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).

Created on August 23, 2014 at 12:34:56. See the history of this page for a list of all contributions to it.