Kernel (category theory)

In category theoryits applicationsother branchesmathematics, kernels aregeneralization ofkernelsgroup homomorphisms andkernelsmodule homomorphismscertain other kernels from algebra. Intuitively,kernel ofmorphism f : A → B is"most general" morphism k : K → A which, when composedf, yields zero.

Note that kernel pairsdifference kernels (aka binary equalisers) sometimes go byname "kernel"; while related, these aren't quitesameandnot discussedthis article.

Tablecontents

1 Definition 2 First properties 3 Examples 4 Relationother categorical concepts 5 Relationshipalgebraic kernels

Definition

Let C becategory. In orderdefinekernel ingeneral category-theoretical sense, C needshave zero morphisms. In that case, if f: A → Ban arbitrary morphismC, thenkernelfan equaliserf andzero morphism from AB. In symbols:

ker f = eq (f, 0_A,B)

To be more explicit,following universal property can be used. A kernelfany morphism k: K → A such that:

• f o k iszero morphism from KB;
• Given any morphism k': K' → A such that f o k' iszero morphism, there isunique morphism u: K' → K such that k o u = k'.

Image:KerCat01.png Image:KerCat02.png

Note thatmany concrete contexts, one would refer toobject K as"kernel", rather thanmorphism k. In those situations, K would besubsetA,that would be sufficientreconstruct k as an inclusion map; innonconcrete case,contrast, we needmorphism kdescribe how K isbe interpreted assubobjectA. One may preferthink ofkernel aspair (K,k) rather than as simply K or k alone.

First properties

Not every morphism needshavekernel, but ifdoes, then all its kernelsisomorphic instrong sense: if k : K → Al : L → Akernelsf : A → B, then there existsunique isomorphism φ : K → L such that l o φ = k.

Every kernel ismonomorphism.

Examples

Kernelsfamiliarmany categories from abstract algebra, such ascategorygroupss orcategory(left) modules overfixed ring (including vector spaces overfixed field). To be explicit, if f: A → B ishomomorphismonethese categories,Kits kernel inusual algebraic sense, then K issubalgebraA andinclusion homomorphism from KA iskernel incategorical sense.

Note that incategorymonoids, category-theoretic kernels exist just asgroups, but these kernels don't carry sufficient informationalgebraic purposes. Therefore,notionkernel studiedmonoid theoryslightly different. Conversely, incategoryringss, thereno kernels incategory-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, therestillnotionkernel studiedring theory. See Relationshipalgebraic kernels below forresolutionthis paradox.

We have plentyalgebraic examples; now we should give exampleskernelscategories from topologyfunctional analysis.

Relationother categorical concepts

The dual conceptthatkernelthatcokernel. That is,kernel ofmorphismits cokernel inopposite category,vice versa.

As mentioned above,kernel istypebinary equaliser, or difference kernel. Conversely, inpreadditive category, every binary equaliser can be constructed askernel. To be specific,equaliser ofmorphisms fg iskernel ofdifference g − f. In symbols:

eq (f,g) = ker (g − f).

Itbecausethis fact that binary equaliserscalled "difference kernels", evennon-preadditive categories where morphsims cannot be subtracted.

Every kernel, like any other equaliser, ismonomorphism. Conversely,monomorphismcalled normal ifiskernelsome morphism. A categorycalled normal if every monomorphismnormal.

Abelian categories,particular,always normal. In this situation,kernel ofcokernelany morphism (which always existsan abelian category) turns outbeimagethat morphism;symbols:

im f = ker coker f (in an abelian category)

When m ismonomorphism,must be its own image; thus, not onlyabelian categories normal, so that every monomorphism iskernel, but we also know which morphismmonomorphism iskernel of,wit, its cokernel. In symbols:

m = ker (coker m) (for monomorphismsan abelian category)

Relationshipalgebraic kernels

Universal algebra definesnotionkernelhomomorphisms between two algebraic structures ofsame kind. This conceptkernel measures how fargiven homomorphismfrom being injective. Theresome overlap between this algebraic notion andcategorical notionkernel since both generalizesituationgroupsmodules mentioned above. In general, however,universal-algebraic notionkernelmore likecategory-theoretic conceptkernel pair. In particular, kernel pairs can be usedinterpret kernelsmonoid theory or ring theorycategory-theoretic terms.