Relative Adjunctions and Density

In the previous post, we encountered relative adjunctions. One curious aspect was that if L is left J-relative adjoint to R:

  • R determines L up to isomorphism.
  • L does not necessarily determine R up to isomorphism.

In this post, we’re going to look at this aspect of relative adjunctions in more detail. We will see that we get better behaviour if J is a dense functor. This is an important aspect of relative monad theory, so it is worth seeing it in action.

Adjoints determine each other

As before, we will warm up by considering the non-relative situation, and look at ordinary adjunctions. In that case, if L \dashv R and L \dashv R', we have natural isomorphisms:

\mathcal{D}(A, R(B)) \cong \mathcal{C}(L(A),B) \cong \mathcal{D}(A,R'(B))

Now we calculate:

[\mathcal{C},\mathcal{D}](F,R) \cong \int_C \mathcal{D}(F(C),R(C)) \cong \int_C \mathcal{D}(F(C),R'(C)) \cong [\mathcal{C},\mathcal{D}](F,R')

The first and final steps use the following standard natural isomorphism relating natural transformations and ends:

[C,D](F,G) \cong \int_C \mathcal{C}(F(C),G(C))

The middle step then simply uses our assumed adjunctions. From our calculated isomorphism, as the Yoneda lemma is full and faithful:

R \cong R',

so L determines R up to isomorphism. It is clear that we could use a symmetrical argument to show that L is fully determined by R. For a reader familiar with these “Yoneda style” arguments, this is a very routine proof, with the relationship between natural transformations and ends allowing us to make the connection between functors and their actions.

Relative adjunctions

Now we have seen a proof relating adjoints for ordinary adjunctions, lets see how we can adapt it to the J-relative setting. As relative adjunctions are asymmetrical, there are two aspects to consider.

Firstly, we assume that both L and L' are left J-relative adjoint to R, so we have natural isomorphisms:

\mathcal{C}(L(A), B) \cong \mathcal{D}(J(A),R(B)) \cong \mathcal{C}(L'(A),B)

We can actually perform an a very similar calculation to before:

[\mathcal{B},\mathcal{C}](L,F) \cong \int_B \mathcal{C}(L(B),F(B)) \cong \int_B \mathcal{C}(L'(B),F(B)) \cong [\mathcal{B},\mathcal{C}](L',F),

and so L \cong L'. So far, so straightforward.

Now lets consider the situation for the right adjoint. We now assume that L is left J-relative adjoint to both R and R', so we have natural isomorphisms:

\mathcal{D}(J(A),R(B)) \cong \mathcal{C}(L(A),B) \cong \mathcal{D}(J(A),R'(B))

We can try and proceed as before, with the first step

[\mathcal{C},\mathcal{D}](F,R) \cong \int_C \mathcal{D}(F(C),R(C)),

but we immediately become stuck, as the only relationships we have involving the functor R also mention J. To make progress, we need something new.

A functor J : \mathcal{B} \rightarrow \mathcal{D} is said to be dense if the restricted Yoneda embedding

\mathcal{D} \rightarrow [\mathcal{B}^{op},\mathsf{Set}]

is full and faithful. Explicitly, this means we have a natural isomorphism:

\mathcal{D}(D,D') \cong [\mathcal{B}^{op},\mathsf{Set}](\mathcal{D}(J(-),D), \mathcal{D}(J(-),D'))

This looks encouraging, as J appears exactly in the position we would hope. With the same assumptions as before, assuming J is dense, we calculate:

\begin{aligned} [\mathcal{C},\mathcal{D}](F,R) &\cong \int_C \mathcal{D}(F(C),R(C)) \\ &\cong \int_C [\mathcal{B}^{op},\mathsf{Set}](\mathcal{D}(J(-),F(C)), \mathcal{D}(J(-),R(C))) \\ &\cong \int_C \int_B \mathsf{Set}(\mathcal{D}(J(B),F(C)), \mathcal{D}(J(B),R(C))) \\ &\cong \int_C \int_B \mathsf{Set}(\mathcal{D}(J(B),F(C)), \mathcal{D}(J(B),R'(C))) \\ &\cong \int_C [\mathcal{B}^{op},\mathsf{Set}](\mathcal{D}(J(-),F(C)), \mathcal{D}(J(-),R'(C))) \\ &\cong \int_C \mathcal{D}(F(C),R'(C)) \\ &\cong [\mathcal{C},\mathcal{D}] (F,R') \end{aligned}

Therefore, R \cong R', so if J is dense, right adjoints are determined up to isomorphism.

Conclusion

This post was a bit more technical than usual. This was so that the role of density in the arguments could be made fully explicit. Along the way we saw that density wasn’t a particularly complicated condition, simply allowing us to make Yoneda style arguments in more restrictive settings. Density fits very naturally into calculations involving relative monads or adjunctions, and was introduced by Ulmer in the same paper as relative adjunctions. Ulmer indicates that Gabriel had also introduced the same notion independently at about the same time. Kelly points out in “Basic Concepts in Enriched Category Theory” that the special case of dense subcategories actually goes back further to Isbell’s “Adequate subcategories” from 1960. That the idea has arisen independently several times provides evidence of its utility.

Further reading: I’ve probably recommended it before, but for users wanting more background on these Yoneda style arguments, I strongly recommended Loregian’s “(Co)end calculus” book.

Acknowledgement: Thanks to Nathanael Arkor for very helpful feedback correcting some erroneous remarks relating to the history of dense functors in earlier versions.

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