DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Intersection of a supplied family of subobjects as its greatest lower bound

Definition

Let $([m_{i}])_{i \in I}$ be a supplied family of subobjects of an object $C$ , indexed by a set $I$ . An intersection of this family is a greatest lower bound in the subobject order of Subobjects and quotient objects form oppositely oriented partially ordered collections: it is a subobject $[p]$ with $[p] \leq [m_{i}]$ for every $i \in I$ , and every subobject $[q]$ satisfying $[q] \leq [m_{i}]$ for all $i$ also satisfies $[q] \leq [p]$ .

This definition does not assert that the intersection exists. When $I$ is empty, its intersection, if it exists, is the greatest subobject of $C$ , represented by $1_{C} : C \to C$ .

Depends on

Subobjects and quotient objects form oppositely oriented partially ordered collections

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

E. Riehl, Category Theory in Context, section 4.7 (standard reference, not scraped)
S. Mac Lane, Categories for the Working Mathematician, section V.8 (standard reference, not scraped)