Subsection 4.2.2 (00CX): Cotensors of Pointed Sets by Sets

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4.2.2 Cotensors of Pointed Sets by Sets

Let $\webleft (X,x_{0}\webright )$ be a pointed set and let $A$ be a set.

Definition 4.2.2.1.1 ▶ Cotensors of Pointed Sets by Sets

The cotensor of $\webleft (X,x_{0}\webright )$ by $A$^[1] is the pointed set ^[2] $A\pitchfork \webleft (X,x_{0}\webright )$ satisfying the following universal property:

We have a bijection

\[ \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]

natural in $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

Remark 4.2.2.1.2 ▶ Unwinding Definition 4.2.2.1.1

The universal property of Definition 4.2.2.1.1 is equivalent to the following one:

We have a bijection

\[ \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright ) \cong \mathsf{Sets}^{\otimes }_{\mathbb {E}_{0}}\webleft (A\times K,X\webright ), \]

natural in $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, where $\mathsf{Sets}^{\otimes }_{\mathbb {E}_{0}}\webleft (A\times K,X\webright )$ is the set defined by

\[ \mathsf{Sets}^{\otimes }_{\mathbb {E}_{0}}\webleft (A\times K,X\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\in \mathsf{Sets}\webleft (A\times K,X\webright )\ \middle |\ \begin{aligned} & \text{for each $a\in A$, we}\\ & \text{have $f\webleft (a,k_{0}\webright )=x_{0}$}\end{aligned} \webright\} . \]

Proof of Remark 4.2.2.1.2.

This follows from the bijection

\[ \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright )\cong \mathsf{Sets}^{\otimes }_{\mathbb {E}_{0}}\webleft (A\times K,X\webright ), \]

natural in $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ constructed in the proof of Remark 4.2.1.1.2.

Construction 4.2.2.1.3 ▶ Construction of Cotensors of Pointed Sets by Sets

Concretely, the cotensor of $\webleft (X,x_{0}\webright )$ by $A$ is the pointed set $A\pitchfork \webleft (X,x_{0}\webright )$ consisting of:

The Underlying Set. The set $A\pitchfork X$ given by

\[ A\pitchfork X\cong \bigwedge _{a\in A}\webleft (X,x_{0}\webright ), \]

where $\bigwedge _{a\in A}\webleft (X,x_{0}\webright )$ is the smash product of the $A$-indexed family $\webleft (\webleft (X,x_{0}\webright )\webright )_{a\in A}$ of Definition 4.6.1.1.1.
The Basepoint. The point $\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]=\webleft [\webleft (x_{0},x_{0},x_{0},\ldots \webright )\webright ]$ of $\bigwedge _{a\in A}\webleft (X,x_{0}\webright )$.

Proof of Construction 4.2.2.1.3.

We claim we have a bijection

\[ \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]

natural in $\webleft (K,k_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.

Map I. We define a map

\[ \Phi _{K}\colon \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )\to \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]

by sending a morphism of pointed sets

\[ \xi \colon \webleft (K,k_{0}\webright )\to \webleft (A\pitchfork X,\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]\webright ) \]

to the map of sets
where

\[ \xi _{a}\colon \webleft (K,k_{0}\webright )\to \webleft (X,x_{0}\webright ) \]

is the morphism of pointed sets defined by

\[ \xi _{a}\webleft (k\webright )=\begin{cases} x^{k}_{a} & \text{if $\xi \webleft (k\webright )\neq \webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$,}\\ x_{0} & \text{if $\xi \webleft (k\webright )=\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$}\end{cases} \]

for each $k\in K$, where $x^{k}_{a}$ is the $a$th component of $\xi \webleft (k\webright )=\webleft [\webleft (x^{k}_{a}\webright )_{a\in A}\webright ]$. Note that:
1. The definition of $\xi _{a}\webleft (k\webright )$ is independent of the choice of equivalence class. Indeed, suppose we have
  \begin{align*} \xi \webleft (k\webright ) & = \webleft [\webleft (x^{k}_{a}\webright )_{a\in A}\webright ]\\ & = \webleft [\webleft (y^{k}_{a}\webright )_{a\in A}\webright ] \end{align*}
  
