PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A zero object supplies a unique compatible system of zero morphisms

Statement

A chosen zero object $0$ in a category supplies a unique compatible system of zero morphisms.

Facts & Assumptions

Given: A zero object $0$ in a category $C$ .

[L1]

A zero object is both initial and terminal, so there are unique arrows $A \to 0$ and $0 \to B$ for every $A, B$ (Initial object, terminal object, and zero object).

[L2]

A system of zero morphisms must absorb composition on either side (Category with zero morphisms).

Proof

technique · direct

1.1

Define $0_{A, B}$ as the composite of the unique arrows $A \to 0$ and $0 \to B$ supplied by [L1].

givenL1

2.1

For $f : A^{'} \to A$ , both $0_{A, B} \circ f$ and $0_{A^{'}, B}$ factor through $0$ using the unique arrow $A^{'} \to 0$ , so they are equal; the same uniqueness proves $g \circ 0_{A, B} = 0_{A, B^{'}}$ .

step 1.1L1L2

3.1

Any compatible zero family must have $0_{A, B} = 0_{0, B} \circ 0_{A, 0}$ , and both factors are the unique arrows to and from $0$ , so the family of step 1.1 is unique.

step 2.1L1L2∎

Depends on

Used by

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)
Haru Park, Category Theory and Homological Algebra, section 2.2 (standard reference, not scraped)