nLab
homotopy category of a dg-category

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

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

for any two objects xx and yy, and composition is induced from the composition law of AA. Here H 0()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.