Diagram chases without elements
It is possible to perform diagram chases even in categories in which the objects are not sets, with a mathematical sleight-of-hand using something called members.
Call an arrow
Each object
Rules for chasing diagrams
For the members in any abelian category:
is a monomorphism if and only if for all , implies ; is a monomorphism if and only if, for all , implies ; is an epimorphism if and only if for each there is with ; is the zero arrow if and only if, for all , ; - A sequence
is exact at if and only if and to every with there exists with ; - (Subtraction) Given
and with , there is a member with ; moreover, any with has and any with has .