  with $x^{k}_{a}\neq y^{k}_{a}$ for some $a\in A$. Then there exist $a_{x},a_{y}\in A$ such that $x^{k}_{a_{x}}=y^{k}_{a_{y}}=x_{0}$. The equivalence relation $\mathord {\sim }$ on $\prod _{a\in A}X$ then forces
  
  \begin{align*} \webleft [\webleft (x^{k}_{a}\webright )_{a\in A}\webright ] & = \webleft [\webleft (x_{0}\webright )_{a\in A}\webright ],\\ \webleft [\webleft (y^{k}_{a}\webright )_{a\in A}\webright ] & = \webleft [\webleft (x_{0}\webright )_{a\in A}\webright ], \end{align*}
  
  however, and $\xi _{a}\webleft (k\webright )$ is defined to be $x_{0}$ in this case.
2. The map $\xi _{a}$ is indeed a morphism of pointed sets, as we have
  \[ \xi _{a}\webleft (k_{0}\webright )=x_{0} \]
  
  since $\xi \webleft (k_{0}\webright )=\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$ as $\xi $ is a morphism of pointed sets and $\xi _{a}\webleft (k_{0}\webright )$, defined to be the $a$th component of $\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$, is equal to $x_{0}$.
Map II. We define a map

\[ \Psi _{K}\colon \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright )\to \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright ), \]

given by sending a map
to the morphism of pointed sets

\[ \xi ^{\dagger }\colon \webleft (K,k_{0}\webright )\to \webleft (A\pitchfork X,\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]\webright ) \]

defined by

\[ \xi ^{\dagger }\webleft (k\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\xi _{a}\webleft (k\webright )\webright )_{a\in A}\webright ] \]

for each $k\in K$. Note that $\xi ^{\dagger }$ is indeed a morphism of pointed sets, as we have

\begin{align*} \xi ^{\dagger }\webleft (k_{0}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\xi _{a}\webleft (k_{0}\webright )\webright )_{a\in A}\webright ]\\ & = x_{0}, \end{align*}

where we have used that $\xi _{a}\in \mathsf{Sets}_{*}\webleft (K,X\webright )$ is a morphism of pointed sets for each $a\in A$.
Naturality of $\Psi $. We need to show that, given a morphism of pointed sets

\[ \phi \colon \webleft (K,k_{0}\webright )\to \webleft (K',k'_{0}\webright ), \]

the diagram

commutes. Indeed, given a map of sets
we have

\begin{align*} \webleft [\Psi _{K}\circ \webleft (\phi ^{*}\webright )_{*}\webright ]\webleft (\xi \webright ) & = \Psi _{K}\webleft (\webleft (\phi ^{*}\webright )_{*}\webleft (\xi \webright )\webright )\\ & = \Psi _{K}\webleft (\webleft (\phi ^{*}\webright )_{*}\webleft ([\mspace {-3mu}[a\mapsto \xi _{a}]\mspace {-3mu}]\webright )\webright )\\ & = \Psi _{K}\webleft (\webleft ([\mspace {-3mu}[a\mapsto \phi ^{*}\webleft (\xi _{a}\webright )]\mspace {-3mu}]\webright )\webright )\\ & = \Psi _{K}\webleft (\webleft ([\mspace {-3mu}[a\mapsto [\mspace {-3mu}[k\mapsto \xi _{a}\webleft (\phi \webleft (k\webright )\webright )]\mspace {-3mu}]]\mspace {-3mu}]\webright )\webright )\\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (\xi _{a}\webleft (\phi \webleft (k\webright )\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = \phi ^{*}\webleft ([\mspace {-3mu}[k'\mapsto \webleft [\webleft (\xi _{a}\webleft (k'\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\webright )\\ & = \phi ^{*}\webleft (\Psi _{K'}\webleft (\xi \webright )\webright )\\ & = \webleft [\phi ^{*}\circ \Psi _{K'}\webright ]\webleft (\xi \webright ). \end{align*}
Naturality of $\Phi $. Since $\Psi $ is natural and $\Psi $ is a componentwise inverse to $\Phi $, it follows from Chapter 8: Categories, Item 2 of Proposition 8.8.6.1.2 that $\Phi $ is also natural.
Invertibility I. We claim that

