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In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra. Intuitively, the kernel of the morphism ''f'' : ''X'' → ''Y'' is the "most general" morphism ''k'' : ''K'' → ''X'' that yields zero when composed with (followed by) ''f''. Note that kernel pairs and difference kernels (aka binary equalisers) sometimes go by the name "kernel"; while related, these aren't quite the same thing and are not discussed in this article. ==Definition== Let C be a category. In order to define a kernel in the general category-theoretical sense, C needs to have zero morphisms. In that case, if ''f'' : ''X'' → ''Y'' is an arbitrary morphism in C, then a kernel of ''f'' is an equaliser of ''f'' and the zero morphism from ''X'' to ''Y''. In symbols: :ker(''f'') = eq(''f'', 0''XY'') To be more explicit, the following universal property can be used. A kernel of ''f'' is an object ''K'' together with a morphism ''k'' : ''K'' → ''X'' such that: * ''f'' ∘ ''k'' is the zero morphism from ''K'' to ''Y''; * Given any morphism ''k''′ : ''K''′ → ''X'' such that ''f'' ∘ ''k''′ is the zero morphism, there is a unique morphism ''u'' : ''K''′ → ''K'' such that ''k'' ∘ ''u'' = ''k. Note that in many concrete contexts, one would refer to the object ''K'' as the "kernel", rather than the morphism ''k''. In those situations, ''K'' would be a subset of ''X'', and that would be sufficient to reconstruct ''k'' as an inclusion map; in the nonconcrete case, in contrast, we need the morphism ''k'' to describe ''how'' ''K'' is to be interpreted as a subobject of ''X''. In any case, one can show that ''k'' is always a monomorphism (in the categorical sense of the word). One may prefer to think of the kernel as the pair (''K'', ''k'') rather than as simply ''K'' or ''k'' alone. Not every morphism needs to have a kernel, but if it does, then all its kernels are isomorphic in a strong sense: if ''k'' : ''K'' → ''X'' and ' : ''L'' → ''X'' are kernels of ''f'' : ''X'' → ''Y'', then there exists a unique isomorphism φ : ''K'' → ''L'' such that ' ∘ φ = ''k''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kernel (category theory)」の詳細全文を読む スポンサード リンク
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