False statementConstruction: AI-adaptedVerification: AI-generatedSession-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.

FALSE: under this page's convention a coend is the colimit of the same twisted-arrow diagram whose limit is the end

Statement

False claim: with $Tw (C)$ and the projection $π$ as fixed on this page (The twisted arrow category and its projection to $C^{o p} \times C$ ), the coend of a functor $T : C^{o p} \times C \to D$ is the colimit over $Tw (C)$ of the very diagram $T π$ whose limit is the end (The end and the coend of a functor $C^{o p} \times C \to D$ , Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

Facts & Assumptions

Given: The walking arrow $C$ , with objects $0$ and $1$ and one non-identity morphism $u : 0 \to 1$ , and its hom-bifunctor as the integrand.

[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 $S e t$ ).

[F4]

The hom-assignment sends $(a, b)$ to $C (a, b)$ , and a morphism of the product category consisting of $h : a^{'} \to a$ and $u : b \to b^{'}$ acts by $C (h, u) : C (a, b) ⟶ C (a^{'}, b^{'}), f ⟼ u \circ f \circ h$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

[L3]

For every locally small category $C$ , the hom-assignment $C (-, -) : C^{o p} \times C \to S e t$ is a functor (The hom-assignment $C (-, -) : C^{o p} \times C \to S e t$ is a bifunctor).

[F1]

The objects of $Tw (C)$ are the morphisms of $C$ , a morphism $f \to g$ is a pair $(a, b)$ with $b f a = g$ , and $π$ sends $f : c \to c^{'}$ to $(c, c^{'})$ (The twisted arrow category and its projection to $C^{o p} \times C$ ).

[F5]

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

[F2]

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 (The end and the coend of a functor $C^{o p} \times C \to D$ ).

[F3]

The opposite category has the same objects and reverses every morphism: $C^{o p} (A, B) = C (B, A)$ (Opposite category $C^{o p}$ ).

[L1]

The wedges over $T$ are the cones over $T π$ , so an end is the limit over the twisted arrow category; the coend is the colimit over $Tw (C)^{o p}$ of the integrand read with the domain and codomain of each arrow interchanged (An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite).

[L2]

For small $C$ and a set-valued integrand, the coend is the disjoint union of the diagonal values modulo the dinaturality relation, generated by the pairs $(c, T (f, 1_{c}) (x))$ and $(c^{'}, T (1_{c^{'}}, f) (x))$ for $f : c \to c^{'}$ and $x \in T (c^{'}, c)$ (A set-valued coend is the disjoint union of the diagonal values modulo the dinaturality relation).

Refutation

technique · direct

1.1

Take $C$ to be the walking arrow and $T = C (-, -)$ , a functor into $S e t$ by [L3]. Its values are $C (0, 0) = {1_{0}}$ , $C (1, 1) = {1_{1}}$ , $C (0, 1) = {u}$ and $C (1, 0) = \emptyset$ . By [F1] the objects of $Tw (C)$ are $1_{0}$ , $1_{1}$ and $u$ ; a morphism $1_{0} \to u$ is the pair $(1_{0}, u)$ and a morphism $1_{1} \to u$ is the pair $(u, 1_{1})$ , while a morphism out of $u$ would need a component in $C (1, 0)$ and there is none. So $Tw (C)$ is a cospan, and $T π$ takes the value ${1_{0}}$ at $1_{0}$ , ${1_{1}}$ at $1_{1}$ and ${u}$ at $u$ .

F1F4F6L3givenconstruct

2.1

The colimit of $T π$ over $Tw (C)$ has one element. A cocone under a cospan is determined by its component at the codomain of the two arrows, the other two components being that one precomposed with the transition maps; so a cocone with apex $X$ is exactly a function ${u} \to X$ , and by [F5] the initial such is ${u}$ itself.

F5step 1.1

2.2

The coend has two elements. By [L2] it is the quotient of ${1_{0}} ⊔ {1_{1}}$ by the relation generated by the pairs indexed by a morphism $f : c \to c^{'}$ and an element $x$ of $T (c^{'}, c)$ ; the only non-identity morphism is $u : 0 \to 1$ , and $T (1, 0) = C (1, 0)$ is empty, so it contributes no generating pair, while identity morphisms contribute only reflexive pairs. The relation is therefore equality and the coend is the two-element set.

F2F4L2step 1.1

3.1

One element is not two, so the coend is not the colimit of $T π$ over $Tw (C)$ and the claim is false. The correct description is [L1]: the coend is the colimit over $Tw (C)^{o p}$ , by [F3] the same objects with every arrow reversed, of the functor sending $f : c \to c^{'}$ to $T (c^{'}, c)$ — domain and codomain interchanged. On this witness that diagram takes the values ${1_{0}}$ , ${1_{1}}$ and $\emptyset$ , its index category is a span, and its colimit is the two-element set, as it must be.

F2F3L1step 2.1step 2.2∎

Remarks

Two changes separate the correct description from the false one, and taking only one of them is what the false claim does. The index category must be reversed and the integrand must be reindexed; on this witness the reindexing is what moves the empty set from the $(1, 0)$ position, where it contributed nothing, to the apex of the diagram, where it stops the two other values from being identified.

The witness is as small as it can be. Any category in which every hom-set is nonempty in both directions would hide the failure, because the two diagrams would then have the same shape of transition maps; the walking arrow is chosen precisely because $C (1, 0)$ is empty.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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

F. Loregian, (Co)end Calculus (arXiv:1501.02503v7), Remark 1.2.3 (standard reference, not scraped)