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    RAPL and LAPC, and the exactness of $\Hom$ and $\tensor$

    T-DEFRAPL

    Right adjoints preserve limits and are left exact; left adjoints preserve colimits and are right exact, so $\Hom$ is left exact and tensor right exact.Suggest a slogan

    (RAPL, LAPC)

    Let \((F,G)\) be an adjoint pair between categories in which limits and colimits exist, with \(F\) the left adjoint. Then right adjoints preserve limits, \[ G\qty( \inverselim_i B_i ) \cong \inverselim_i G(B_i) , \] and left adjoints preserve colimits, \[ F\qty( \directlim_i A_i ) \cong \directlim_i F(A_i) . \]

    The proof is a universal property check: applying \(G\) to a diagram \(B_i\) gives a diagram \(G(B_i)\), and adjointness converts a cone on \(G(B_i)\) with vertex \(T\) into a cone on \(B_i\) with vertex \(F(T)\), hence a unique map \(F(T) \to \inverselim B_i\), hence a unique \(T \to G(\inverselim B_i)\). That is exactly the universal property of \(\inverselim G(B_i)\). The colimit statement is the same argument run the other way.

    The payoff is the exactness of the tensor-hom pair, with no diagram chase. Kernels are limits and cokernels are colimits, so for the adjunction with \(\wait\tensor_A N\) on the left and \(\Hom_A(N,\wait)\) on the right: given \(0 \to M' \to M \to M''\), one has \(M' \cong \ker(M \to M'')\), so \[ \Hom(N,M') \cong \Hom(N, \ker(M \to M'')) \cong \ker\qty( \Hom(N,M) \to \Hom(N,M'') ) , \] which is left exactness of \(\Hom_A(N,\wait)\). Symmetrically, \(\wait \tensor_A N\) preserves the cokernel and is therefore right exact.

    For sheaves, \(\pi^{-1}\) and \(\pi^*\) are left adjoints and hence right exact, while \(\pi_*\) is a right adjoint and hence left exact; the failure of exactness of \(\pi_*\) is measured by its higher direct images.

    State RAPL and LAPC.
    Deduce that $\Hom_A(N,\wait)$ is left exact and $\wait\tensor_A N$ is right exact.
    Why does knowing which side of an adjunction a functor sits on settle its exactness?

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