Exact dg categories I : Foundations

We introduce the notion of exact dg category, which provides a differential graded enhancement of Nakaoka--Palu's notion of extriangulated category. We give a definition in complete analogy with Quillen's but where the category of kernel-cokernel pairs is replaced with a more sophisticated homotopy category. We introduce the notion of stable dg category, and prove that the H⁰-category of an exact dg category A is triangulated if and only if A is stable. We illustrate our theory with several examples including the homotopy category of two-term complexes and Amiot's fundamental domain for generalized cluster categories.