Difference core

A difference kernel , even Egalisator or after the English name equalizer called, is a generalization of the mathematical concept of core to any categories .

definition

Two morphisms are given in one category . A difference kernel of and is a morphism with the following properties: ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle i\colon Z\rightarrow X}$

• ${\displaystyle f\circ i=g\circ i}$ and
• for every morphism to which applies there is exactly one morphism such that . ${\displaystyle i'\colon Z'\to X}$ ${\displaystyle f\circ i'=g\circ i'}$ ${\displaystyle c\colon Z'\to Z}$ ${\displaystyle i'=i\circ c}$

${\displaystyle {\begin{array}{ccccc}Z'&&&&\\\downarrow ^{c}&\searrow ^{i'}&&&\\Z&{\xrightarrow[{i}]{}}&X&{\underset {f}{\overset {g}{\rightrightarrows }}}&Y\\\end{array}}}$

Examples

• In the categories set of sets, top of topological spaces , - mod of the link modules over a ring , the inclusion mapping is in the situation of the above definition ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle i\colon \{x\in X\mid f(x)=g(x)\}\hookrightarrow X}$

a difference core. Especially in the latter category is

${\displaystyle \{x\in X\mid f(x)=g(x)\}=\{x\in X\mid (f-g)(x)=0\}}$

automatically a sub-module that coincides with the core of the difference , which explains the term difference core . ${\displaystyle f-g}$

• In the categories of groups , Abelian groups , vector spaces or rings , the difference kernel of two morphisms is given by the difference kernel of the underlying set mappings.
• Has the considered category zero objects and is in the situation above definition of zero morphism , so is a difference of core and nothing more than a kernel of . Each core is thus an example of a difference core. ${\displaystyle g=0_{XY}}$ ${\displaystyle X\rightarrow Y}$ ${\displaystyle f}$ ${\displaystyle 0_{XY}}$ ${\displaystyle f}$

Remarks

• Difference cores are not clearly determined. But are in the situation above definition and two differential kernels and , it follows from the uniqueness property that there is a uniquely determined isomorphism with there. Difference kernels are therefore determined up to (unambiguous) isomorphism, which is why one often speaks of the difference kernel and denotes it. ${\displaystyle i\colon Z\rightarrow X}$ ${\displaystyle {\tilde {i}}\colon {\tilde {Z}}\rightarrow X}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle c\colon {\tilde {Z}}\rightarrow Z}$ ${\displaystyle {\tilde {i}}=i\circ c}$ ${\displaystyle \mathrm {ker} (f,g)}$
• In another linguistic inaccuracy, the object is called the difference core. The actually meant morphism is then always an obvious inclusion figure that can go unmentioned. ${\displaystyle Z}$
• It is said that a category has difference kernels if there is one difference kernel for every two morphisms . Referred to in the examples above categories Set , Top and - Mod appear to have difference cores. The subcategory Set _{2 of} the at least two-element sets of Set has no difference cores. ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle R}$
• Difference kernels are monomorphisms . The reverse is generally not true. Those monomorphisms that appear as difference kernels are called regular .

Equivalent description

A difference kernel of two morphisms in any category can also be described as the sub-object of which is characterized by the following equivalent properties : ${\displaystyle f,g\colon X\to Y}$ ${\displaystyle i\colon \ker(f,g)\to X}$ ${\displaystyle X}$

${\displaystyle \operatorname {Hom} (T,\ker(f,g))\cong \ker(\operatorname {Hom} (T,f),\operatorname {Hom} (T,g))}$

in which

${\displaystyle \operatorname {Hom} (T,f)\colon \operatorname {Hom} (T,X)\to \operatorname {Hom} (T,Y)}$

${\displaystyle \operatorname {Hom} (T,f)(t):=ft}$

and the difference kernel on the right-hand side is the difference kernel described above in the category of sets, not that in the category under consideration.

Furthermore, the isomorphism in point 2 should of course be in , that is: Let's call the family of isomorphisms ${\displaystyle T}$

${\displaystyle \varphi _{T}\colon \operatorname {Hom} (T,\ker(f,g))\to \ker(\operatorname {Hom} (T,f),\operatorname {Hom} (T,g))}$

then it holds for all and all for whom the following expression is defined that ${\displaystyle a\colon T_{0}\to T}$ ${\displaystyle t}$

${\displaystyle \varphi _{T_{0}}(ta)=\varphi _{T}(t)a}$

See also

• Differential coke

Individual evidence

1. ↑ B. Pareigis: Categories and Functors , BG Teubner (1969), Chapter 1.9: Difference kernels and cores
2. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, definition 16.2
3. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, examples 16.9
4. ↑ Horst Herrlich, George E. Strecker: Category Theory , Allyn and Bacon Inc. 1973, sentence 16.4