\[ \Psi _{K}\circ \Phi _{K}=\text{id}_{\mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )}. \]

Indeed, given a morphism of pointed sets

\[ \xi \colon \webleft (K,k_{0}\webright )\to \webleft (A\pitchfork X,\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]\webright ) \]

we have

\begin{align*} \webleft [\Psi _{K}\circ \Phi _{K}\webright ]\webleft (\xi \webright ) & = \Psi _{K}\webleft (\Phi _{K}\webleft (\xi \webright )\webright )\\ & = \Psi _{K}\webleft ([\mspace {-3mu}[a\mapsto \xi _{a}]\mspace {-3mu}]\webright )\\ & = \Psi _{K}\webleft ([\mspace {-3mu}[a'\mapsto \xi _{a'}]\mspace {-3mu}]\webright )\\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (\mathrm{ev}_{a}\webleft ([\mspace {-3mu}[a'\mapsto \xi _{a'}\webleft (k\webright )]\mspace {-3mu}]\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (\xi _{a}\webleft (k\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}].\end{align*}

Now, we have two cases:
1. If $\xi \webleft (k\webright )=\webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$, we have
  \begin{align*} \webleft [\Psi _{K}\circ \Phi _{K}\webright ]\webleft (\xi \webright ) & = \cdots \\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (\xi _{a}\webleft (k\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[k\mapsto \xi \webleft (k\webright )]\mspace {-3mu}]\\ & = \xi .\end{align*}
2. If $\xi \webleft (k\webright )\neq \webleft [\webleft (x_{0}\webright )_{a\in A}\webright ]$ and $\xi \webleft (k\webright )=\webleft [\webleft (x^{k}_{a}\webright )_{a\in A}\webright ]$ instead, we have
  \begin{align*} \webleft [\Psi _{K}\circ \Phi _{K}\webright ]\webleft (\xi \webright ) & = \cdots \\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (\xi _{a}\webleft (k\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[k\mapsto \webleft [\webleft (x^{k}_{a}\webright )_{a\in A}\webright ]]\mspace {-3mu}]\\ & = [\mspace {-3mu}[k\mapsto \xi \webleft (k\webright )]\mspace {-3mu}]\\ & = \xi .\end{align*}
In both cases, we have $\webleft [\Psi _{K}\circ \Phi _{K}\webright ]\webleft (\xi \webright )=\xi $, and thus we are done.
Invertibility II. We claim that

\[ \Phi _{K}\circ \Psi _{K}=\text{id}_{\mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright )}. \]

Indeed, given a morphism $\xi \colon A\to \mathsf{Sets}_{*}\webleft (K,X\webright )$, we have

\begin{align*} \webleft [\Phi _{K}\circ \Psi _{K}\webright ]\webleft (\xi \webright ) & = \Phi _{K}\webleft (\Psi _{K}\webleft (\xi \webright )\webright )\\ & = \Phi _{K}\webleft ([\mspace {-3mu}[k\mapsto \webleft [\webleft (\xi _{a}\webleft (k\webright )\webright )_{a\in A}\webright ]]\mspace {-3mu}]\webright )\\ & = [\mspace {-3mu}[a\mapsto [\mspace {-3mu}[k\mapsto \xi _{a}\webleft (k\webright )]\mspace {-3mu}]]\mspace {-3mu}]\\ & = \xi \end{align*}

This finishes the proof.

