Adj in nLab

Contents

Contents

Idea

The 2-category AdjAdj is the free-standing adjunction (walking adjunction).

A 2-functor Adj→KAdj \to K is an adjunction in the 2-category KK. These 2-functors form one version of the 2-category of adjunctions of KK.

Definition

AdjAdj is the 2-category freely generated by

• two objects: aa and bb,

• two morphisms: L:a→bL: a \to b and R:b→aR: b \to a,

• and two 2-morphisms, called the “unit” and “counit”: i:1 a→LRi: 1_a \to L R and e:RL→1 be: R L \to 1_b, satisfying two relations, called the “triangle equations”.

The restrictions of the free-standing adjunction, AdjAdj, to the sub-2-categories spanned by one endpoint, aa, or the other, bb, define the free-standing monad and the free-standing comonad.

• 2-category of adjunctions

• adjoint logic

• Hegelian taco

References

• C. Auderset, Adjonction et monade au niveau des 2-categories, Cahiers de Top. et Géom. Diff. XV-1 (1974), 3-20. (numdam)

• John Baez, This Week’s Finds in Mathematical Physics (Week 174), (TWF174)

• Kevin Coulembier, Ross Street, Michel van den Bergh, Freely adjoining monoidal duals, arXiv:2004.09697 (2020). (abstract)

• Dieter Pumplün, Eine Bemerkung über Monaden und adjungierte Funktoren, Math. Annalen 185 (1970), 329-377.

• Stephen Schanuel and Ross Street, The free adjunction, Cah. Top. Géom. Diff. 27 (1986), 81-83. (numdam)