Category theory; homological algebra Definition

Zero object

Builds on Category Morphism Initial and terminal objects Trivial group

Definition

Let $\mathcal{C}$ be a category. An object $0$ of $\mathcal{C}$ is a zero object if it is both an initial object and a terminal object. So for every object $X$ there is exactly one morphism $0 \to X$ and exactly one morphism $X \to 0$ .

Remarks

A zero object is unique up to unique isomorphism, because an initial object is. A category with a zero object is called pointed.

Example: in the category of groups, the trivial group $\{e\}$ is a zero object. The only homomorphism $\{e\} \to G$ sends $e$ to the identity of $G$ , and the only map $G \to \{e\}$ is the constant map, which is a homomorphism. The same check shows that the zero module is a zero object in the category of modules over a ring. It also shows that a one-point set is a zero object in the category of pointed sets.

Non-example: in the category of sets, a one-point set $\{*\}$ is terminal but not initial, since there are two functions $\{*\} \to \{0, 1\}$ . The empty set is initial but not terminal, since there is no function from a non-empty set to it. So the category of sets has no zero object.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Category concept
Initial and terminal objects concept
Morphism concept
Trivial group example
Zero object definition

Tag	`C0GMFP0RF9`
URI	`m2://C0GMFP0RF9`
Kind	definition
Writing	W0 stub
MSC	18A30, 18A20
Defines	Zero object
Uses	Category, Morphism, Initial and terminal objects, Trivial group

Zero object

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here