# Why homotopy category is not abelian?

Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are equivalent class that identify maps that are homotopic. The result is Ch(A) is an abelian category but HoCh(A) is no longer abelian.

I even do not know how to start? Which condition of abelian category will fail in this case? Maybe some map in the category does not have kernel?

• @Eric: given the homological algebra tag, presumably the homotopy category of chain complexes. – Qiaochu Yuan Sep 28 '15 at 22:25
• – user99914 Sep 28 '15 at 22:27
• @EricWofsey Sorry about that, I have edited the question. – user198206 Sep 28 '15 at 22:27
• @JohnMa Thanks, actually I have read that but I have not learn triangulated category and do not quite understand that answer...... – user198206 Sep 28 '15 at 22:29
• For abelian groups, I've posted a concrete counterexample for $\operatorname{K}(\mathsf{Ab})$ not being abelian. – Takumi Murayama Sep 28 '15 at 22:43