PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-13

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Fully faithful functors reflect limits and colimits

Statement

Every fully faithful functor reflects every limit and every colimit, without a smallness restriction on the diagram for which the relevant cone is defined.

Facts & Assumptions

Given: A fully faithful functor $F : C \to D$ , a diagram $D : J \to C$ , and a cone $λ$ whose image is limiting.

[F1]

Reflection means that a source cone is limiting whenever its image is limiting (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[F2]

Full faithfulness means every hom-map is bijective (Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors).

[L1]

Colimit reflection is the formal dual of limit reflection (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

Proof

technique · universal property

1.1

For a cone $ξ$ over $D$ , the limiting property of $F λ$ gives a unique $h : F (X) \to F (L)$ satisfying $F (λ_{j}) h = F (ξ_{j})$ . Fullness in [F2] gives $u : X \to L$ with $F (u) = h$ .

F2given

2.1

Faithfulness applied to $F (λ_{j} u) = F (λ_{j}) h = F (ξ_{j})$ gives $λ_{j} u = ξ_{j}$ , so $u$ is a cone morphism.

F2step 1.1

3.1

If $v$ is another factor, then $F (v)$ is a factor through the limiting cone $F λ$ , hence $F (v) = h = F (u)$ ; faithfulness gives $v = u$ . Thus $λ$ is limiting, which is reflection in [F1].

F1F2step 1.1step 2.1

4.1

Apply the identical argument in opposite categories. By [L1], full faithfulness remains hom-set bijectivity and the conclusion is reflection of colimits.

F2L1step 1.1step 2.1step 3.1∎

Depends on

Used by

Dependency tree · two levels

10 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

E. Riehl, Category Theory in Context, Lemma 3.4.3 (standard reference, not scraped)