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| In [[mathematics]], especially [[homological algebra]], a '''differential graded category''' or '''DG category''' for short, is a [[category (mathematics)|category]] whose morphism sets are endowed with the additional structure of a differential graded ''Z''-module.
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| In detail, this means that <math>Hom(A,B)</math>, the morphisms from any object ''A'' to another object ''B'' of the category is a direct sum <math>\oplus_{n \in \mathbf Z}Hom_n(A,B)</math> and there is a differential ''d'' on this graded group, i.e. for all ''n'' a linear map <math>d: Hom_n(A,B) \rightarrow Hom_{n+1}(A,B)</math>, which has to satisfy <math>d \circ d = 0</math>. This is equivalent to saying that <math>Hom(A,B)</math> is a [[cochain complex]]. Furthermore, the composition of morphisms
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| <math>Hom(A,B) \otimes Hom(B,C) \rightarrow Hom(A,C)</math> is required to be a map of complexes, and for all objects ''A'' of the category, one requires <math>d(id_A) = 0</math>.
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| == See also ==
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| * [[Differential algebra]]
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| * [[Graded (mathematics)]]
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| * [[Graded category]]
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| ==Examples==
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| * Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all <math>\mathrm{Hom}_n(-,-)</math> vanish for ''n'' ≠ 0) and trivial differential (''d'' = 0).
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| * A little bit more sophisticated is the category of complexes <math>C(\mathcal A)</math> over an additive category <math>\mathcal A</math>. By definition, <math>\mathrm{Hom}_{C(\mathcal A), n} (A, B)</math> is the group of maps <math>A \rightarrow B[n]</math> which do ''not'' need to respect the differentials of the complexes ''A'' and ''B'', i.e. <math>\mathrm{Hom}_{C(\mathcal A), n} (A, B) = \Pi_{l \in \mathbf Z} \mathrm{Hom}(A_l, B_{l+n})</math>. The differential of such a morphism <math>f = (f_l : A_l \rightarrow B_{l+n})</math> of degree ''n'' is defined to be <math>f_{l+1} \circ d_A + (-1)^{n+1} d_B \circ f_l</math>, where <math>d_A, d_B</math> are the differentials of ''A'' and ''B'', respectively. This applies to the category of complexes of [[quasi-coherent sheaf|quasi-coherent sheaves]] on a [[scheme (mathematics)|scheme]] over a ring.
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| * A DG-category with one object is the same as a DG-ring . DG-ring over a field is called DG-algebra, or [[differential graded algebra]].
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| ==Further properties==
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| The category of small dg-categories can be endowed with a [[model category]] structure such that weak equivalences are those functors that induce an equivalence of [[derived category|derived categories]].<ref>{{Citation | last1=Tabuada | first1=Gonçalo | title=Invariants additifs de DG-catégories | url=http://dx.doi.org/10.1155/IMRN.2005.3309 | doi=10.1155/IMRN.2005.3309 | year=2005 | journal=International Mathematics Research Notices | issn=1073-7928 | issue=53 | pages=3309–3339}}</ref>
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| Given a dg-category ''C'' over some ring ''R'', there is a notion of smoothness and properness of ''C'' that reduces to the usual notions of [[smooth morphism|smooth]] and [[proper morphism]]s in case ''C'' is the category of quasi-coherent sheaves on some scheme ''X'' over ''R''.
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| ==References==
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| <references />
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| * {{Citation | last1=Keller | first1=Bernhard | title=Deriving DG categories | url=http://www.numdam.org/numdam-bin/fitem?id=ASENS_1994_4_27_1_63_0 | mr=1258406 | year=1994 | journal=Annales Scientifiques de l'École Normale Supérieure. Quatrième Série | issn=0012-9593 | volume=27 | issue=1 | pages=63–102}}
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| {{DEFAULTSORT:Differential Graded Category}}
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| [[Category:Homological algebra]]
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