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tags: category theory, maths,
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# Doctrinal Adjunctions 1

(under construction) ... ([index](https://hackmd.io/@alexhkurz/Hy1oUrS4u))

Next:
- [Double Adjunctions](https://hackmd.io/bynCe7RFQYWwgiN2RlYN9A) ... [Doctrinal Adjunction 2](https://hackmd.io/@alexhkurz/Bk7akF02Y)

Music: Sean Rowe: [1952 Vincent Black Lightning](https://www.youtube.com/watch?v=CrGOs1a1lOk).

---

In the legendary volume LNM 420, Kelly investigates under the title of Doctrinal Adjunctions (p.257 ff) the question when an adjunction
$$L\dashv R:\mathcal A\to\mathcal B$$

can be lifted to an adjunction
$$L\dashv R:Alg(S) \to Alg(T)$$

His setting is more general than what we pursue here. We are simply interested in the case where $\mathcal A$ and $\mathcal B$ are ordinary categories and $S$ and $T$ are functors. This can also be seen as a simplification of Street's formal theory of monads where one replaces monads by mere endofunctors.

## Example: Monoidal Categories

The paradigmatic is the following (even if it does not quite fit the algebras-for-a-functor set-up). Let $L\dashv R:\mathcal A \to \mathcal B$ be an ordinary adjunction between monoidal categories. Then the adjunction lifts to a monoidal adjunction iff $L$ is a strong monoidal functor.

The reason for this result is the following. First, for an adjunction to be monoidal, both functors $L$ and $R$ need to be lax monoidal.[^laxmonoidal] Second, being adjoint, $R$ is lax monoidal iff $L$ is oplax monoidal. Third, a functor that is lax and oplax monoidal is strong monoidal.

The theorems below carry some technical baggage, but are a variation of the same idea.

## Coalgebras

The 2-category $cEndo(\mathcal C)$ is defined as follows.

**Definition:** Let $\mathcal C$ denote any 2-category. The category
$$cEndo(\mathcal C)$$

has as objects $(X,S)$ endo-1-cells $S$ on $X$. Arrows $(U,\phi):(X,S)\to (Y,T)$ are given by a 1-cell $U:X\to Y$ and a 2-cell
$$\phi:US\to TU.$$

A 2-cell $\sigma:(U,\phi)\to (U',\phi')$ is a 2-cell $\sigma:U\to U'$ such that $T\sigma\circ\phi=\phi'\circ\sigma S$.

**Theorem:** $(L,\phi)\dashv (R,\psi)$ in the 2-category $cEndo(\mathcal C)$ iff $L\dashv R$ in $\mathcal C$ and $\psi$ is iso and $\phi$ is the adjoint transpose of $\psi^{-1}$.

**Corollary:** Let $L\dashv R:A\to B$ be an adjunction in $\mathcal C$ and $S$ and $T$ endofunctors on $A$ and $B$, respectively. The adjunction lifts to an adjunction $Coalg(S)\to Coalg(T)$ iff there is a natural iso $\psi:RS\to TR$.

For the next example, recall that there is a $\sf Pos$-enriched adjunction
$$C\dashv D:\sf Set\to Pos$$

where $D$ is the discrete functor and $C$ the connected-components functor.

**Example**: Let $T:\sf Set\to Set$ and $\overline T:\sf Pos\to Pos$ its posetification. We have
$$\overline TD\cong DT.$$

Therefore $C\dashv D$ lifts to a $\sf Pos$-enriched adjunction $Coalg(T)\to Coalg(\overline T)$. In particular, $D$ preserves all (enriched) limits and the final coalgebra. More informally, the final coalgebra of a set-functor coincides with the final coalgebra of its posetification.

**Remark:** In the example above, we can replace $\sf Pos$ by $\sf Pre$ and the posetification by the preordification. The next example only works for the preordification.