Proposition 4.2.2.1.4 ▶ Properties of Cotensors of Pointed Sets by Sets

Let $\webleft (X,x_{0}\webright )$ be a pointed set and let $A$ be a set.

Functoriality. The assignments $A,\webleft (X,x_{0}\webright ),\webleft (A,\webleft (X,x_{0}\webright )\webright )$ define functors
\begin{gather*} \begin{aligned} A\pitchfork - & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ -\pitchfork X & \colon \mathsf{Sets}^{\mathsf{op}} \to \mathsf{Sets}_{*}, \end{aligned}\\ -_{1}\pitchfork -_{2} \colon \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{gather*}

In particular, given:
- A map of sets $f\colon A\to B$;
- A pointed map $\phi \colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright )$;
the induced map

\[ f\odot \phi \colon A\pitchfork X\to B\pitchfork Y \]

is given by

\[ \webleft [f\odot \phi \webright ]\webleft (\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (\phi \webleft (x_{f\webleft (a\webright )}\webright )\webright )_{a\in A}\webright ] \]

for each $\webleft [\webleft (x_{a}\webright )_{a\in A}\webright ]\in A\pitchfork X$.
Adjointness I. We have an adjunction
witnessed by a bijection
\[ \mathsf{Sets}^{\mathsf{op}}_{*}\webleft (A\pitchfork X,K\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]

i.e. by a bijection

\[ \mathsf{Sets}_{*}\webleft (K,A\pitchfork X\webright )\cong \mathsf{Sets}\webleft (A,\mathsf{Sets}_{*}\webleft (K,X\webright )\webright ), \]

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Adjointness II. We have an adjunctions
witnessed by a bijection
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (A\odot X,Y\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,A\pitchfork Y\webright ), \]

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
As a Weighted Limit. We have
\[ A\pitchfork X\cong \text{lim}^{\webleft [A\webright ]}\webleft (X\webright ), \]

where in the right hand side we write:
- $A$ for the functor $A\colon \text{pt}\to \mathsf{Sets}$ picking $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$;
- $X$ for the functor $X\colon \text{pt}\to \mathsf{Sets}_{*}$ picking $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Iterated Cotensors. We have an isomorphism of pointed sets
\[ A\pitchfork \webleft (B\pitchfork X\webright )\cong \webleft (A\times B\webright )\pitchfork X, \]

natural in $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $\webleft (X,x_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Commutativity With Homs. We have natural isomorphisms
\begin{align*} A\pitchfork \mathsf{Sets}_{*}\webleft (X,-\webright ) & \cong \mathsf{Sets}_{*}\webleft (A\odot X,-\webright ),\\ A\pitchfork \mathsf{Sets}_{*}\webleft (-,Y\webright ) & \cong \mathsf{Sets}_{*}\webleft (-,A\pitchfork Y\webright ). \end{align*}
The Cotensor Evaluation Map. For each $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, we have a map
\[ \mathrm{ev}^{\pitchfork }_{X,Y}\colon X\to \mathsf{Sets}_{*}\webleft (X,Y\webright )\pitchfork Y, \]

natural in $X,Y\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, and given by

\[ \mathrm{ev}^{\pitchfork }_{X,Y}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (f\webleft (x\webright )\webright )_{f\in \mathsf{Sets}_{*}\webleft (X,Y\webright )}\webright ] \]

for each $x\in X$.
The Cotensor Coevaluation Map. For each $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ and each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have a map
\[ \mathrm{coev}^{\pitchfork }_{A,X}\colon A\to \mathsf{Sets}_{*}\webleft (A\pitchfork X,X\webright ), \]

natural in $X\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$ and $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, and given by

\[ \mathrm{coev}^{\pitchfork }_{A,X}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\webleft [\webleft (x_{b}\webright )_{b\in A}\webright ]\mapsto x_{a}]\mspace {-3mu}] \]

for each $a\in A$.