Differential graded category

 ( Homological Algebra ) 

In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded $\backslash Z$-module.

In detail, this means that $\backslash operatorname\{Hom\}(A,B)$, the morphisms from any object A to another object B of the category is a direct sum

$\backslash bigoplus\_\{n\; \backslash in\; \backslash Z\}\backslash operatorname\{Hom\}\_n(A,B)$

and there is a differential d on this graded group, i.e., for each n there is a linear map

$d\backslash colon\; \backslash operatorname\{Hom\}\_n(A,B)\; \backslash rightarrow\; \backslash operatorname\{Hom\}\_\{n+1\}(A,B)$,

which has to satisfy $d\; \backslash circ\; d\; =\; 0$. This is equivalent to saying that $\backslash operatorname\{Hom\}(A,B)$ is a cochain complex. Furthermore, the composition of morphisms $\backslash operatorname\{Hom\}(A,B)\; \backslash otimes\; \backslash operatorname\{Hom\}(B,C)\; \backslash rightarrow\; \backslash operatorname\{Hom\}(A,C)$ is required to be a map of complexes, and for all objects A of the category, one requires $d(\backslash operatorname\{id\}\_A)\; =\; 0$.

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Examples

• Any additive category may be considered to be a DG-category by imposing the trivial grading (i.e. all $\backslash mathrm\{Hom\}\_n(-,-)$ vanish for $n\backslash ne\; 0$) and trivial differential ($d=0$).
• A little bit more sophisticated is the category of complexes $C(\backslash mathcal\; A)$ over an additive category $\backslash mathcal\; A$. By definition, $\backslash operatorname\{Hom\}\_\{C(\backslash mathcal\; A),\; n\}\; (A,\; B)$ is the group of maps $A\; \backslash rightarrow\; Bn$ which do not need to respect the differentials of the complexes A and B, i.e.,

:$\backslash mathrm\{Hom\}\_\{C(\backslash mathcal\; A),\; n\}\; (A,\; B)\; =\; \backslash prod\_\{l\; \backslash in\; \backslash Z\}\; \backslash mathrm\{Hom\}(A\_l,\; B\_\{l+n\})$.

The differential of such a morphism $f\; =\; (f\_l\; \backslash colon\; A\_l\; \backslash rightarrow\; B\_\{l+n\})$ of degree n is defined to be

:$f\_\{l+1\}\; \backslash circ\; d\_A\; +\; (-1)^\{n+1\}\; d\_B\; \backslash circ\; f\_l$,

where $d\_A,\; d\_B$ are the differentials of A and B, respectively. This applies to the category of complexes of quasi-coherent sheaves on a scheme over a ring.

• A DG-category with one object is the same as a DG-ring. A DG-ring over a field is called DG-algebra, or differential graded algebra.

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Further properties The category of small category dg-categories can be endowed with a model category structure such that weak equivalences are those functors that induce an equivalence of derived category.

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 and in case C is the category of quasi-coherent sheaves on some scheme X over R.

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Relation to triangulated categories A DG category C is called pre-triangulated if it has a suspension functor $\backslash Sigma$ and a class of distinguished triangles compatible with the suspension, such that its homotopy category Ho( C) is a triangulated category. A triangulated category T is said to have a dg enhancement C if C is a pretriangulated dg category whose homotopy category is equivalent to T.See for a survey of existence and unicity results of dg enhancements dg enhancements. 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 stable homotopy category can be shown not to arise from a dg category in this way. However, various positive results do exist, for example the derived category D( A) of a Grothendieck abelian category A admits a unique dg enhancement.

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See also

• Differential algebra
• Graded (mathematics)
• Graded category
• Derivator

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External links

• dg-category in nLab

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