(adjoint)
Suppose \(\mca\) and \(\mcb\) are categories with functors \(F: \mca \to \mcb\) and \(G: \mcb \to \mca\). \(F\) and \(G\) are adjoint if there is a bijection \[ \tau_{AB}: \Mor_{\mcb}(F(A), B) \to \Mor_{\mca}(A, G(B)) \] for all \(A \in \mca\) and \(B \in \mcb\), natural in both arguments. We call \(F\) the left adjoint and \(G\) the right adjoint.
Naturality means that for \(f: A' \to A\) in \(\mca\) the square formed by \(\tau_{AB}\), \(\tau_{A'B}\) and the two restriction maps along \(f\) commutes, and similarly for \(g: B \to B'\) in \(\mcb\).
A left adjoint of \(G\), if it exists, is unique up to unique natural isomorphism, and likewise for right adjoints. Right adjoints preserve limits and left adjoints preserve colimits; so an additive left adjoint between abelian categories is right exact, and an additive right adjoint is left exact. Since \(\wait \tensor_A N\) is left adjoint to \(\Hom_A(N,\wait)\), the first is right exact and the second left exact.
Dependencies
D-DEFNATTRNatural transformations and natural isomorphismsuses
Backlinks
T-DEFRAPLRAPL and LAPC, and the exactness of $\Hom$ and $\tensor$usesT-DEFADJPRThe standard adjoint pairsusesD-DEFTENSThe tensor product of modules, and extension of scalarsuses