Fun profunctor facts - companions, conjoints, collages

The companion and conjoint of a \(\mathcal{V}\) functor, \(\mathcal{P}\xrightarrow{F}\mathcal{Q}\)

• Companion, denoted \(\mathcal{P}\overset{\hat F}\nrightarrow \mathcal{Q}\), is defined \(\hat{F}(p,q):=\mathcal{Q}(F(p),q)\)

• Conjoint, denoted \(\mathcal{Q}\overset{\check{F}}\nrightarrow\mathcal{P}\)

Linked by

• Companion and conjoint of identity: referenced
• Companion and conjoint of addition: referenced
• Exercise 4-36: referenced
• Exercise 4-38: referenced
• Exercise 4-41: referenced
• Exercise 4-41: referenced

A \(\mathcal{V}\) adjunction.

A pair of \(\mathcal{V}\) functors, \(\mathcal{P}\overset{F}{\underset{G}\rightleftarrows} \mathcal{Q}\) such that: \(\forall p\in \mathcal{P}, q \in \mathcal{Q}: \mathcal{P}(p,G(q)) \cong \mathcal{Q}(F(p),q)\)

The collage of a \(\mathcal{V}\) profunctor, \(\mathcal{X}\overset{\phi}\nrightarrow \mathcal{Y}\)

• A \(\mathcal{V}\) category, denoted \(\mathbf{Col}(\phi)\)

  • \(Ob(\mathbf{Col}(\phi)):=\)?\(Ob(\mathcal{X})\sqcup Ob(\mathcal{Y})\)

  • \(\mathbf{Col}(\phi)(a,b) :=\) ?\(\begin{cases}\mathcal{X}(a,b) & a,b \in \mathcal{X} \\ \phi(a,b)& a \in \mathcal{X}, b \in \mathcal{Y} \\ \varnothing & a \in \mathcal{Y}, b \in \mathcal{X} \\ \mathcal{Y}(a,b) & a,b \in \mathcal{Y} \end{cases}\)

• There are obvious functors \(\mathcal{X}\xrightarrow{i_\mathcal{X}}\mathbf{Col}(\phi)\) and \(\mathcal{Y}\xrightarrow{i_\mathcal{Y}}\mathbf{Col}(\phi)\) sending each object to “itself" called collage inclusions

Linked by

• Exercise 4-44: referenced

• For any preorder \(\mathcal{P}\), there is an identity functor \(\mathcal{P}\xrightarrow{id}\mathcal{P}\)

• Its ?companion and conjoint agree: \(\hat{id}=\check{id}=\mathcal{P}\overset{U_\mathcal{P}}\nrightarrow \mathcal{P}\) equivalent to the ?unit profunctor.

• Consider the addition function on real numbers.

• It is monotonic, since \((a,b,c)\leq(a',b',c')\) means \(a+b+c\leq a'+b'+c'\)

• Therefore it has a companion and a conjoint

  • The companion \(\mathbb{R}\times\mathbb{R}\times\mathbb{R}\overset{\hat +}\nrightarrow \mathbb{R}\) sends (a,b,c,d) to true iff ?\(a+b+c\leq d\).

Linked by

• Exercise 4-38: referenced

TODO

Check that the companion \(\hat{id}\) of the identity functor is actually the unit profunctor.

Solution(1)

TODO

Considering the addition example, what is the conjoint of this addition function?

Solution(0)

TODO

Let \(\mathcal{V}\) be a skeletal quantale and \(\mathcal{P}\overset{F}{\underset{G}\rightleftarrows} \mathcal{Q}\) be \(\mathcal{V}\) functors. Show that \(F \dashv G\) iff the companion of the former equals the conjoint of the latter, i.e. \(\hat F = \check G\)

Solution(0)

TODO

Draw a Hasse diagram for the collage of the profunctor:

Solution(0)

TODO