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 x with codomain a∈A a member of a, written x∈ma, and define x≑y for two members of a to mean there are epimorphisms u,v with xy=yv. One can check this is an equivalence relation on the set of members of a. We can then think of members of a as equivalence classes of arrows to a, with this relation.

Each object a has a zero member (the equivalence class of the zero arrow 0β†’a). Each member x∈ma also has a "negative", denoted βˆ’x.

Rules for chasing diagrams

For the members in any abelian category:

  1. f:aβ†’b is a monomorphism if and only if for all x∈ma, fx≑0 implies x≑0;
  2. f:aβ†’b is a monomorphism if and only if, for all x,xβ€²βˆˆma, fx≑fxβ€² implies x≑xβ€²;
  3. g:bβ†’c is an epimorphism if and only if for each z∈mc there is y∈mb with gy≑z;
  4. h:rβ†’s is the zero arrow if and only if, for all x∈mr, hx≑0;
  5. A sequence aβ†’fbβ†’gc is exact at b if and only if gf=0 and to every y∈mb with gy≑0 there exists x∈ma with fx≑y;
  6. (Subtraction) Given g:bβ†’c and x,y∈mb with gx≑gy, there is a member z∈mb with gz≑0; moreover, any f:bβ†’d with fx≑0 has fy≑fz and any h:bβ†’a with hy≑0 has hxβ‰‘βˆ’hz.