Diagram chases without elements

#category_theory/abelian_categories

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 $a \in A$ a member of $a$ , written $x \in_{m} a$ , and define $x \equiv y$ for two members of $a$ to mean there are epimorphisms $u, v$ with $x y = y v$ . 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 $0 \to a$ ). Each member $x \in_{m} a$ also has a "negative", denoted $- x$ .

Rules for chasing diagrams

For the members in any abelian category:

$f : a \to b$ is a monomorphism if and only if for all $x \in_{m} a$ , $f x \equiv 0$ implies $x \equiv 0$ ;
$f : a \to b$ is a monomorphism if and only if, for all $x, x^{'} \in_{m} a$ , $f x \equiv f x^{'}$ implies $x \equiv x^{'}$ ;
$g : b \to c$ is an epimorphism if and only if for each $z \in_{m} c$ there is $y \in_{m} b$ with $g y \equiv z$ ;
$h : r \to s$ is the zero arrow if and only if, for all $x \in_{m} r$ , $h x \equiv 0$ ;
A sequence $a \overset{f}{\to} b \overset{g}{\to} c$ is exact at $b$ if and only if $g f = 0$ and to every $y \in_{m} b$ with $g y \equiv 0$ there exists $x \in_{m} a$ with $f x \equiv y$ ;
(Subtraction) Given $g : b \to c$ and $x, y \in_{m} b$ with $g x \equiv g y$ , there is a member $z \in_{m} b$ with $g z \equiv 0$ ; moreover, any $f : b \to d$ with $f x \equiv 0$ has $f y \equiv f z$ and any $h : b \to a$ with $h y \equiv 0$ has $h x \equiv - h z$ .

(UNDER CONSTRUCTION)