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352:). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The
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of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism
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1181:
Awodey, s. (1996). "Structure in
Mathematics and Logic: A Categorical Perspective".
482:
1552:
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1434:
1320:. Grundlehren der mathematischen Wissenschaften. Vol. 332. pp. 35–69.
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1019:
349:
46:
1202:
Baez, John C. (2004). "Quantum
Quandaries: A Category-Theoretic Perspective".
1194:
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Carter, J. Scott; Crans, Alissa; Elhamdadi, Mohamed; Saito, Masahico (2008).
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387:, the diagonal morphism can be simply described by its action on elements
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Popescu, Nicolae; Popescu, Liliana (1979). "Categories and functors".
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restrictions that the image of the diagonal map will fail to satisfy.
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27:
For the particular instance of the notion in algebraic geometry, see
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857:{\displaystyle \delta _{b}\circ \tau _{l}=\operatorname {id} _{b}}
346:
33:
231:{\displaystyle \pi _{k}\circ \delta _{a}=\operatorname {id} _{a}}
1364:
Muro, Fernando (2016). "Homotopy units in A-infinity algebras".
798:{\displaystyle \delta _{b}\colon b\sqcup b{\stackrel {}{\to }}b}
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at that element. However, most notions of sequence spaces have
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682:
84:
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539:{\displaystyle \mathbb {R} \rightarrow \mathbb {R} ^{2}}
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1397:"Some categorical properties of complex spaces Part II"
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Bulletin of the
Faculty of Education, Chiba University
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Kashiwara, Msakia; Schapira, Pierre (2006). "Limits".
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wikibooks:Category Theory/(Co-)cones and (co-)limits
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Publications du Département de Mathématiques (Lyon)
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172:{\displaystyle \delta _{a}:a\rightarrow a\times a}
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1022:if and only if the codiagonal is an isomorphism.
1249:"Cohomology of Categorical Self-Distributivity"
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1285:Faith, Carl (1973). "Product and Coproduct".
1204:The Structural Foundations of Quantum Gravity
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649:The dual notion of a diagonal morphism is a
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1497:"Lectures on basic homological algebra"
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505:. The reason for the name is that the
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333:-th component. The existence of this
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638:; each element maps to the constant
1509:"Categories for Me [note]"
595:
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935:is the injection morphism to the
580:. The diagonal morphism into the
550:is given by the line that is the
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1495:Herscovich, Estanislao (2020).
1007:{\displaystyle {\mathcal {C}}}
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690:{\displaystyle {\mathcal {C}}}
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356:of a diagonal morphism in the
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92:{\displaystyle {\mathcal {C}}}
1:
1478:"Categories for Me, and You?"
898:{\displaystyle l\in \{1,2\}.}
311:canonical projection morphism
272:{\displaystyle k\in \{1,2\},}
1567:. You can help Knowledge by
1454:"Petit guide des catégories"
990:be a morphism in a category
1435:10.1007/978-94-009-9550-5_1
1297:10.1007/978-3-642-80634-6_4
603:{\displaystyle X^{\infty }}
1635:
1546:
1094:Popescu & Popescu 1979
26:
1507:Laurent, Olivier (2013).
1395:Masakatsu, Uzawa (1972).
928:{\displaystyle \tau _{l}}
719:{\displaystyle b\sqcup b}
128:exists, there exists the
121:{\displaystyle a\times a}
1476:Aubert, Clément (2019).
1343:Mitchell, Barry (1965).
983:{\displaystyle f:X\to Y}
337:is a consequence of the
302:{\displaystyle \pi _{k}}
1326:10.1007/3-540-27950-4_3
1195:10.1093/philmat/4.3.209
1183:Philosophia Mathematica
1563:-related article is a
1366:Trans. Amer. Math. Soc
1317:Categories and Sheaves
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1346:Theory of Categories
1206:. pp. 240–265.
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1452:Pupier, R. (1964).
1291:. pp. 83–109.
1278:2006math......7417C
1222:2004quant.ph..4040B
1152:co-Diagonal in nlab
653:. For every object
573:{\displaystyle y=x}
385:concrete categories
1429:. pp. 1–148.
1349:. Academic Press.
1062:Carter et al. 2008
1037:Diagonal embedding
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1444:978-94-009-9552-9
1388:10.1090/tran/6545
1356:978-0-12-499250-4
1335:978-3-540-27949-5
1306:978-3-642-80636-0
1239:978-0-19-926969-3
948:{\displaystyle l}
789:
666:{\displaystyle b}
631:{\displaystyle X}
498:{\displaystyle x}
420:{\displaystyle a}
400:{\displaystyle x}
366:Cartesian product
326:{\displaystyle k}
130:diagonal morphism
65:{\displaystyle a}
16:(Redirected from
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1269:math/0607417
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1175:Bibliography
1159:
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1123:Laurent 2013
1101:
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808:satisfying
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485:formed from
483:ordered pair
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182:satisfying
181:
129:
49:, for every
40:
18:Diagonal map
1540:ncatlab.org
1528:ncatlab.org
1020:epimorphism
644:convergence
350:isomorphism
47:mathematics
1608:Categories
1487:1910.05172
1464:(1): 1–18.
1078:Faith 1973
1048:References
699:coproducts
697:where the
618:valued in
427:. Namely,
99:where the
1614:Morphisms
1416:0577-6856
1410:: 83–93.
1379:1111.2723
1165:Muro 2016
1014:with the
975:→
917:τ
875:∈
833:τ
829:∘
820:δ
763:→
753:⊔
747::
738:δ
711:⊔
614:into the
612:injection
596:∞
548:real line
522:→
469:⟩
457:⟨
436:δ
376:, namely
291:π
249:∈
207:δ
203:∘
194:π
164:×
158:→
143:δ
113:×
72:in every
1026:See also
640:sequence
378:equality
370:relation
335:morphism
74:category
1288:Algebra
1274:Bibcode
1218:Bibcode
1016:pushout
546:on the
372:on the
368:, is a
364:of the
360:, as a
313:to the
309:is the
101:product
1441:
1414:
1353:
1332:
1303:
1236:
1018:is an
908:where
481:, the
374:domain
362:subset
282:where
51:object
1559:This
1512:(PDF)
1500:(PDF)
1482:arXiv
1400:(PDF)
1374:arXiv
1264:arXiv
1252:(PDF)
1208:arXiv
552:graph
507:image
354:image
347:up to
341:that
1565:stub
1439:ISBN
1412:ISSN
1351:ISBN
1330:ISBN
1301:ISBN
1234:ISBN
958:Let
864:for
383:For
238:for
1431:doi
1384:doi
1370:368
1322:doi
1293:doi
1226:doi
1191:doi
41:In
1610::
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1130:^
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220:id
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1484::
1462:1
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893:.
890:}
887:2
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878:{
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768:[
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626:X
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565:=
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518:R
493:x
466:x
463:,
460:x
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448:x
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264:}
261:2
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255:1
252:{
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152::
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