structural:expansion, linguistic:translation
Generalizations

Lawvere's fixed-point theorem provides for a broad generalization of Cantor's theorem to any category with finite products in the following way: let C {\displaystyle {\mathcal {C}}} be such a category, and let 1 {\displaystyle 1} be a terminal object in C {\displaystyle {\mathcal {C}}}. Suppose that Y {\displaystyle Y} is an object in C {\displaystyle {\mathcal {C}}} and that there exists an endomorphism α: Y → Y {\displaystyle \alpha:Y\to Y} that does not have any fixed points; that is, there is no morphism y: 1 → Y {\displaystyle y:1\to Y} that satisfies α ∘ y = y {\displaystyle \alpha \circ y=y}. Then there is no object T {\displaystyle T} of C {\displaystyle {\mathcal {C}}} such that a morphism f: T × T → Y {\displaystyle f:T\times T\to Y} can parameterize all morphisms T → Y {\displaystyle T\to Y}. In other words, for every object T {\displaystyle T} and every morphism f: T × T → Y {\displaystyle f:T\times T\to Y}, an attempt to write maps T → Y {\displaystyle T\to Y} as maps of the form f (−, x): T → Y {\displaystyle f(-,x):T\to Y} must leave out at least one map T → Y {\displaystyle T\to Y}.