symmetric monoidal (∞,1)-category of spectra
Doctrinal Adjunction is the title of a 1974 paper (Kelly) that gives conditions under which adjoint morphisms in a 2-category , and additionally the unit and counit, may be lifted to the category - for some 2-monad on .
Here - is the 2-category of strict -algebras, lax T-morphisms, and -transformations, but the result works as well for pseudo algebras.
The proof (Kelly) relies solely on the properties of the mate correspondence.
For the unit and counit of the adjunction to be -transformations, and hence for the adjunction to live in -, it is necessary and sufficient that have an inverse that makes into a lax -morphism, and hence into a strong -morphism.
Again, the proof hinges on the properties of mates: we take the conditions for the unit and counit to be -transformations and pass to mates wrt and . Noting that is the mate of , the conditions are seen to be equivalent to requiring that and are mutually inverse.
It follows that
in - if and only if in and has inverse = the mate of .
Doctrinal adjunction can be stated cleanly in terms of double categories. Namely, for any 2-monad there is a double category -Alg whose objects are -algebras, whose horizontal arrows are lax -morphisms, whose vertical arrows are colax -morphisms, and whose 2-cells are 2-cells in the base 2-category that make a certain cube commute. The horizontal 2-category of this double category is -, and its vertical 2-category is -. There is an obvious forgetful double functor , where is the double category of squares or “quintets” in .
It is straightforward to verify that a conjunction in the double category -Alg is precisely an adjunction in between -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 -Alg has a companion precisely when it is a strong (= pseudo) -morphism. The two central results of Kelly’s paper can then be stated as:
The forgetful double functor creates conjunctions. I.e. given a horizontal arrow in and a left conjoint of (i.e. a left adjoint of in ), there is a unique left conjoint of in lying over .
Let be a vertical arrow in (i.e. a colax -morphism) and let and be horizontal arrows (i.e. lax -morphisms). Then from any two of the following three data we can uniquely construct the third.
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 -algebras.
The above theorem then asserts
For two adjoint functors between monoidal categories, is oplax monoidal precisely if is lax monoidal.
See oplax monoidal functor for more details.
The following article explains the double category perspective: