TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ rests on unproved material (inherited)

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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.
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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.

A set-valued coend is the disjoint union of the diagonal values modulo the dinaturality relation

Statement

Let $C$ be a small category (Small, locally small, and large categories) and let $T : C^{o p} \times C \to S e t$ be a functor (Sets and functions form the large locally small category $S e t$ ). Write

$∐_{c} T (c, c) = {(c, x) : c \in Ob (C), x \in T (c, c)}$

for the disjoint union of the diagonal values, and let $\sim$ be the least equivalence relation on it (Equivalence relation, equivalence class, and the quotient set $A / \sim$ ) containing

$(c, T (f, 1_{c}) (x)) \sim (c^{'}, T (1_{c^{'}}, f) (x)) for every f : c \to c^{'} and every x \in T (c^{'}, c) .$

Then $T$ has a coend, and it is the disjoint union of the diagonal values modulo the dinaturality relation (The end and the coend of a functor $C^{o p} \times C \to D$ ):

$\int^{c} T (c, c) = (∐_{c} T (c, c)) / \sim, ρ_{c} (x) = [(c, x)] .$

Both generating elements are named with their summands: the pair is generated by an element $x$ of the off-diagonal value $T (c^{'}, c)$ , pushed into the summand at $c$ by $T (f, 1_{c})$ and into the summand at $c^{'}$ by $T (1_{c^{'}}, f)$ .

Facts & Assumptions

Given: A small category $C$ and a functor $T : C^{o p} \times C \to S e t$ .

[F1]

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]

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

[L2]

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).

[F5]

The coproduct of an indexed family regarded as a diagram on the discrete category $I$ is its colimit, with injections $ι_{i}$ through which every family $f_{i} : A_{i} \to X$ factors by a unique copairing (Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations).

[L1]

For a small $C$ whose displayed coproducts exist, a coend is the coequalizer of two maps between coproducts, namely of the two morphisms determined on the $f$ -summand by $T (f, 1_{c})$ into the summand at $c$ and by $T (1_{c^{'}}, f)$ into the summand at $c^{'}$ , the $f$ -summand being $T (c^{'}, c)$ (An end is the equalizer of two products, and a coend the coequalizer of two coproducts).

[F6]

A coequalizer of $g, h : A ⇉ B$ is a morphism $q : B \to Q$ satisfying $q g = q h$ such that, whenever $k : B \to X$ satisfies $k g = k h$ , there is a unique $u : Q \to X$ with $u q = k$ (Equalizers and coequalizers as limits and colimits of a parallel pair).

[F2]

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

[F3]

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 $C^{o p} \times C \to D$ ).

Proof

technique · direct

1.1

Since $C$ is small, the objects and the morphisms of $C$ form sets, so both families in question are set-indexed. A coproduct is a colimit of a discrete diagram, and on a discrete index category the only morphisms are identities, so the least equivalence relation of [L2] is equality and the colimit is the tagged union itself. Hence $∐_{c} T (c, c)$ and $∐_{f : c \to c^{'}} T (c^{'}, c)$ exist in $S e t$ and are the displayed disjoint unions.

F1F4F5L2

2.1

By [L1] the coend of $T$ is the coequalizer, if it exists, of the two functions $Λ, P : ∐_{f : c \to c^{'}} T (c^{'}, c) \to ∐_{c} T (c, c)$ given on the summand at $f : c \to c^{'}$ by $Λ (f, x) = (c, T (f, 1_{c}) (x))$ and $P (f, x) = (c^{'}, T (1_{c^{'}}, f) (x))$ , where $x$ ranges over $T (c^{'}, c)$ .

F3L1step 1.1

3.1

The quotient map $q : ∐_{c} T (c, c) \to (∐_{c} T (c, c)) / \sim$ coequalises $Λ$ and $P$ , because every pair $(Λ (f, x), P (f, x))$ is one of the generating pairs of $\sim$ . If $k$ is a function on $∐_{c} T (c, c)$ with $k Λ = k P$ , then ${(s, t) : k (s) = k (t)}$ is an equivalence relation containing every generating pair, so it contains $\sim$ ; hence $k$ is constant on classes and factors as $k = \overset{ˉ}{k} q$ for a unique $\overset{ˉ}{k}$ , uniqueness because $q$ is surjective. So $q$ is a coequalizer of $Λ$ and $P$ .

F2F6step 2.1

4.1

Therefore the coequalizer of step 2.1 exists, and by [L1] and [F3] the coend of $T$ is the quotient set of step 3.1, with the initial cowedge given by $ρ_{c} (x) = [(c, x)]$ .

F3L1step 3.1∎

Remarks

The source of a generating pair is the off-diagonal value $T (c^{'}, c)$ , indexed by $f : c \to c^{'}$ running the other way; this is the same swap that appears in the description of a coend as a colimit over $Tw (C)^{o p}$ (An end is a limit over the twisted arrow category, and a coend is a colimit over its opposite). If $T (c^{'}, c)$ is empty for some pair, that summand contributes no generating pair at all, and the corresponding identifications simply do not happen.

Identity morphisms contribute nothing: at $f = 1_{c}$ both $T (1_{c}, 1_{c})$ are the identity of $T (c, c)$ and the generating pair is $((c, x), (c, x))$ , which every equivalence relation contains already.

Depends on

Used by

The coend of the hom-bifunctor Example
The tensor product of monoid sets as a coend Example
FALSE: under this page's convention a coend is the colimit of the same twisted-arrow diagram whose limit is the end False statement

Dependency tree · two levels

26 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 1.2.4 (standard reference, not scraped)
E. Riehl, Categorical Homotopy Theory, (7.1.6) (standard reference, not scraped)