▁a ▁pre - abel ian ▁category ▁ ▁has ▁a ▁nod al ▁decomposition , ▁then ▁there ▁exist ▁morph isms ▁ ▁and ▁ ▁which ▁( be ing ▁not ▁necessarily ▁is omorph isms ) ▁connect ▁the ▁nod al ▁decomposition ▁with ▁the ▁basic ▁decomposition ▁by ▁the ▁following ▁ident ities : ▁ ▁C ategories ▁with ▁nod al ▁decomposition ▁▁ ▁A ▁category ▁ ▁is ▁called ▁a ▁category ▁with ▁nod al ▁decomposition ▁if ▁each ▁morph ism ▁ ▁has ▁a ▁nod al ▁decomposition ▁in ▁. ▁This ▁property ▁plays ▁an ▁important ▁role ▁in ▁construct ing ▁en velop es ▁and ▁ref in ements ▁in ▁. ▁▁ ▁In ▁an ▁ab elian ▁category ▁ ▁the ▁basic ▁decomposition ▁▁ ▁is ▁always ▁nod al . ▁As ▁a ▁cor ollary , ▁all ▁ab elian ▁categories ▁have ▁nod al ▁decomposition . ▁▁ ▁If ▁a ▁pre - abel ian ▁category ▁ ▁is ▁linear ly ▁complete , ▁well - power ed ▁in ▁strong ▁mon omorph isms ▁and ▁co - well - power ed ▁in ▁strong ▁ep im orph isms , ▁then ▁▁ ▁has ▁nod al ▁decomposition . ▁ ▁More ▁generally , ▁suppose ▁a ▁category ▁ ▁is ▁linear ly ▁complete , ▁well - power ed ▁in ▁strong ▁mon omorph isms , ▁co - well - power ed ▁in ▁strong ▁ep im orph isms , ▁and ▁in ▁addition ▁strong ▁ep im orph isms ▁disc ern ▁mon omorph isms ▁in ▁, ▁and , ▁du ally , ▁strong ▁mon omorph isms ▁disc ern ▁ep im orph isms ▁in ▁, ▁then ▁▁ ▁has ▁nod al ▁decomposition . ▁ ▁The ▁category ▁Ste ▁of ▁s tere ot ype