DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties

Definition

Let $D : J \to C$ be a diagram. A limit of $D$ is a terminal object $(L, λ)$ of $Cone (D)$ (Constant diagrams, cones, cocones, and their morphisms, Initial object, terminal object, and zero object). It is written

$L = \lim_{j \in J} D (j), λ_{j} : L \to D (j) .$ Explicitly, for every cone $(X, ξ)$ there exists a unique morphism $u : X \to L$ such that $λ_{j} u = ξ_{j}$ for every $j$ . The diagram has a limit when such a cone exists. A colimit of $D$ is an initial object $(Q, ρ)$ of $Cocone (D)$ , written $Q = {colim}_{j \in J} D (j), ρ_{j} : D (j) \to Q .$

Explicitly, for every cocone $(X, ξ)$ there exists a unique morphism $u : Q \to X$ such that $u ρ_{j} = ξ_{j}$ for every $j$ . The diagram has a colimit when such a cocone exists.

Depends on

Dependency tree · 2 levels

Sources

E. Riehl, Category Theory in Context, Definitions 3.1.6 and 3.1.11