doctrinal adjunction


2-Category theory

Higher algebra



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

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



Let fuf \dashv u be an adjunction in some 2-category KK and let TT be a 2-monad on KK.

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

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


For the unit and counit of the adjunction fuf \dashv u to be TT-transformations, and hence for the adjunction to live in TT-Alg lAlg_l, it is necessary and sufficient that f˜\tilde f have an inverse f¯\bar f that makes (f,f¯)(f,\bar f) into a lax TT-morphism, and hence (f,f¯)(f,\bar f) into a strong TT-morphism.

Again, the proof hinges on the properties of mates: we take the conditions for the unit and counit to be TT-transformations and pass to mates wrt TfTuT f \dashv T u and 111 \dashv 1. Noting that f˜\tilde f is the mate of u¯\bar u, the conditions are seen to be equivalent to requiring that f¯\bar f and f˜\tilde f are mutually inverse.

It follows that


(f,f¯)(u,u¯)(f, \bar f) \dashv (u, \bar u) in TT-Alg lAlg_l if and only if fuf \dashv u in KK and f¯\bar f has inverse f˜\tilde f = the mate of u¯\bar u.

In terms of double categories

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

It is straightforward to verify that a conjunction in the double category TT-Alg is precisely an adjunction in KK between TT-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 TT-Alg has a companion precisely when it is a strong (= pseudo) TT-morphism. The two central results of Kelly’s paper can then be stated as:

  1. The forgetful double functor U:TAlgSq(K)U\colon T \mathbf{Alg} \to \mathbf{Sq}(K) creates conjunctions. I.e. given a horizontal arrow uu in TAlgT \mathbf{Alg} and a left conjoint ff of U(u)U(u) (i.e. a left adjoint of uu in KK), there is a unique left conjoint of uu in TAlgT\mathbf{Alg} lying over ff.

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

    1. Data making ff' a horizontal companion of ff;
    2. Data making uu a right conjoint of ff;
    3. Data making ff' a left adjoint of uu 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 TT-algebras.


Monoidal categories

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

The above theorem then asserts


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

See oplax monoidal functor for more details.


The following article explains the double category perspective:

Revised on January 28, 2017 18:29:50 by David Roberts (