Recall that the forgetful functor from preorders is an ordinary right adjoint
$$D\dashv V:\sf Pre\to Set.$$

**Example**: Let $T:\sf Set\to Set$ and $\overline T:\sf Pre\to Pre$ a functor with
$$V \overline T\cong TV.$$

Therefore $D\dashv V$ lifts to an ordinary adjunction $Coalg(\overline T)\to Coalg(T)$. In particular, the carrier of the final $\overline T$-coalgebra has a canonical $T$-coalgebra structure, which is the final $T$-coalgebra. For example, this applies to the power functor $\sf Pre\to Pre$ equipped with the bisimulation (Egli-Milner) order or either of the forward/backward simulation orders.

This is not the case for the next example.

**Remark:** In particular, the example above applies if $\overline T$ is the preordification of $T$. On the other hand, if $\overline T$ is the posetification, we only know that there is an arrow $TV\to V\overline T$, which is, for example in case of the powerset functor $T$, a quotient but not an isomorphism.

## Remarks on the Proof

Let us come back to

$$L\dashv R:\mathcal A\to\mathcal B$$

In order to lift $L,R$ to

$$L\dashv R:Coalg(S) \to Coalg(T)$$

we need $\phi:LS\to TS$ and $\psi:RT\to SR$.

...

## Algebras

Algebras are co-coalgebras. We write out the dual for future reference.

The 2-category $aEndo(\mathcal C)$ is obtained from $cEndo(\mathcal C)$ by reversing 2-cells.

**Definition:** Let $\mathcal C$ denote any 2-category. The category
$$aEndo(\mathcal C)$$

has as objects $(X,S)$ endo-1-cells $S$ on $X$. Arrows $(U,\phi):(X,S)\to (Y,T)$ are given by a 1-cell $U:X\to Y$ and a 2-cell
$$\phi:TU\to US.$$

A 2-cell $\sigma:(U,\phi)\to (U',\phi')$ is a 2-cell $\sigma:U\to U'$ such that $\sigma S\circ\phi'=\phi'\circ T\sigma$.

**Remark:** Let $\mathcal C$ be the category of locally-small categories. Then the condition that $\phi$ in $(U,\phi)$ is natural, ensures that $U$ is a functor from $S$-algebras to $T$-algebras. The condition $\sigma S\circ\phi'=\phi'\circ T\sigma$ on $\sigma$ ensures that $\sigma$ gives rise to a $T$-algebra morphism from the $U$-image to the $U'$-image of a given $S$-algebra.

**Theorem:** $(L,\phi)\dashv (R,\psi)$ in the 2-category $aEndo(\mathcal C)$ iff $L\dashv R$ in $\mathcal C$ and $\phi$ is iso and $\psi$ is the adjoint transpose of $\phi^{-1}$.

**Corollary:** Let $L\dashv R:A\to B$ be an adjunction in $\mathcal C$ and $S$ and $T$ endofunctors on $A$ and $B$, respectively. The adjunction lifts to an adjunction $Alg(S)\to Alg(T)$ iff there is a natural iso $\phi:TL\to LS$.

## References

The main references for this material are

- Kelly: Doctrinal Adjunctions. [Lecture Notes in Mathematics 420](https://www.dropbox.com/s/eccdua5edu05axb/LNM%20420.pdf?dl=0). 1974.
- Street: [The Formal Theory of Monads](https://www.sciencedirect.com/science/article/pii/0022404972900199). 1972.

The nLab has good articles on [doctrinal adjunction](https://ncatlab.org/nlab/show/doctrinal+adjunction) and [monoidal adjunction](https://ncatlab.org/nlab/show/monoidal+adjunction).

Further references and more details, in particular on the coalgebra situation, are in Section 4 of

- Balan, Kurz, Velebil: [Extending Set-Functors to Generalised Metric Spaces](https://lmcs.episciences.org/5132/pdf), 2019.

[^laxmonoidal]: The condition of a functor $F$ being lax monoidal is best remembered as "lax monoidal functors preserve monoids". More explicitely, if the domain of $F$ carries a monoidal structure $(I,\otimes)$ then $F$ is lax monoidal if there are natural transformations $FX\otimes FY\to F(X\otimes Y)$ and $I\to FI$.