4.3.1 Foundations

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

Definition 4.3.1.1.1 ▶ The Left Tensor Product of Pointed Sets

The left tensor product of pointed sets is the functor[1]

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

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.
  • ${\mathbf{\beta }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}}}\colon \mathsf{Sets}_{*}\times \mathsf{Sets}\xrightarrow {\cong }\mathsf{Sets}\times \mathsf{Sets}_{*}$ 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 4.2.1.1.6.

Remark 4.3.1.1.2 ▶ Unwinding Definition 4.3.1.1.1: 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:

  1. Pointed maps $f\colon X\lhd Y\to Z$.
  2. 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 4.3.1.1.3 ▶ Unwinding Definition 4.3.1.1.1: 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 4.3.1.1.4 ▶ 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 $y\in Y$.

Notation 4.3.1.1.5 ▶ Elements of Left Tensor Products of Pointed Sets

We write[2] $x\lhd y$ for the element $\webleft [\webleft (y,x\webright )\webright ]$ of

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

Remark 4.3.1.1.6 ▶ Basepoints of Left Tensor Products of Pointed Sets

Employing the notation introduced in Notation 4.3.1.1.5, 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 4.3.1.1.7 ▶ Properties of Left Tensor Products of Pointed Sets

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

  1. Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto X\lhd Y$ define functors
    \begin{gather*} \begin{aligned} X\lhd - & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -\lhd Y & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ \end{aligned}\\ -_{1}\lhd -_{2} \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{gather*}

    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 $x\lhd y\in X\lhd Y$.

  2. 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 4.3.2.1.1.

  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 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 4.3.1.1.7.
Item 1: Functoriality
Clear.
Item 2: Adjointness I
This follows from Item 3 of Proposition 4.2.1.1.6.
Item 3: Adjointness II
For $X\lhd -$ to admit a right adjoint would require it to preserve colimits by of . 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 4.2.1.1.6.

Remark 4.3.1.1.8 ▶ 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 4.3.1.1.7 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 by Item 3 of Proposition 4.3.1.1.7, no functor can.[3]

Footnotes

[1] Further Notation: Also written $\lhd _{\mathsf{Sets}_{*}}$.
[2] Further Notation: Also written $x\lhd _{\mathsf{Sets}_{*}}y$.
[3] The functor $\textbf{Sets}_{*}\webleft (X,-\webright )$ is instead right adjoint to $X\wedge -$, the smash product of pointed sets of Definition 4.5.1.1.1. See Item 2 of Proposition 4.5.1.1.9.
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