TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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Weighting by a representable evaluates the diagram

Statement

Let $J$ be small, let $M$ be locally small (Small, locally small, and large categories), let $F : J \to M$ be a diagram and let $j_{0}$ be an object of $J$ .

Limit clause. For the covariant representable weight $W = J (j_{0}, -) : J \to S e t$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category, Presheaves, covariantly and contravariantly representable functors, and representations), the weighted limit exists and is the value of the diagram (Set-weighted limits and colimits):

${J (j_{0}, -), F} = F (j_{0}) .$

Colimit clause. For the contravariant representable weight $W = J (-, j_{0}) : J^{o p} \to S e t$ , the weighted colimit exists and

$J (-, j_{0}) ⋆ F = F (j_{0}) .$

Facts & Assumptions

Given: A small $J$ , a locally small $M$ , a diagram $F : J \to M$ and an object $j_{0}$ of $J$ .

[F5]

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

[F4]

The covariant hom-assignment $C (a, -)$ sends $u : b \to c$ to $u_{*} : C (a, b) ⟶ C (a, c), f ⟼ u \circ f$ , and the contravariant hom-assignment $C (-, a)$ sends $u$ to precomposition (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

[F6]

A representation of a functor is an object with a natural isomorphism from the corresponding hom-functor; The pair $(R, θ)$ is a representation of $F$ , and $R$ is a representing object (Presheaves, covariantly and contravariantly representable functors, and representations).

[F1]

A weighted limit ${W, F}$ is an object that represents the functor sending an object to the set of natural transformations from the weight, that is $M (m, {W, F}) ≅ [J, S e t] (W, M (m, F -))$ naturally in $m$ ; the weighted colimit is characterised by $M (W ⋆ F, m) ≅ [J^{o p}, S e t] (W, M (F -, m))$ (Set-weighted limits and colimits).

[L1]

For locally small $J$ the evaluation maps $E_{a, G} : Nat (C (a, -), G) \overset{≅}{\to} G (a)$ are bijections natural in both variables; in particular, for $η : G \Rightarrow G^{'}$ , $E_{a, G^{'}} (η \circ α) = η_{a} (E_{a, G} (α))$ (The Yoneda bijection $Nat (C (a, -), F) ≅ F (a)$ is natural in both $a$ and $F$ ).

[L2]

For locally small $J$ , an object $a$ and a presheaf $P$ , evaluation at the identity gives a bijection $E_{P}^{a} : Nat (C (-, a), P) \overset{≅}{\to} P (a), E_{P}^{a} (α) = α_{a} (1_{a})$ , natural in both variables (For a presheaf $P$ , $Nat (C (-, a), P) ≅ P (a)$ naturally in $a$ and $P$ ).

[L3]

Two representing objects of one functor are joined by a unique compatible isomorphism (Representing objects are unique up to a unique isomorphism compatible with their universal elements).

Proof

technique · direct

1.1

Fix an object $m$ of $M$ . The functor $M (m, F -) : J \to S e t$ is the covariant hom-functor of [F4] composed with $F$ , and it takes values in sets because $M$ is locally small. By [L1] applied in $J$ , which is locally small by [F5], evaluation at $1_{j_{0}}$ is a bijection from $Nat (J (j_{0}, -), M (m, F -))$ to $M (m, F j_{0})$ .

F1F4F5L1

2.1

That bijection is natural in $m$ . A morphism $h : m^{'} \to m$ induces the natural transformation $M (h, F -) : M (m, F -) \Rightarrow M (m^{'}, F -)$ , and the naturality of [L1] in its functor variable gives $E_{j_{0}} (M (h, F -) \circ α) = M (h, F j_{0}) (E_{j_{0}} (α))$ , which is precisely compatibility with precomposition by $h$ . Hence $F (j_{0})$ represents $m \mapsto [J, S e t] (J (j_{0}, -), M (m, F -))$ in the sense of [F6], so by [F1] it is a weighted limit ${J (j_{0}, -), F}$ , unique up to the unique compatible isomorphism by [L3].

F1F6L1L3step 1.1

3.1

For the colimit clause the weight $J (-, j_{0})$ and the diagram $M (F -, m)$ are both presheaves on $J$ , so [L2] gives a bijection from $Nat (J (-, j_{0}), M (F -, m))$ to $M (F j_{0}, m)$ , evaluation at $1_{j_{0}}$ again. Its naturality in the presheaf variable makes it natural in $m$ , now with respect to postcomposition, so $F (j_{0})$ represents $m \mapsto [J^{o p}, S e t] (J (-, j_{0}), M (F -, m))$ and by [F1] it is the weighted colimit $J (-, j_{0}) ⋆ F$ . The two clauses use the published Yoneda statement of matching variance, and neither is obtained from the other.

F1F6L2L3step 1.1step 2.1∎

Remarks

The two clauses give the same object $F (j_{0})$ from weights of opposite variance, and that is not an accident: a representable weight concentrates all the weighting at one object of the index category, and both the limit and the colimit then have nothing left to take. Which representable does it is fixed by the variance, and swapping the two weights would ask for a natural transformation between functors of opposite variance.

A representable weight and the constant singleton weight are both cases in which the weighted limit can be named without computing anything. For a general weight the object is described instead by A weighted limit is an ordinary limit over the category of elements of the weight, and a weighted colimit an ordinary colimit over it, which replaces $J$ by the category of elements of the weight. For the covariant representable weight that category has $(j_{0}, 1_{j_{0}})$ as an initial object, since a morphism out of it to $(c, g)$ is a morphism $f : j_{0} \to c$ with $f = g$ ; for the contravariant one the same pair is terminal. A limit over a category with an initial object, and a colimit over one with a terminal object, is the value there, which is why the weighted object collapses to $F (j_{0})$ .

Depends on

Used by

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Sources

G. M. Kelly, Basic Concepts of Enriched Category Theory (TAC Reprints 10), (3.10) (standard reference, not scraped)
E. Riehl, Categorical Homotopy Theory, Example 7.1.4 (standard reference, not scraped)