjointness II. The functor

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

does not admit a right adjoint.
4.
Adjointness III. We have a ${\text{忘}}$-relative adjunction
witnessed by a bijection of sets

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\lhd Y,Z)\cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(|Y|,\mathsf{Sets}_{*}(X,Z)) \]

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

Proof of Proposition 7.3.1.1.7.

Item 1: Functoriality

This follows from the definition of $\lhd $ as a composition of functors (Definition 7.3.1.1.1).

Item 2: Adjointness I

This follows from Item 3 of Proposition 7.2.1.1.6.

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 \mathrm{pt}& \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}|\mathrm{pt}|\odot X\\ & \cong X\\ & \ncong \mathrm{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 7.2.1.1.6.

Remark 7.3.1.1.8 ▶ 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 7.3.1.1.7 states that we have a natural bijection

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\lhd Y,Z)\cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(|Y|,\mathsf{Sets}_{*}(X,Z)), \]

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

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\lhd Y,Z)\cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(Y,\boldsymbol {\mathsf{Sets}}_{*}(X,Z)), \]

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

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

to satisfy

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

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 $\boldsymbol {\mathsf{Sets}}_{*}(X,-)$ can’t be a right adjoint for $X\lhd -$, and as shown by Item 3 of Proposition 7.3.1.1.7, no functor can.¹

¹The functor $\boldsymbol {\mathsf{Sets}}_{*}(X,-)$ is instead right adjoint to $X\wedge -$, the smash product of pointed sets of Definition 7.5.1.1.1. See Item 2 of Proposition 7.5.1.1.10.

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