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1299:. dg enhancements of an exact functor between triangulated categories are defined similarly. In general, there need not exist dg enhancements of triangulated categories or functors between them, for example
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920:{\displaystyle \mathrm {Hom} _{C({\mathcal {A}}),n}(A,B)=\prod _{l\in \mathbb {Z} }\mathrm {Hom} (A_{l},B_{l+n})}
1420:
Alberto
Canonaco; Paolo Stellari (2017), "A tour about existence and uniqueness of dg enhancements and lifts",
1300:
446:{\displaystyle \operatorname {Hom} (A,B)\otimes \operatorname {Hom} (B,C)\rightarrow \operatorname {Hom} (A,C)}
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and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho(
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structure such that weak equivalences are those functors that induce an equivalence of
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for a survey of existence and unicity results of dg enhancements dg enhancements.
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may be considered to be a DG-category by imposing the trivial grading (i.e. all
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177:{\displaystyle \bigoplus _{n\in \mathbb {Z} }\operatorname {Hom} _{n}(A,B)}
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is a pretriangulated dg category whose homotopy category is equivalent to
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A DG-category with one object is the same as a DG-ring. A DG-ring over a
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1434:
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whose morphism sets are endowed with the additional structure of a
1361:
Tabuada, Gonçalo (2005), "Invariants additifs de DG-catégories",
731:{\displaystyle \operatorname {Hom} _{C({\mathcal {A}}),n}(A,B)}
1159:, respectively. This applies to the category of complexes of
1099:{\displaystyle f_{l+1}\circ d_{A}+(-1)^{n+1}d_{B}\circ f_{l}}
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A little bit more sophisticated is the category of complexes
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is called pre-triangulated if it has a suspension functor
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492:{\displaystyle d(\operatorname {id} _{A})=0}
367:. Furthermore, the composition of morphisms
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191:on this graded group, i.e., for each
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1197:dg-categories can be endowed with a
932:The differential of such a morphism
325:. This is equivalent to saying that
1244:Relation to triangulated categories
1474:(1994), "Deriving DG categories",
1404:: CS1 maint: unflagged free DOI (
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643:{\displaystyle C({\mathcal {A}})}
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107:, the morphisms from any object
1422:Journal of Geometry and Physics
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1541:Categories in category theory
1313:Grothendieck abelian category
588:) and trivial differential (
187:and there is a differential
57:{\displaystyle \mathbb {Z} }
26:differential graded category
1280:. A triangulated category
1182:differential graded algebra
1144:{\displaystyle d_{A},d_{B}}
69:In detail, this means that
1557:
650:over an additive category
318:{\displaystyle d\circ d=0}
1180:is called DG-algebra, or
1151:are the differentials of
1301:stable homotopy category
1265:{\displaystyle \Sigma }
581:{\displaystyle n\neq 0}
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1161:quasi-coherent sheaves
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115:of the category is a
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28:, often shortened to
1333:Graded (mathematics)
1328:Differential algebra
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1208:Given a dg-category
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1536:Homological algebra
1521:dg-category in nLab
1491:10.24033/asens.1689
1444:2017JGP...122...28C
607:{\displaystyle d=0}
22:homological algebra
1284:is said to have a
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1189:Further properties
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1371:(53): 3309–3339,
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1480:, SĂ©rie 4,
562:vanish for
195:there is a
34:DG category
30:dg-category
18:mathematics
1530:Categories
1435:1605.00490
1349:References
1003:of degree
197:linear map
117:direct sum
1500:0012-9593
1460:119326832
1428:: 28–52,
1393:119162782
1385:1073-7928
1343:Derivator
1260:Σ
1084:∘
1052:−
1033:∘
972:→
959::
862:∈
855:∏
773:which do
749:→
711:
573:≠
547:−
541:−
426:
420:→
402:
396:⊗
378:
336:
304:∘
262:
243:→
225:
212::
157:
137:∈
130:⨁
80:
1400:citation
1322:See also
1228:in case
503:Examples
38:category
1508:1258406
1440:Bibcode
1311:) of a
1276:) is a
1167:over a
785:, i.e.,
64:-module
36:, is a
1506:
1498:
1458:
1391:
1383:
1222:smooth
1165:scheme
1111:where
1456:S2CID
1430:arXiv
1389:S2CID
1236:over
1195:small
1178:field
1163:on a
363:is a
1496:ISSN
1418:See
1406:link
1381:ISSN
1369:2005
1224:and
1169:ring
1155:and
781:and
508:Any
24:, a
1486:doi
1448:doi
1426:122
1373:doi
1291:if
775:not
683:Hom
423:Hom
399:Hom
375:Hom
333:Hom
247:Hom
216:Hom
148:Hom
77:Hom
32:or
16:In
1532::
1504:MR
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1494:,
1482:27
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