CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A colimit of a set-valued functor is the set of connected components of its category of elements

Statement

Let $J$ be a small category (Small, locally small, and large categories) and let $D : J \to S e t$ be a functor (Sets and functions form the large locally small category $S e t$ ). Let $\int D$ be its category of elements (The category of elements of a covariant functor or a presheaf) and let $π_{0} (\int D)$ be the quotient of its set of objects by the least equivalence relation (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ) containing every pair $(a, b)$ for which there is a morphism $a \to b$ ; two objects lie in the same class exactly when they are joined by a finite zigzag of morphisms, which is the connectedness condition of Isomorphism, groupoid, and connected category.

Then

${colim}_{j \in J} D (j) = π_{0} (\int D)$

(Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties), the colimiting cocone sending $x \in D (j)$ to the class of the object $(j, x)$ .

Facts & Assumptions

Given: A small category $J$ and a functor $D : J \to S e t$ .

[F5]

A category is small when both $Ob (C)$ and $Mor (C)$ are sets. (Small, locally small, and large categories).

[F6]

Sets as objects and functions as morphisms form a large locally small category $S e t$ (Sets and functions form the large locally small category $S e t$ ).

[F1]

The category of elements $\int G$ of a functor $G : C \to S e t$ has objects $(c, x)$ with $c \in C$ and $x \in G (c)$ ; and a morphism $(c, x) \to (d, y)$ given by a morphism $f : c \to d$ in $C$ satisfying $G (f) (x) = y$ (The category of elements of a covariant functor or a presheaf).

[L1]

Every small diagram $D : J \to S e t$ has a colimit; it is the quotient of the tagged union $S = {(j, x) : x \in D (j)}$ by the least equivalence relation containing $(j, x) \sim (k, D (u) (x))$ for $u : j \to k$ (Set has all small colimits, realized as a quotient of a set-indexed disjoint union).

[F2]

A binary relation on a set is an equivalence relation when it is reflexive, symmetric and transitive; the quotient set is the set of its classes (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ).

[F3]

A category is connected when it is nonempty and any two objects can be joined by a finite zigzag of morphisms, with successive arrows allowed to point in either direction (Isomorphism, groupoid, and connected category).

[F4]

A colimit of $D$ is an initial cocone: explicitly, for every cocone $(X, ξ)$ there exists a unique morphism $u : Q \to X$ such that $u ρ_{j} = ξ_{j}$ for every $j$ . (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

Proof

technique · direct

1.1

By [F1] the objects of $\int D$ are exactly the pairs $(j, x)$ with $x \in D (j)$ , which is exactly the tagged union named in [L1]; and a morphism $(j, x) \to (k, y)$ of $\int D$ exists precisely when some $u : j \to k$ has $D (u) (x) = y$ , that is, precisely for the pairs that generate the equivalence relation of [L1].

F1F5F6L1

2.1

The least equivalence relation containing the generating pairs of [L1] is therefore the least equivalence relation on the objects of $\int D$ containing every pair joined by a morphism, and by [F2] its classes are the classes defining $π_{0} (\int D)$ . Two objects lie in one class exactly when a finite chain of generating pairs, each used in either direction, joins them, which is the finite-zigzag condition of [F3].

F2F3step 1.1

3.1

By [L1] the colimit of $D$ is the quotient of the tagged union by that relation, with cocone components sending $x \in D (j)$ to the class of $(j, x)$ ; by step 2.1 that quotient is $π_{0} (\int D)$ , and by [F4] the universal property of the colimit is the one asserted. If $J$ is empty, or every $D (j)$ is empty, both sides are the empty set.

F4L1step 2.1∎

Remarks

Nothing about the category of elements is used beyond its objects and the existence of its morphisms: the identification is between the generating pairs of the published $S e t$ -colimit construction and the morphisms of $\int D$ , and everything else is the same quotient read twice.

The empty case is not an exception. A category with no objects has no connected components, and a diagram of empty sets has the empty set as its colimit, so both sides are empty; connectedness requires nonemptiness, but a set of components does not.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

25 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

G. M. Kelly, Basic Concepts of Enriched Category Theory (TAC Reprints 10), (3.35) (standard reference, not scraped)