Relation To Other Categorical Concepts
The dual concept to that of kernel is that of cokernel. That is, the kernel of a morphism is its cokernel in the opposite category, and vice versa.
As mentioned above, a kernel is a type of binary equaliser, or difference kernel. Conversely, in a preadditive category, every binary equaliser can be constructed as a kernel. To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In symbols:
- eq (f,g) = ker (g − f).
It is because of this fact that binary equalisers are called "difference kernels", even in non-preadditive categories where morphisms cannot be subtracted.
Every kernel, like any other equaliser, is a monomorphism. Conversely, a monomorphism is called normal if it is the kernel of some morphism. A category is called normal if every monomorphism is normal.
Abelian categories, in particular, are always normal. In this situation, the kernel of the cokernel of any morphism (which always exists in an abelian category) turns out to be the image of that morphism; in symbols:
- im f = ker coker f (in an abelian category)
When m is a monomorphism, it must be its own image; thus, not only are abelian categories normal, so that every monomorphism is a kernel, but we also know which morphism the monomorphism is a kernel of, to wit, its cokernel. In symbols:
- m = ker (coker m) (for monomorphisms in an abelian category)
Read more about this topic: Kernel (category Theory)
Famous quotes containing the words relation to, relation, categorical and/or concepts:
“The whole point of Camp is to dethrone the serious. Camp is playful, anti-serious. More precisely, Camp involves a new, more complex relation to “the serious.” One can be serious about the frivolous, frivolous about the serious.”
—Susan Sontag (b. 1933)
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“We do the same thing to parents that we do to children. We insist that they are some kind of categorical abstraction because they produced a child. They were people before that, and they’re still people in all other areas of their lives. But when it comes to the state of parenthood they are abruptly heir to a whole collection of virtues and feelings that are assigned to them with a fine arbitrary disregard for individuality.”
—Leontine Young (20th century)
“It is impossible to dissociate language from science or science from language, because every natural science always involves three things: the sequence of phenomena on which the science is based; the abstract concepts which call these phenomena to mind; and the words in which the concepts are expressed. To call forth a concept, a word is needed; to portray a phenomenon, a concept is needed. All three mirror one and the same reality.”
—Antoine Lavoisier (1743–1794)