nLab doctrinal adjunction

Idea

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

Here $T$-$Alg_l$ is the 2-category of strict $T$-algebras, lax T-morphisms, and $T$-transformations, but the result works as well for pseudo algebras.

The term ‘doctrinal’ refers to the concept doctrine, in particular to its definition as a 2-monad.

Statement

Theorem

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

There is a bijection between 2-morphisms $\bar u$ making $(u,\bar u)$ a lax $T$-morphism and 2-morphisms $\tilde f$ making $(f,\tilde f)$ a colax $T$-morphism; it is given by taking mates with respect to the adjunctions $T f \dashv T u$ and $f \dashv u$.

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

Proposition

For the unit and counit of the adjunction $f \dashv u$ to be $T$-transformations, and hence for the adjunction to live in $T$-$Alg_l$, it is necessary and sufficient that $\tilde f$ have an inverse $\bar f$ that makes $(f,\bar f)$ into a lax $T$-morphism, and hence $(f,\bar f)$ 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 $T f \dashv T u$ and $1 \dashv 1$. Noting that $\tilde f$ is the mate of $\bar u$, the conditions are seen to be equivalent to requiring that $\bar f$ and $\tilde f$ are mutually inverse.

It follows that

Proposition

$(f, \bar f) \dashv (u, \bar u)$ in $T$-$Alg_l$ if and only if $f \dashv u$ in $K$ and $\bar f$ has inverse $\tilde f$ = the mate of $\bar u$.

Dually, one has:

Proposition

Let $(f, \bar f)$ and $(u, \bar u)$ be colax morphisms of $T$-algebras. Suppose $f \dashv u$ as functors.

1. There is a bijection between lax structures on $f$ and colax structures on $u$.
2. The adjunction $f \vdash u$ lifts to an adjunction $(f, \bar f) \vdash (u, \bar u)$ if and only if the lax structure $\tilde u$ on $u$ induced by $(f,\bar f)$ is inverse to $\bar u$ (meaning, in particulat, that $(u,\bar u)$ is strong).

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; see double category of algebras. The horizontal 2-category of this double category is $T$-$Alg_l$, and its vertical 2-category is $T$-$Alg_c$. There is an obvious forgetful double functor $T \mathbf{Alg} \to \mathbf{Sq}(K)$, where $\mathbf{Sq}(K)$ 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 $U\colon T \mathbf{Alg} \to \mathbf{Sq}(K)$ creates conjunctions. I.e. given a horizontal arrow $u$ in $T \mathbf{Alg}$ 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 $T\mathbf{Alg}$ lying over $f$.

2. Let $f\colon A\to B$ be a vertical arrow in $T \mathbf{Alg}$ (i.e. a colax $T$-morphism) and let $f'\colon A\to B$ and $u\colon B\to A$ 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 $f'$ a horizontal companion of $f$;
   2. Data making $u$ a right conjoint of $f$;
   3. Data making $f'$ 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

Corollary

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

Moreover:

Corollary

For two lax monoidal functors $L \dashv R$ between monoidal categories to be monoidally adjoint it is necessary and sufficient that:

1. $L \dashv R$ as functors
2. $L$ is strong monoidal, and the inverse of the lax monoidal structure on $L$ is the mate of the lax monoidal structure of $R$.

See at oplax monoidal functor and at monoidal adjunction for more details.

Related concepts

• monoidal adjunction

References

• Max Kelly, Doctrinal Adjunction Lecture Notes in Mathematics, (1974), Volume 420/1974, doi:10.1007/BFb0063105

The following article explains the double category perspective:

• Mike Shulman, Comparing composites of left and right derived functors, New York Journal of Mathematics, Volume 17 (2011) pp 75-125, (arXiv:0706.2868, journal abstract page)

Doctrinal adjunction is related to monadicity in:

• Stephen Lack, Morita contexts as lax functors, Applied Categorical Structures 22.2 (2014): 311-330.