- Kernel (category theory)
In
category theory and its applications to other branches ofmathematics , kernels are a generalization of the kernels ofgroup homomorphism s and the kernels ofmodule homomorphism s and certain other kernels from algebra. Intuitively, the kernel of themorphism "f" : "X" → "Y" is the "most general" morphism "k" : "K" → "X" which, when composed with "f", yields zero.Note that
kernel pair s anddifference kernel s (aka binaryequaliser s) 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 morphism s.In that case, if "f" : "X" → "Y" is an arbitrarymorphism 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 any morphism "k" : "K" → "X" such that:
* "f" o "k" is the zero morphism from "K" to "Y";*
Given any morphism "k"′ : "K"′ → "X" such that "f" o "k"′ is the zero morphism, there is aunique morphism "u" : "K"′ → "K" such that "k" o "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 aninclusion map ; in the nonconcrete case, in contrast, we need the morphism "k" to describe "how" "K" is to be interpreted as asubobject of "X". In any case, one can show that "k" is always amonomorphism (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" : "L" → "X" are kernels of "f" : "X" → "Y", then there exists a unique
isomorphism φ : "K" → "L" such that "l" o φ = "k".Examples
Kernels are familiar in many categories from
abstract algebra , such as the category of groups or the category of (left) modules over a fixed ring (includingvector space s over a fixed field).To be explicit, if "f" : "X" → "Y" is ahomomorphism in one of these categories, and "K" is its kernel in the usual algebraic sense, then "K" is asubalgebra of "X" and the inclusion homomorphism from "K" to "X" is a kernel in the categorical sense.Note that in the category of
monoid s, category-theoretic kernels exist just as for groups, but these kernels don't carry sufficient information for algebraic purposes.Therefore, the notion of kernel studied in monoid theory is slightly different.Conversely, in thecategory of rings , there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms.Nevertheless, there is still a notion of kernel studied in ring theory.See Relationship to algebraic kernels below for the resolution of this paradox."We have plenty of algebraic examples; now we should give examples of kernels in categories from
topology andfunctional analysis ."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 theopposite category , and vice versa.As mentioned above, a kernel is a type of binary equaliser, or
difference kernel .Conversely, in apreadditive 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 thecokernel 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)Relationship to algebraic kernels
Universal algebra defines a notion of kernel for homomorphisms between twoalgebraic structure s of the same kind.This concept of kernel measures how far the given homomorphism is from beinginjective .There is some overlap between this algebraic notion and the categorical notion of kernel since both generalize the situation of groups and modules mentioned above.In general, however, the universal-algebraic notion of kernel is more like the category-theoretic concept ofkernel pair .In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic terms.
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