In Topoi Goldblatt uses category theory to explore the logical foundations of mathematics, while using logic as the motivation for learning category theory. (The alternative approach from algebraic topology is largely ignored. Sheaves get a brief mention around page 100, but are only used in the last third of the book, while functors and natural transformations are only touched on.

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The definition of linear morphisms is generalized to the natural number object. Properties of such morphisms are investigated. The necessary and sufficient condition of monocity of a linear morphism of an arbitrary topos is formulated. Linear monomorphisms are demonstrated to be split. Two proofs of complementarity and, accordingly, decidability of linear monomorphisms are proposed.

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1. Generalization of Linear Morphisms on N in Topoi

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Abstract

The definition of linear morphisms is generalized to the natural number object. Properties of such morphisms are investigated. The necessary and sufficient condition of monocity of a linear morphism of an arbitrary topos is formulated. Linear monomorphisms are demonstrated to be split. Two proofs of complementarity and, accordingly, decidability of linear monomorphisms are proposed.

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