PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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 $\mathcal C$ .

[L1]

A zero object is both initial and terminal, so there are unique arrows $A\to0$ 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 $A\to0$ 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 $A'\to0$ , so they are equal; the same uniqueness proves $g\circ0_{A,B}=0_{A,B'}$ .

step 1.1L1L2

3.1

Any compatible zero family must have $0_{A,B}=0_{0,B}\circ0_{A,0}$ , 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

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 7 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)