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

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

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 $\tilde{f}$ is the mate of $\overline{u}$, the conditions are seen to be equivalent to requiring that $\overline{f}$ and $\tilde{f}$ are mutually inverse.

It follows that

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 $T\mathrm{Alg}\to \mathrm{Sq}(K)$, where $\mathrm{Sq}(K)$ 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 $U:T\mathrm{Alg}\to \mathrm{Sq}(K)$ creates conjunctions. I.e. given a horizontal arrow $u$ in $T\mathrm{Alg}$ and a left conjoint $f$ of $U(u)$ (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

• a lax $T$-morphism is a lax monoidal functor;

• an oplax $T$-morphism is an oplax monoidal functor.

The above theorem then asserts

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.