PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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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.

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Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects

Statement

For the unique diagram $D : \emptyset \to C$ , an object is the apex of a limiting cone exactly when it is terminal in $C$ . It is the apex of a colimiting cocone exactly when it is initial in $C$ .

Facts & Assumptions

Given: The empty diagram $D : \emptyset \to C$ .

[F1]

A limit is a terminal cone and a colimit is an initial cocone (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[F2]

An object is terminal when every object has exactly one morphism to it, and initial when it has exactly one morphism to every object (Initial object, terminal object, and zero object).

[L1]

Limits in a category are colimits of the dual diagram in the opposite category (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

Proof

technique · universal property

1.1

A cone over $D$ with apex $X$ has no legs, and every morphism $X \to Y$ is a cone morphism because there are no compatibility equations.

given

2.1

Hence a cone with apex $T$ is terminal in the cone category if and only if for every $X$ there is exactly one morphism $X \to T$ . By [F1] and [F2], this is equivalent to $T$ being both a limit apex and a terminal object.

F1F2step 1.1

3.1

The empty category is its own opposite. Applying [L1] to step 2.1 turns the limiting assertion into the assertion that an empty-diagram colimit is an initial object, in both directions.

L1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 8 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, Example 3.1.14 (standard reference, not scraped)