nLab Adj

Idea

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

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

Definition

$Adj$ is the 2-category freely generated by

• two objects: $a$ and $b$,

• two morphisms: $L: a \to b$ and $R: b \to a$,

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

The restrictions of the free-standing adjunction, $Adj$, to the sub-2-categories spanned by one endpoint, $a$, or the other, $b$, define the free-standing monad and the free-standing comonad.

Related entries

• 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)