TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A reflective subcategory has every ambient colimit, obtained by reflecting an ambient colimit

Statement

Let $R ⊣ I$ exhibit $A$ as a reflective full subcategory of $C$ . If $D : J \to A$ and $I D$ has a colimit $L$ in $C$ , then $D$ has a colimit in $A$ , represented by $R (L)$ . In particular, the inclusion need not preserve this colimit.

Facts & Assumptions

Given: A reflection $R ⊣ I$ (Reflective full subcategory and reflector), a diagram $D : J \to A$ , and a colimiting cocone $λ_{j} : I (D_{j}) \to L$ in $C$ .

[L1]

The counit $ε_{A} : R I (A) \to A$ of a reflection is an isomorphism for every $A \in A$ (The counit of a reflection is an isomorphism).

[L2]

A left adjoint sends every existing colimiting cocone to a colimiting cocone (Left adjoints preserve every colimit that exists).

[L3]

A colimit is a cocone through which every cocone factors uniquely (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

Proof

technique · direct

1.1

Apply $R$ to the ambient cocone. By [L2], $(R (L), R (λ_{j}))$ is colimiting for $R I D$ ; precomposing its legs with the inverses $ε_{D_{j}}^{- 1} : D_{j} \to R I (D_{j})$ of the isomorphisms supplied by [L1] transports it to a cocone with legs $R (λ_{j}) \circ ε_{D_{j}}^{- 1} : D_{j} \to R (L)$ , also for the empty indexing category.

L1L2

2.1

Transport across isomorphisms preserves the existence and uniqueness clauses in [L3], so the resulting cocone is a colimit of $D$ in $A$ . The construction applies the reflector to $L$ and does not assert that $I (R (L))$ is isomorphic to $L$ , so it does not assert preservation by the inclusion.

step 1.1L1L3∎

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: 22 results over 9 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, section 4.5 (standard reference, not scraped)