False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ rests on unproved material (inherited)

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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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‡ 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.

FALSE: every functor on $C^{o p} \times C$ has an end

Statement

False claim: every functor $T : C^{o p} \times C \to D$ has an end (The end and the coend of a functor $C^{o p} \times C \to D$ ).

Facts & Assumptions

Given: The discrete category $C$ on the set $N$ of natural numbers, the full subcategory $D$ of $S e t$ whose objects are the finite sets, and the functor $T$ with $T (c, c^{'}) = {0, 1}$ for every pair of objects.

[F4]

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 $S e t$ ).

[F3]

A subcategory has a subclass of the objects and, for each pair, a subclass of the morphisms; The subcategory is full when $A (A, B) = C (A, B)$ for every pair of its objects (Subcategory and full subcategory).

[F6]

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

[F8]

A wedge from $d$ to $T$ is a family $ω_{c} : d \to T (c, c)$ with $T (1_{c}, f) \circ ω_{c} = T (f, 1_{c^{'}}) \circ ω_{c^{'}}$ for every $f : c \to c^{'}$ (Wedges and cowedges, and the categories they form).

[F1]

An end of $T$ is a terminal object of the category of wedges over $T$ and a coend an initial object of the category of cowedges under $T$ ; in short, an end is a terminal wedge and a coend an initial cowedge, so an end is a wedge through which every wedge factors by exactly one morphism (The end and the coend of a functor $C^{o p} \times C \to D$ ).

[L1]

For small $C$ and a target where the displayed objects exist, an end is the equalizer of two products, the first indexed by the objects of $C$ and the second by its morphisms (An end is the equalizer of two products, and a coend the coequalizer of two coproducts).

[F2]

A product of $(A_{i})_{i \in I}$ is an object $P$ with projections $p_{i}$ such that every family $f_{i} : X \to A_{i}$ has a unique pairing $⟨ f_{i} ⟩_{i \in I} : X \to P, p_{i} ⟨ f_{i} ⟩ = f_{i} (i \in I)$ (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

[F5]

A set $A$ is finite when $A \approx n$ for some $n \in N$ , and then $∣ A ∣$ is that unique $n$ (The cardinality $∣ A ∣$ of a finite set).

[L2]

For every $n \in N$ there is no injection $σ (n) \to n$ (The pigeonhole principle on $N$ ).

[F7]

A category is complete when it has all small limits; Completeness and cocompleteness do not assert the existence of limits or colimits of large diagrams (Finite, small, and large limits and colimits; complete and cocomplete categories).

Refutation

technique · direct

1.1

Let $C$ be discrete on $N$ , so its only morphisms are identities and it is small by [F6]; let $D$ be the full subcategory of $S e t$ on the finite sets, which is a category by [F3] and [F4]; and let $T$ send every pair of objects to the two-element set ${0, 1}$ and every morphism to an identity, which is a functor because every morphism of $C^{o p} \times C$ is an identity. The index category is deliberately small, so that smallness of the index is not what is at issue.

F3F4F6givenconstruct

2.1

A wedge over $T$ with vertex $X$ is an unconstrained family: by [F8] the wedge equation is imposed only at morphisms of $C$ , and all of those are identities, at which it reads $ω_{c} = ω_{c}$ . So a wedge with vertex $X$ is exactly a family of functions $X \to {0, 1}$ indexed by $N$ , and by [L1] and [F2] an end of $T$ is exactly a product of the diagonal values in $D$ .

F2F8L1step 1.1

3.1

No object of $D$ has that property. Suppose $E$ were an end, with $∣ E ∣ = n$ by [F5]. Take the vertex to be a one-element set, which is an object of $D$ ; the wedges with that vertex are the families $(ϵ_{c})_{c \in N}$ with $ϵ_{c} \in {0, 1}$ , and the $σ (n)$ families that are $1$ at exactly one of the numbers $0, \dots, n$ and $0$ elsewhere are pairwise distinct. By [F1] each factors through $E$ by exactly one morphism from a one-element set, that is by exactly one element of $E$ , and distinct wedges give distinct elements; this is an injection $σ (n) \to E$ , hence an injection $σ (n) \to n$ , which [L2] forbids. So $T$ has no end and the claim is false.

F1F2F5L2step 2.1

4.1

A large index category is a second and independent way for an end to fail, since the equalizer description of [L1] would then ask for a product over a proper class, and [F7] records that completeness asserts nothing about diagrams that are not small. The refutation above does not use that route: its index category is small, and what fails is the target.

F7step 3.1∎

Remarks

The witness turns on the target, not on the index. Taking $D$ to be all of $S e t$ would make the end exist, since the required product is then available; taking the diagonal values to be one-element sets would also make it exist, since the product of one-element sets is a one-element set. It is the combination of infinitely many two-element values with a target closed under nothing infinite that removes the end.

The correct sufficient condition is on this page: Ends exist over a small index category in a complete target, and coends in a cocomplete one asks for a small index category and a complete target, and the witness above has the first without the second.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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

B. Richter, From Categories to Homotopy Theory (author's draft), Proposition 4.5.3 (standard reference, not scraped)

FALSE: every functor on Cop×CCop×C has an end