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that must be satisfied if the equation is to have a solution, and its dimension is the number of independent constraints that must be satisfied for the equation to have a solution.
512:
1114:
991:{\displaystyle {\begin{aligned}\operatorname {im} (f)&=\ker(\operatorname {coker} f),\\\operatorname {coim} (f)&=\operatorname {coker} (\ker f).\end{aligned}}}
1253:
The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space
1427:. A map is injective if and only if its kernel is trivial, and a map is surjective if and only if its cokernel is trivial, or in other words, if
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must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the
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settings, such as with bounded linear operators between
Hilbert spaces, one typically has to take the
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44:
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1173:{\displaystyle 0\to \ker T\to V{\overset {T}{\longrightarrow }}W\to \operatorname {coker} T\to 0.}
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In particular, every abelian category is normal (and conormal as well). That is, every
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is the freedom in a solution. The cokernel may be expressed via the real valued map
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As with all universal constructions the cokernel, if it exists, is unique
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Additionally, the cokernel can be thought of as something that "detects"
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775:, it makes sense to add and subtract morphisms. In such a category, the
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1362:, (one degree of freedom). The kernel may be expressed as the subspace
848:{\displaystyle \operatorname {coeq} (f,g)=\operatorname {coker} (g-f).}
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756:{\displaystyle \operatorname {coker} (f)=H/\operatorname {im} (f).}
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Formally, one may connect the kernel and the cokernel of a map
627:) if it is the cokernel of some morphism. A category is called
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of the domain (it maps to the domain), while the cokernel is a
16:
Quotient space of a codomain of a linear map by the map's image
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456:
1924:
1008:
can be written as the kernel of some morphism. Specifically,
518:
19:"Coker (mathematics)" redirects here. For other uses, see
1338:(one constraint), and in that case the solution space is
787:(if it exists) is just the cokernel of their difference:
363:
One can define the cokernel in the general framework of
191:
that one is seeking to solve, the cokernel measures the
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These can be interpreted thus: given a linear equation
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Explicitly, this means the following. The cokernel of
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in a solution, if one exists. This is elaborated in
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755:
1058:{\displaystyle m=\ker(\operatorname {coker} (m))}
2136:
53:but its sources remain unclear because it lacks
1077:that an equation must satisfy, as the space of
1073:The cokernel can be thought of as the space of
147:. The dimension of the cokernel is called the
1491:
862:(a special kind of preadditive category) the
355:of the image before passing to the quotient.
283:with respect to this property. Often the map
174:of the codomain (it maps from the codomain).
2119:
2109:
1865:
1498:
1484:
1423:in the same way that the kernel "detects"
631:if every epimorphism is normal (e.g. the
568:, then there exists a unique isomorphism
84:Learn how and when to remove this message
1454:Categories for the Working Mathematician
2137:
1275:As a simple example, consider the map
617:. Conversely an epimorphism is called
473:for this diagram, i.e. any other such
1864:
1517:
1479:
1268:is simply the dimension of the space
1220:, and its dimension is the number of
599:Like all coequalizers, the cokernel
358:
25:
1505:
1012:is the kernel of its own cokernel:
13:
1457:, Second Edition, 1978, p. 64
210:More generally, the cokernel of a
166:, hence the name: the kernel is a
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1471:, 2014, p. 82, p. 139 footnote 8.
1331:to have a solution, we must have
291:itself is called the cokernel of
275:of the category, and furthermore
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688:is normal, the cokernel is just
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177:Intuitively, given an equation
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1241:the cokernel is the space of
488:can be obtained by composing
1469:Aurora Modern Math Originals
1272:the dimension of the image.
1068:
7:
1793:Constructions on categories
1416:to there being a solution.
1201:the kernel is the space of
638:
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1900:Higher-dimensional algebra
1465:Category Theory in Context
265:such that the composition
164:kernels of category theory
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465:. Moreover, the morphism
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525:, or more precisely: if
39:This article includes a
1710:Cokernels and quotients
1633:Universal constructions
1308:. Then for an equation
492:with a unique morphism
241:bounded linear operator
68:more precise citations.
