---
tag: PBE65H0506
kind: definition
name: Pseudo-element
msc: ["18E10"]
defines: 1WV2THA6NY
uses: [Y9CZ3MTNF1, 859F0TFVA3, RVXDSY04KG, YF9F9EG6M3]
sources:
  - url: https://en.wikipedia.org/wiki/Abelian_category
---

## Definition

Let $\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $A$ be an object of $\mathcal{A}$.
Consider the [[morphisms|859F0TFVA3]] with codomain $A$.
Two such morphisms $x \colon X \to A$ and $y \colon Y \to A$ are *equivalent*, written $x \sim y$, if there are an object $Z$ and [[epimorphisms|RVXDSY04KG]] $u \colon Z \to X$ and $v \colon Z \to Y$ with $x \circ u = y \circ v$.
This is an equivalence relation.
A *pseudo-element* of $A$ is an equivalence class for $\sim$; one writes $x \in_p A$.
For a morphism $f \colon A \to B$ and $x \in_p A$, the pseudo-element $f(x) \in_p B$ is the class of $f \circ x$.
The *zero pseudo-element* of $A$ is the class of a [[zero morphism|YF9F9EG6M3]] $0 \to A$.

## Remarks

Transitivity of $\sim$ uses that pullbacks of epimorphisms are epimorphisms in an abelian category.
A morphism $x$ is equivalent to a zero morphism exactly when $x = 0$.
The basic rules are these.
A morphism $f \colon A \to B$ is a monomorphism if and only if $f(x) = 0$ implies $x = 0$.
It is an epimorphism if and only if every $y \in_p B$ is $f(x)$ for some $x \in_p A$.
A sequence $A \xrightarrow{f} B \xrightarrow{g} C$ is exact if and only if $g \circ f = 0$ and every $y \in_p B$ with $g(y) = 0$ is $f(x)$ for some $x \in_p A$.

Example: in the category of abelian groups, let $x, y \colon \Z \to \Z$ be $x(n) = n$ and $y(n) = -n$.
Take $Z = \Z$, $u = \id$ and $v = y$, both epimorphisms; then $x \circ u = y \circ v$, so $x \sim y$.
More generally, two morphisms into an abelian group $A$ are equivalent exactly when they have the same image, so the pseudo-elements of $A$ correspond to the subgroups of $A$.

Non-example: the morphisms $x(n) = n$ and $w(n) = 2n$ from $\Z$ to $\Z$ are not equivalent.
If $x \circ u = w \circ v$ with $u, v$ epimorphisms, then $x$ and $w$ have the same image, but these images are $\Z$ and $2\Z$.