Cartesian closedness of lax comma categories
Maria Manuel Clementino, Fernando Lucatelli Nunes
Abstract
For a finitely complete category X, we prove that the lax comma category Cat//X is cartesian closed if and only if its free coproduct completion Fam(X) is cartesian closed. The proof uses a change-of-base theorem: postcomposition with a functor induces a left adjoint between lax comma categories precisely when that functor is a left multiadjoint. We describe the exponentials and derive examples from known results on family categories.
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