False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-26

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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

1.1F3F4F6givenconstruct

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.

2.1F2F8L1step 1.1

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 $D$ .

3.1F1F2F5L2step 2.1

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.

4.1F7step 3.1∎

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.

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