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walking adjoint equivalence

Contents

Idea

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

Definition

Definition

The free-standing adjoint equivalence is the unique (up to isomorphism) 2-category π’œβ„°\mathcal{AE} with exactly two objects 00 and 11, 1-morphisms freely generated by an arrow i:0β†’1i: 0 \rightarrow 1, and an arrow j:1β†’0j: 1 \rightarrow 0, and 2-morphisms generated by 2-isomorphisms ΞΉ 0:j∘iβ‰…id(0)\iota_{0}: j \circ i \cong id(0) and ΞΉ 1:i∘jβ‰…id(1)\iota_{1}: i \circ j \cong id(1), subject to the equalities

id(i)=(ΞΉ 1i)∘(iΞΉ 0 βˆ’1),id(j)=(ΞΉ 0 βˆ’1j)∘(jΞΉ 1)id(i) = (\iota_1 i)\circ (i \iota_0^{-1}), id(j) =(\iota_0^{-1} j) \circ (j \iota_1)

Remark

The arrow i:0β†’1i: 0 \rightarrow 1 is an adjoint equivalence.

Remark

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

Representing of adjoint equivalences

π’œβ„°\mathcal{AE} is the model for all adjoint equivalences in all 2-categories. In other words, any adjoint equivalence in a 2-category π’œ\mathcal{A} is just a 2-functor from π’œβ„°\mathcal{AE}:

Proposition

Let π’ž\mathcal{C} be a 2-category (weak or strict). Let ff be a 1-arrow of π’ž\mathcal{C} which is part of an adjoint equivalence. Then there is a 2-functor F:π’œβ„°β†’π’žF: \mathcal{AE} \rightarrow \mathcal{C} such that the arrow i:0β†’1i:0 \rightarrow 1 of ℐ\mathcal{I} maps under FF to ff.

Proof

Immediate from the definitions.

Last revised on July 5, 2020 at 06:39:12. See the history of this page for a list of all contributions to it.