PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A functor that creates limits of a given shape lifts their existence and preserves the created limits, and dually for colimits

Statement

If $F : C \to D$ creates limits of shape $J$ , then for every $D : J \to C$ whose image has a limit, $D$ has a limit and $F$ preserves it. The dual assertion holds for created colimits.

Facts & Assumptions

Given: A functor $F$ creating $J$ -limits and a diagram $D$ such that $F D$ has a limit.

[F1]

Creation lifts a target limiting cone, up to a cone isomorphism, to a limiting source cone and includes reflection (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[L1]

Preservation of a chosen limit is equivalent to invertibility of its canonical comparison (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).

[L2]

Creation and preservation of colimits are dual to their limit forms (A limiting cone for a diagram is exactly a colimiting cocone for the formally dual diagram in the opposite category).

Proof

technique · unpacking the definition

1.1

Choose a limiting cone $μ$ over $F D$ . By the existence clause of [F1], there is a limiting cone $λ$ over $D$ and a cone isomorphism $F λ ≅ μ$ . Hence $D$ has a limit.

F1given

2.1

The cone isomorphism in step 1.1 is precisely an invertible canonical comparison after identifying its direction by the target universal property. By [L1], $F$ preserves $λ$ .

L1step 1.1

3.1

Because any limiting source cone is uniquely compatibly isomorphic to $λ$ , its image is also limiting. Thus preservation does not depend on the chosen source limit.

F1step 1.1step 2.1

4.1

Reverse every arrow. By [L2], steps 1.1, 2.1, and 3.1 prove existence and preservation for created colimits.

L2step 1.1step 2.1step 3.1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 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, Definition 3.4.7 and Proposition 3.4.9 (standard reference, not scraped)