Item 1 (00GW) — The Clowder Project

Table of Contents
Part 2: Sets
Chapter 7: Tensor Products of Pointed Sets
Section 7.5: The Smash Product of Pointed Sets
Subsection 7.5.8: The Diagonal
Item 1 (cite)

2 tags refer to this tag

7.5.8 The Diagonal

Definition 7.5.8.1.1 ▶ The Diagonal of $\wedge $

The diagonal of the smash product of pointed sets is the natural transformation

whose component

\[ \Delta ^{\wedge }_{X}\colon \webleft (X,x_{0}\webright )\to \webleft (X\wedge X,x_{0}\wedge x_{0}\webright ) \]

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

in $\mathsf{Sets}_{*}$, and thus by

\[ \Delta ^{\wedge }_{X}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x\wedge x \]

for each $x\in X$.

Proof of Definition 7.5.8.1.1.

Being a Morphism of Pointed Sets

We have

\[ \Delta ^{\wedge }_{X}\webleft (x_{0}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x_{0}\wedge x_{0}, \]

and thus $\Delta ^{\wedge }_{X}$ is a morphism of pointed sets.

Naturality

We need to show that, given a morphism of pointed sets

\[ f\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ), \]

the diagram

commutes. Indeed, this diagram acts on elements as

and hence indeed commutes, showing $\Delta ^{\wedge }$ to be natural.

Proposition 7.5.8.1.2 ▶ Properties of the Diagonal of $\wedge $

Let $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$.

1.
Monoidality. The diagonal

\[ \Delta ^{\wedge }\colon \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\Longrightarrow {\wedge }\circ {\Delta ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}}}, \]

of the smash product of pointed sets is a monoidal natural transformation:
1. (a)
  Compatibility With Strong Monoidality Constraints. For each $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$, the diagram
  commutes.
2. (b)
  Compatibility With Strong Unitality Constraints. The diagrams
  commute, i.e. we have
  
  \begin{align*} \Delta ^{\wedge }_{S^{0}} & = \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}}\\ & = \rho ^{\mathsf{Sets}_{*},-1}_{S^{0}}, \end{align*}
  
  where we recall that the equalities
  
  \begin{align*} \lambda ^{\mathsf{Sets}_{*}}_{S^{0}} & = \rho ^{\mathsf{Sets}_{*}}_{S^{0}},\\ \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}} & = \rho ^{\mathsf{Sets}_{*},-1}_{S^{0}}\end{align*}
  
  are always true in any monoidal category by , of .

The Diagonal of the Unit. The component

\[ \Delta ^{\wedge }_{S^{0}} \colon S^{0} \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }S^{0}\wedge S^{0} \]

of $\Delta ^{\wedge }$ at $S^{0}$ is an isomorphism.

Proof of Proposition 7.5.8.1.2.

Item 1: Monoidality

We claim that $\Delta ^{\wedge }$ is indeed monoidal:

1.
Item 1a: Compatibility With Strong Monoidality Constraints: We need to show that the diagram
commutes. Indeed, this diagram acts on elements as
and hence indeed commutes.
2.
Item 1b: Compatibility With Strong Unitality Constraints: As shown in the proof of Definition 7.5.5.1.1, the inverse of the left unitor of $\mathsf{Sets}_{*}$ with respect to to the smash product of pointed sets at $\webleft (X,x_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\mathsf{Sets}_{*}\webright )$ is given by

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

for each $x\in X$, so when $X=S^{0}$, we have

\begin{align*} \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}}\webleft (0\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}1\wedge 0,\\ \lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}}\webleft (1\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}1\wedge 1. \end{align*}

But since $1\wedge 0=0\wedge 0$ and

\begin{align*} \Delta ^{\wedge }_{S^{0}}\webleft (0\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}0\wedge 0,\\ \Delta ^{\wedge }_{S^{0}}\webleft (1\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}1\wedge 1, \end{align*}

it follows that we indeed have $\Delta ^{\wedge }_{S^{0}}=\lambda ^{\mathsf{Sets}_{*},-1}_{S^{0}}$.

This finishes the proof.

Item 2: The Diagonal of the Unit

This follows from Item 1 and the invertibility of the left/right unitor of $\mathsf{Sets}_{*}$ with respect to $\wedge $, proved in the proof of Definition 7.5.5.1.1 for the left unitor or the proof of Definition 7.5.6.1.1 for the right unitor.

Statistics Cite

Noticed something off, or have any comments? Feel free to reach out!

•Email
•Discord

You can also use the contact form below: