DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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.

Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations

Definition

Let $(A_{i})_{i \in I}$ be an indexed family of objects (An indexed family $(A_{i})_{i \in I}$ is a function with domain $I$ ; ${A_{i} : i \in I}$ is its range), regarded as a diagram on the discrete category $I$ . Its product is its limit (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties): an object $P = \prod_{i \in I} A_{i}$ with projections $p_{i} : P \to A_{i}$ such that every family $f_{i} : X \to A_{i}$ has a unique pairing

$⟨ f_{i} ⟩_{i \in I} : X \to P, p_{i} ⟨ f_{i} ⟩ = f_{i} (i \in I) .$

Its coproduct is its colimit: an object $Q = ∐_{i \in I} A_{i}$ with injections $ι_{i} : A_{i} \to Q$ such that every family $f_{i} : A_{i} \to X$ has a unique copairing

$[f_{i}]_{i \in I} : Q \to X, [f_{i}]_{i \in I} ι_{i} = f_{i} (i \in I) .$

The empty product is therefore terminal and the empty coproduct initial. A one-object product or coproduct is canonically the object itself. A product or coproduct need not exist.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 9 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

E. Riehl, Category Theory in Context, Definitions 3.1.9 and 3.1.13 (standard reference, not scraped)