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The generalisation of this to more than two morphisms is straightforward; simply use a larger diagram with more morphisms in it. The degenerate case of only one morphism is also straightforward; then
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The notion of difference kernel also makes sense in a category-theoretic context. The terminology "difference kernel" is common throughout category theory for any binary equaliser. In the case of a
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of zero under that function; that is not true in all situations. However, the terminology "difference kernel" has no other meaning.
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category, including the categories where difference kernels are used, as well as the category of sets itself, the object
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539:{\displaystyle \operatorname {Eq} ({\mathcal {F}}):=\{x\in X\mid \forall f,g\in {\mathcal {F}},\;f(x)=g(x)\}.}
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of morphisms which appear in the diagram. With this view, we see that if there are no morphisms involved,
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equaliser, that is an equaliser of exactly two morphisms. However, if the category in question is
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morphisms is slightly subtle: one might initially draw the diagram as consisting of the objects
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Of course, all of this presumes an algebraic context where the kernel of a function is the
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that is an equaliser of some set of morphisms. Some authors require more strictly that
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and no morphisms. This is incorrect, however, since the limit of such a diagram is the
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which generates examples of equalisers in the category of finite sets. Written by
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A binary equaliser (that is, an equaliser of just two functions) is also called a
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can always be taken to be the ordinary notion of equaliser, and the morphism
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325:{\displaystyle \operatorname {Eq} (f,g):=\{x\in X\mid f(x)=g(x)\}.}
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does not make an appearance and the equaliser diagram consists of
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alone. The limit of this diagram is then any isomorphism between
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Set of arguments where two or more functions have the same value
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in the category in question, and the equaliser is simply the
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holds in a given category, then that category is said to be
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In more explicit terms, the equaliser consists of an object
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It can be proved that any equaliser in any category is a
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that can be constructed from equalisers and products.
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120:Learn how and when to remove this message
354:The definition above used two functions
140:is a set of arguments where two or more
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640:. As an even more degenerate case, let
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790:. These objects and morphisms form a
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549:This equaliser may be written as Eq(
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854:{\displaystyle f\circ eq=g\circ eq}
672:. This may also be denoted DiffKer(
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1082:{\displaystyle f\circ m=g\circ m}
908:{\displaystyle f\circ m=g\circ m}
398:such that, given any two members
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1001:{\displaystyle m:O\rightarrow X}
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700:: The difference kernel of
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782:are morphisms from
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43:This article
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1946:Categorified
1850:n-categories
1801:Key concepts
1639:Direct limit
1622:Coequalizers
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1540:Yoneda lemma
1446:Key concepts
1436:Key concepts
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1338:
1324:
1307:, a special
1270:
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1240:(a category
1235:
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652:, since the
649:
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586:
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185:
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150:solution set
137:
131:
116:
107:
97:
90:
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76:
64:
52:Please help
47:verification
44:
1914:-categories
1890:Kan complex
1880:Tricategory
1862:-categories
1752:Subcategory
1510:Exponential
1478:Preadditive
1473:Pre-abelian
1285:Coequaliser
1129:isomorphism
1127:can be any
1008:is said to
976:A morphism
817:satisfying
732:with value
382:, then the
200:. Then the
164:Definitions
134:mathematics
110:August 2024
2062:Set theory
2056:Categories
1932:3-category
1922:2-category
1895:∞-groupoid
1870:Bicategory
1617:Coproducts
1577:Equalizers
1483:Bicategory
1358:References
931:such that
761:categories
688:), or Ker(
614:degenerate
561:, ...) if
220:such that
80:newspapers
1981:Symmetric
1926:2-functor
1666:Relations
1589:Pullbacks
1364:Equalizer
1205:. If the
1074:∘
1062:∘
993:→
945:∘
919:morphism
900:∘
888:∘
843:∘
828:∘
646:empty set
628:}. Since
622:singleton
490:∈
478:∀
475:∣
469:∈
444:
418:) equals
384:equaliser
287:∣
281:∈
254:
228:) equals
202:equaliser
190:functions
142:functions
138:equaliser
2041:Glossary
2021:Category
1995:n-monoid
1948:concepts
1604:Colimits
1572:Products
1525:Morphism
1468:Concrete
1463:Additive
1453:Category
1336:(1998).
1305:Pullback
1279:See also
1256:) = Ker(
1242:enriched
1231:complete
1207:converse
1180:codomain
1010:equalise
923: :
869: :
809: :
741:preimage
716:−
692:−
364:finitely
154:equation
2031:Outline
1990:n-group
1955:2-group
1910:Strict
1900:∞-topos
1696:Modules
1634:Pushout
1582:Kernels
1515:Functor
1458:Abelian
1366:at the
1211:regular
1178:is the
1156:product
1092:In any
792:diagram
680:), Ker(
644:be the
94:scholar
1977:Traced
1960:2-ring
1690:Fields
1676:Groups
1671:Magmas
1559:Limits
1287:, the
1227:binary
1114:subset
917:unique
710:kernel
180:. Let
152:of an
96:
89:
82:
75:
67:
1971:-ring
1858:Weak
1842:Topos
1686:Rings
1343:(PDF)
1316:Notes
1309:limit
1225:be a
1112:as a
877:, if
796:limit
620:be a
612:As a
426:) in
370:is a
236:) in
146:equal
144:have
136:, an
101:JSTOR
87:books
1661:Sets
1289:dual
1194:and
1162:and
1150:and
1031:and
778:and
770:and
734:zero
704:and
402:and
358:and
208:and
184:and
178:sets
172:and
168:Let
73:news
1505:End
1495:CCC
1371:Lab
1158:of
1135:to
1116:of
1108:of
1051:if
786:to
406:of
394:of
378:to
372:set
216:of
204:of
196:to
188:be
176:be
132:In
56:by
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1332:;
1268:.
1260:-
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1198:.
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1139:.
1125:eq
1120:.
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1089:.
966:.
927:→
873:→
813:→
807:eq
763:.
736:.
684:,
676:,
660:.
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601:=
593:,
589:,
557:,
553:,
460::=
441:Eq
410:,
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339:,
272::=
251:Eq
1975:(
1968:n
1966:E
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1924:(
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1872:(
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954:m
951:=
948:u
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