False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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FALSE: A reflective inclusion creates colimits

Statement

False claim. The inclusion of every reflective full subcategory creates all small colimits.

Facts & Assumptions

Given: The inclusion $I : A ↪ S e t$ of the full subcategory whose only object is a fixed singleton $1$ .

[L1]

A full subcategory is reflective when its inclusion has a left adjoint (Reflective full subcategory and reflector).

[L2]

Ordinary creation of a colimit requires every target colimiting cocone to be isomorphic to the image of a source colimiting cocone (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[L3]

The empty colimit is an initial object (Limits of empty diagrams are terminal objects, and colimits of empty diagrams are initial objects).

[L4]

For locally small categories, an adjunction determines hom-set bijections natural in both variables, and conversely every such natural family of bijections determines a unique unit and counit satisfying the triangle identities, hence a unique adjunction structure (Under local smallness, transposition gives the natural hom-set bijection, and conversely).

Refutation

technique · counterexample

1.1

The constant functor $R : S e t \to A$ at $1$ is left adjoint to $I$ , since the unique maps $X \to 1$ give hom-set bijections ${Hom}_{A} (R X, 1) ≅ {Hom}_{S e t} (X, I 1)$ natural in $X$ ; by the converse clause of [L4] these determine a unit and counit satisfying the triangle identities, hence an adjunction $R ⊣ I$ . Thus $I$ is a reflective inclusion by [L1].

L1L4

1.2

The empty diagram in $S e t$ has colimit $\emptyset$ , whereas the empty diagram in $A$ has colimit $1$ , by [L3]. Since $I (1) = 1 ≇ \emptyset$ , the ambient colimiting cocone is not isomorphic to the image of any cocone with an apex in $A$ .

2.1

This violates the creation requirement [L2], so a reflective inclusion need not create colimits.

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 28 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, corollary 4.5.15 (standard reference, not scraped)