### Context

#### 2-Category theory

2-category theory

## Structures on 2-categories

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

Doctrinal Adjunction is the title of a 1974 paper (Kelly) that gives conditions under which adjoint morphisms $f⊣u$ in a 2-category $K$, and additionally the unit and counit, may be lifted to the category $T$-${\mathrm{Alg}}_{l}$ for some 2-monad $T$ on $K$.

Here $T$-${\mathrm{Alg}}_{l}$ is the 2-category of strict $T$-algebras, lax T-morphisms, and $T$-transformations, but the result works as well for pseudo algebras.

## Statement

###### Theorem

Let $f⊣u$ be an adjunction in some 2-category $K$ and let $T$ be a 2-monad on $K$.

There is a bijection between 2-morphisms $\overline{u}$ making $\left(u,\overline{u}\right)$ a lax $T$-morphism and 2-morphisms $\stackrel{˜}{f}$ making $\left(f,\stackrel{˜}{f}\right)$ a colax $T$-morphism; it is given by taking mates with respect to the adjunctions $Tf⊣Tu$ and $f⊣u$.

The proof (Kelly) relies solely on the properties of the mate correspondence.

###### Proposition

For the unit and counit of the adjunction $f⊣u$ to be $T$-transformations, and hence for the adjunction to live in $T$-${\mathrm{Alg}}_{l}$, it is necessary and sufficient that $\stackrel{˜}{f}$ have an inverse $\overline{f}$ that makes $\left(f,\overline{f}\right)$ into a lax $T$-morphism, and hence $\left(f,\overline{f}\right)$ into a strong $T$-morphism.

Again, the proof hinges on the properties of mates: we take the conditions for the unit and counit to be $T$-transformations and pass to mates wrt $Tf⊣Tu$ and $1⊣1$. Noting that $\stackrel{˜}{f}$ is the mate of $\overline{u}$, the conditions are seen to be equivalent to requiring that $\overline{f}$ and $\stackrel{˜}{f}$ are mutually inverse.

It follows that

###### Proposition

$\left(f,\overline{f}\right)⊣\left(u,\overline{u}\right)$ in $T$-${\mathrm{Alg}}_{l}$ if and only if $f⊣u$ in $K$ and $\overline{f}$ has inverse $\stackrel{˜}{f}$ = the mate of $\overline{u}$.

## In terms of double categories

Doctrinal adjunction can be stated cleanly in terms of double categories. Namely, for any 2-monad $T$ there is a double category $T$-Alg whose objects are $T$-algebras, whose horizontal arrows are lax $T$-morphisms, whose vertical arrows are colax $T$-morphisms, and whose 2-cells are 2-cells in the base 2-category $K$ that make a certain cube commute. The horizontal 2-category of this double category is $T$-${\mathrm{Alg}}_{l}$, and its vertical 2-category is $T$-${\mathrm{Alg}}_{c}$. There is an obvious forgetful double functor $T\mathrm{Alg}\to \mathrm{Sq}\left(K\right)$, where $\mathrm{Sq}\left(K\right)$ is the double category of squares or “quintets” in $K$.

It is straightforward to verify that a conjunction in the double category $T$-Alg is precisely an adjunction in $K$ between $T$-algebras whose left adjoint is colax, whose right adjoint is lax, and for which the lax and colax structure maps are mates under the adjunction – i.e. a “doctrinal adjunction” in the above sense. Furthermore, an arrow in $T$-Alg has a companion precisely when it is a strong (= pseudo) $T$-morphism. The two central results of Kelly’s paper can then be stated as:

1. The forgetful double functor $U:T\mathrm{Alg}\to \mathrm{Sq}\left(K\right)$ creates conjunctions. I.e. given a horizontal arrow $u$ in $T\mathrm{Alg}$ and a left conjoint $f$ of $U\left(u\right)$ (i.e. a left adjoint of $u$ in $K$), there is a unique left conjoint of $u$ in $T\mathrm{Alg}$ lying over $f$.

2. Let $f:A\to B$ be a vertical arrow in $T\mathrm{Alg}$ (i.e. a colax $T$-morphism) and let $f\prime :A\to B$ and $u:B\to A$ be horizontal arrows (i.e. lax $T$-morphisms). Then from any two of the following three data we can uniquely construct the third.

1. Data making $f\prime$ a horizontal companion of $f$;
2. Data making $u$ a right conjoint of $f$;
3. Data making $f\prime$ a left adjoint of $u$ in the horizontal 2-category.

Of these, the second is actually a general statement about companions and conjoints in any double category. Of course, the first is a special property of the forgetful double functor from the double category of $T$-algebras.

## Examples

### Monoidal categories

Let $K=$ Cat and $T$ the 2-monad whose 2-algebras are monoidal categories. Then

The above theorem then asserts

###### Corollary

For two adjoint functors $\left(L⊣R\right)$ between monoidal categories, $L$ is oplax monoidal precisely if $R$ is lax monoidal.

See oplax monoidal functor for more details.

## References

• Max Kelly, Doctrinal Adjunction Lecture Notes in Mathematics, (1974), Volume 420/1974
• Mike Shulman, Comparing composites of left and right derived functors (2011) NYJM explains the double category perspective.
Revised on October 22, 2012 22:37:30 by Urs Schreiber (82.169.65.155)