This book in categorial proof theory formulates in terms of category theory a generalization close to linear algebra of the notions of distributive lattice and Boolean algebra. These notions of distributive lattice category and Boolean category codify a plausible nontrivial notion of identity of proofs in classical propositional logic, which is in accordance with Gentzens cut-elimination procedure for multiple-conclusion sequents modified by admitting new principles called union of proofs and zero proofs. It is proved that these notions of category are coherent in the sense that there is a...