1867:Higher category theory
1613:Natural transformation
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452:such that the diagram
400:and the zero morphism
314:, the cokernel of the
298:In many situations in
21:Coker (disambiguation)
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554:are two cokernels of
2155:Isomorphism theorems
1736:Algebraic categories
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773:preadditive category
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647:, the cokernel of a
1905:Homotopy hypothesis
1583:Commutative diagram
1350:, or equivalently,
287:is understood, and
1618:Universal property
1222:degrees of freedom
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649:group homomorphism
645:category of groups
633:category of groups
613:is necessarily an
392:is defined as the
201:degrees of freedom
41:list of references
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1815:Opposite category
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1449:Saunders Mac Lane
1396:: given a vector
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779:of two morphisms
680:. In the case of
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1810:Kleisli category
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369:zero morphisms
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304:abelian groups
302:, such as for
245:Hilbert spaces
158:Cokernels are
123:quotient space
102:linear mapping
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1884:Key concepts
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1623:Yoneda lemma
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60:Please help
52:
1997:-categories
1973:Kan complex
1963:Tricategory
1945:-categories
1835:Subcategory
1593:Exponential
1561:Preadditive
1556:Pre-abelian
1461:Emily Riehl
1421:surjections
1414:obstruction
1289:, given by
1243:constraints
1207:homogeneous
1075:constraints
777:coequalizer
615:epimorphism
523:isomorphism
394:coequalizer
349:topological
193:constraints
66:introducing
2139:Categories
2015:3-category
2005:2-category
1978:∞-groupoid
1953:Bicategory
1700:Coproducts
1660:Equalizers
1566:Bicategory
1442:References
1425:injections
1197:to solve,
1087:solutions.
2064:Symmetric
2009:2-functor
1749:Relations
1672:Pullbacks
1209:equation
1203:solutions
1165:→
1159:
1153:→
1142:⟶
1134:→
1128:
1122:→
1069:Intuition
1041:
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834:−
825:
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739:
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544:′ :
521:a unique
478:′ :
471:universal
281:universal
207:, below.
205:intuition
168:subobject
2124:Glossary
2104:Category
2078:n-monoid
2031:concepts
1687:Colimits
1655:Products
1608:Morphism
1551:Concrete
1546:Additive
1536:Category
1302:) = (0,
686:subgroup
666:quotient
655: :
639:Examples
629:conormal
625:conormal
604: :
573: :
559: :
530: :
497: :
469:must be
463:commutes
443: :
425: :
408: :
383: :
377:morphism
373:cokernel
333:quotient
322: :
256: :
243:between
235:between
231:(e.g. a
229:category
227:in some
218: :
212:morphism
137:codomain
112: :
98:cokernel
2114:Outline
2073:n-group
2038:2-group
1993:Strict
1983:∞-topos
1779:Modules
1717:Pushout
1665:Kernels
1598:Functor
1541:Abelian
1412:is the
1368:, 0) ⊆
1205:to the
1105:by the
868:coimage
672:by the
664:is the
643:In the
353:closure
339:by the
331:is the
312:modules
271:is the
162:to the
135:of the
121:is the
62:improve
2060:Traced
2043:2-ring
1773:Fields
1759:Groups
1754:Magmas
1642:Limits
1083:kernel
858:In an
693:modulo
620:normal
371:. The
237:groups
149:corank
2054:-ring
1941:Weak
1925:Topos
1769:Rings
1432:= im(
1390:) → (
1356:) + (
1321:) = (
1270:minus
1218:) = 0
1156:coker
1038:coker
961:coker
920:coker
864:image
822:coker
771:In a
710:coker
583:with
519:up to
375:of a
347:. In
341:image
239:or a
195:that
129:/ im(
100:of a
47:, or
1744:Sets
1360:, 0)
1352:(0,
1233:) =
1192:) =
939:coim
866:and
798:coeq
783:and
623:(or
539:and
186:) =
160:dual
96:The
1588:End
1578:CCC
1382:: (
1336:= 0
1245:on
1125:ker
1100:→
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1029:ker
970:ker
911:ker
668:of
396:of
343:of
335:of
310:or
279:is
268:q f
151:of
139:of
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1325:,
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1298:,
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889:im
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659:→
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596:.
588:=
586:q'
577:→
563:→
548:→
534:→
507::
501:→
482:→
447:→
429:→
417:.
412:→
405:XY
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295:.
260:→
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155:.
116:→
51:,
43:,
2058:(
2051:n
2049:E
2011:)
2007:(
1995:n
1959:)
1955:(
1943:n
1785:)
1781:(
1775:)
1771:(
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1492:t
1485:v
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1430:W
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1370:V
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1364:(
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1354:b
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1346:b
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1323:a
1319:y
1315:x
1313:(
1311:T
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1300:y
1296:x
1294:(
1292:T
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1260:T
1256:W
1247:w
1235:w
1231:v
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1227:T
1216:v
1214:(
1212:T
1194:w
1190:v
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1102:W
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1006:m
982:.
979:)
976:f
967:(
958:=
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908:=
901:)
898:f
895:(
872:f
843:.
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837:f
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828:(
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813:g
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807:f
804:(
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732:/
728:H
725:=
722:)
719:f
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670:H
661:H
657:G
653:f
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402:0
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182:(
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23:.
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