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: Every reflective subcategory is closed under ambient colimits

Statement

False claim. If a full subcategory is reflective, then the colimit in the ambient category of every diagram valued in the subcategory again lies in the subcategory.

Facts & Assumptions

Given: The full subcategory $A \subseteq S e t$ 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]

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

[L3]

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 $R : S e t \to A$ with value $1$ is left adjoint to the inclusion: both ${Hom}_{A} (1, 1)$ and ${Hom}_{S e t} (X, 1)$ are singletons, naturally in $X$ . By the converse clause of [L3] that natural family of bijections determines a unit and counit satisfying the triangle identities, hence an adjunction $R ⊣ I$ , and [L1] then makes $A$ reflective.

L1L3

1.2

In $A$ , the object $1$ is initial and is therefore the empty colimit. In $S e t$ the empty colimit is $\emptyset$ by [L2], and $\emptyset$ is not an object of $A$ .

2.1

Thus an ambient colimit of a diagram valued in a reflective subcategory need not remain in that subcategory, refuting the claim.

step 1.1step 1.2∎

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: 21 results over 8 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)