PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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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.

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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

A reflective inclusion need not preserve even the empty colimit Counterexample
Additive category Definition
The singleton set and trivial group are terminal, while the empty set and trivial group are initial Example
FALSE: A reflective inclusion creates colimits False statement
FALSE: colimits in Grp are computed by taking the Set-colimit of the underlying diagram False statement
FALSE: Every reflective subcategory is closed under ambient colimits False statement
The empty biproduct is a zero object Proposition
Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals Theorem

Dependency tree · two levels

7 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, Example 3.1.14 (standard reference, not scraped)