Differential coke

The difference coker is a mathematical term from the subfield of category theory . It is the dual concept to the difference core . Alternative names are Koegalizer or, based on the English name, Koequalizer . The spellings with "c", that is, difference coker , coegalizer or coequalizer , are also common.

definition

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

• ${\displaystyle q\circ f=q\circ g}$
• If there is also a morphism , then there is a clearly determined morphism with . ${\displaystyle q'\colon Y\rightarrow Z'}$ ${\displaystyle q'\circ f=q'\circ g}$ ${\displaystyle c\colon Z\rightarrow Z'}$ ${\displaystyle q'=c\circ q}$

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

Examples

• In the category Set of the sets or the category Top of the topological spaces are as in the above definition. Let the smallest equivalence relation containing all pairs be assumed. Then the identification map is a differential coker of and . ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle Q}$ ${\displaystyle Y}$ ${\displaystyle (f(x),g(x)),\ x\in X}$ ${\displaystyle q\rightarrow Y/Q}$ ${\displaystyle f}$ ${\displaystyle g}$
• In the category - Mod of the link modules over a ring , in the situation of the above definition, let the sub-module of all differences generated by . Then the quotient mapping is a difference coker of and . So this is nothing else than the coke core of the difference , which explains the term differential coke. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle Q}$ ${\displaystyle f(x)-g(x),\ x\in X}$ ${\displaystyle Y}$ ${\displaystyle q\rightarrow Y/Q}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle f-g}$
• Has the considered category zero objects and is in the situation above definition of zero morphism , so is a Differenzkokern of and nothing but a cokernel of . Each coke core is thus an example of a differential coke core. ${\displaystyle g=0_{XY}}$ ${\displaystyle X\rightarrow Y}$ ${\displaystyle f}$ ${\displaystyle 0_{XY}}$ ${\displaystyle f}$

Remarks

• Differential cokes are not clearly determined. But are in the situation above definition and two Differenzkokerne of and it follows from the uniqueness property that there is a uniquely determined isomorphism with there. Differential cokes are therefore determined up to (unambiguous) isomorphism, which is why one often speaks of the differential coke. ${\displaystyle q\colon Y\rightarrow Z}$ ${\displaystyle {\tilde {q}}\colon Y\rightarrow {\tilde {Z}}}$ ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle c\colon Z\rightarrow {\tilde {Z}}}$ ${\displaystyle {\tilde {q}}=c\circ q}$
• In another linguistic inaccuracy, the object is called the differential coke. The actually meant morphism is then always an obvious quotient mapping and is therefore not mentioned. ${\displaystyle Z}$
• It is said that a category has differential cokers if there is one differential coker for every two morphisms . Referred to in the examples above categories Set , Top and - Mod apparently have Differenzkokerne. ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle R}$
• The difference cores of a category are exactly the difference cores of the dual category .
• A morphism is a difference kernel of if and only if the diagram ${\displaystyle q\colon Y\rightarrow Z}$ ${\displaystyle f,g\colon X\rightarrow Y}$

  ${\displaystyle {\begin{array}{ccc}X&{\xrightarrow {f}}&Y\\\downarrow ^{g}&&\downarrow ^{q}\\Y&{\xrightarrow {q}}&Z\\\end{array}}}$

is a pushout .

• Each differential coker is an epimorphism . The reverse is generally not true, those epimorphisms that occur as differential cokers are called regular .

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, Example 16.3 (2)
4. ^ H. Schubert: Categories II, Akademie-Verlag Berlin 1970, sentence 18.4.3