nLab walking adjoint equivalence

1. Idea

The walking adjoint equivalence (as in “walking structure”) or free-standing adjoint equivalence is the 2-category (in fact a (2,1)-category) which ‘represents’ adjoint equivalences in a 2-category. It is a categorification of the free-standing adjoint isomorphism?.

2. Definition

Definition 2.1. The free-standing adjoint equivalence is the unique (up to isomorphism) 2-category $\mathcal{AE}$ with exactly two objects $0$ and $1$, 1-morphisms freely generated by an arrow $i: 0 \rightarrow 1$, and an arrow $j: 1 \rightarrow 0$, and 2-morphisms generated by 2-isomorphisms $\iota_{0}: j \circ i \cong id(0)$ and $\iota_{1}: i \circ j \cong id(1)$, subject to the equalities

$$id(i) = (\iota_1 i)\circ (i \iota_0^{-1}), id(j) =(\iota_0^{-1} j) \circ (j \iota_1)$$

Remark 2.2. The arrow $i: 0 \rightarrow 1$ is an adjoint equivalence.

Remark 2.3. The free-standing adjoint equivalence is a (2,1)-category.

3. Representing of adjoint equivalences

$\mathcal{AE}$ is the model for all adjoint equivalences in all 2-categories. In other words, any adjoint equivalence in a 2-category $\mathcal{A}$ is just a 2-functor from $\mathcal{AE}$:

Proposition 3.1. Let $\mathcal{C}$ be a 2-category (weak or strict). Let $f$ be a 1-arrow of $\mathcal{C}$ which is part of an adjoint equivalence. Then there is a 2-functor $F: \mathcal{AE} \rightarrow \mathcal{C}$ such that the arrow $i:0 \rightarrow 1$ of $\mathcal{I}$ maps under $F$ to $f$.

Proof. Immediate from the definitions.  ▮

4. Related entries

• Adj
• walking equivalence