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

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A comma-category projection strictly creates the limits preserved by the functor

Statement

Let $U : A \to C$ , fix $C \in C$ , and let $Π : (C ↓ U) \to A$ be the projection. If a diagram $D : J \to (C ↓ U)$ has a projected limit $(L, p_{j})$ in $A$ and $U$ preserves that limit, then there is a unique structure arrow $λ : C \to U (L)$ making $(L, λ)$ the limit of $D$ in the comma category. Thus $Π$ strictly creates every limit of $Π D$ that $U$ preserves, including the empty limit.

Facts & Assumptions

Given: A diagram $D$ in $(C ↓ U)$ , whose objects have structure arrows $λ_{j} : C \to U (A_{j})$ , and a limiting cone $(L, p_{j})$ of $Π D$ preserved by $U$ .

[L1]

A comma morphism $h : (A, α) \to (B, β)$ satisfies $β = U (h) \circ α$ (Comma category, slice category, and coslice category).

[L2]

A limiting cone has a unique mediating map from every cone, with the same clause for the empty diagram (Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties).

[L3]

Strict creation means that every target limiting cone has a unique lift with exactly the same apex and legs, and the lifted cone is limiting (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

Proof

technique · direct

1.1

The arrows $λ_{j} : C \to U (A_{j})$ form a cone over $U Π D$ . Since $U (L)$ with legs $U (p_{j})$ is limiting, [L2] gives a unique $λ : C \to U (L)$ satisfying $U (p_{j}) λ = λ_{j}$ for every $j$ . By [L1], the same legs $p_{j}$ are comma morphisms from $(L, λ)$ .

L1L2

2.1

Given any comma cone with apex $(B, β)$ , the projected limit supplies a unique $h : B \to L$ with $p_{j} h$ equal to its legs. Both $U (h) β$ and $λ$ have the same composites with every $U (p_{j})$ , so uniqueness of the preserved limit gives $U (h) β = λ$ ; hence $h$ is the unique comma morphism. The lifted cone is limiting.

step 1.1L1L2

3.1

The arrow $λ$ in step 1.1 is forced by the projected apex and legs, so the lift is unique on the nose. For an empty indexing category, preservation of the terminal object gives the unique map $C \to U (L)$ by the same limit property. Thus all strict-creation clauses in [L3] hold, including the degenerate and empty diagrams.

step 1.1step 2.1L2L3∎

Depends on

Used by

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, lemma 4.7.2 (standard reference, not scraped)
T. Leinster, Basic Category Theory, lemma A.2 (standard reference, not scraped)