nLab homotopy category of a dg-category

Let $A$ be a dg-category, so that one has a mapping complex of morphisms $Map_A(x,y)$ between any two objects $x$ and $y$. In analogy with simplicially enriched categories, the homotopy category of $A$ is the preadditive category $ho(A)$ whose objects are the objects of $A$, morphism groups are given by

(1)$$\Hom_{ho(A)}(x, y) = H^0(Map_A(x,y))$$

for any two objects $x$ and $y$, and composition is induced from the composition law of $A$. Here $H^0(-)$ denotes the zeroth cohomology group of the complex (which is by convention graded cohomologically).

Related entries

• homotopy category of an (infinity,1)-category