CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 reflective subcategory of a complete category is complete

Statement

If $A$ is a reflective full subcategory of a complete category $C$ , then $A$ is complete: every small diagram in $A$ has a limit.

Facts & Assumptions

Given: A reflective full inclusion $I : A \to C$ and a complete category $C$ .

[L1]

A reflective inclusion creates every ambient limit in the ordinary isomorphism-invariant sense (A reflective inclusion creates every ambient limit in the ordinary isomorphism-invariant sense).

[L2]

A category is complete when every small diagram in it has a limit, including the empty diagram (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · direct

1.1givenL2

Let $D$ be any small diagram in $A$ . Completeness of $C$ gives a limit of $I D$ .

2.1step 1.1L1L2∎

By [L1], the inclusion creates from that ambient limit a limit of $D$ in $A$ . This applies to every small $D$ , including the empty diagram, so [L2] proves that $A$ is complete.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

E. Riehl, Category Theory in Context, corollary 4.5.15 (standard reference, not scraped)