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We construct an "almost involution" assigning a new DG-category to a givenone, and use this construction in order to recover, say, the abelian categoryof graded modules over the graded ring $R^*$ from the DG-category of DG-modulesover a DG-ring $(R^*,d)$. This provides an appropriate technical background forthe definition and discussion of abelian and exact DG-categories. In thesetting of exact DG-categories, derived categories of the second kind aredefined in the maximal natural generality. We develop the related abstractcategory-theoretic language and use it in order to formulate and prove severalfull-and-faithfulness theorems for triangulated functors induced by theinclusions of fully exact DG-subcategories. Such functors are fully faithfulfor derived categories of the second kind more often than for the conventionalderived categories. Examples and applications range from the categories ofcomplexes in abelian/exact categories to matrix factorization categories, andfrom curved DG-modules over curved DG-rings to quasi-coherent CDG-modules overquasi-coherent CDG-quasi-algebras over schemes.

Exact DG-categories and fully faithful triangulated inclusion functors