nLab
strict 2-adjunction

Given strict 2-categories, AA and CC, and strict 2-functors F:ACF:A\to C and U:CAU:C\to A, a strict 2-adjunction is given one of the following two equivalent means:

  • an isomorphism of categories C(Fa,c)A(a,Uc)C(F a,c)\cong A(a,U c) for each object aa in AA and object cc in CC, which is strict 2-natural both in aa and in cc;

  • a pair of strict 2-natural 2-transformations of 2-functors: unit η:Id AUF\eta : Id_A \to U F, and counit ϵ:FUId B\epsilon : F U\to Id_B, satisfying the triangle identities strictly. Note that this is an ordinary adjunction internal to the 2-category 2Cat2Cat of Cat-enriched categories, strict 2-functors, and strict 2-natural transformations.

There are also more relaxed forms of 2-adjunction, involving weak 2-categories, weak 2-functors, and/or weak 2-natural transformations; see 2-adjunction.

Revised on January 16, 2018 11:16:23 by Tim Porter (95.148.217.11)