TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The solution-set condition at an object is exactly a jointly weakly initial set in its comma category

Statement

Let $U : A \to C$ and fix $C \in C$ . A supplied family $(η_{i} : C \to U (A_{i}))_{i \in I}$ is a solution set at $C$ if and only if the corresponding supplied set of objects $(A_{i}, η_{i})$ is jointly weakly initial in the comma category $(C ↓ U)$ .

Facts & Assumptions

Given: A functor $U : A \to C$ , an object $C$ , and a supplied set-indexed family $η_{i} : C \to U (A_{i})$ as in The solution-set condition for a functor, stated object by object.

[L1]

An object of $(C ↓ U)$ is an arrow $f : C \to U (A)$ , and a morphism $(A_{i}, η_{i}) \to (A, f)$ is a map $h : A_{i} \to A$ satisfying $f = U (h) \circ η_{i}$ (Comma category, slice category, and coslice category).

[L2]

A set of objects is jointly weakly initial exactly when every target receives a morphism from one member of that set (Weakly initial object and jointly weakly initial set).

Proof

technique · direct

1.1L1L2

If $(η_{i})$ is a solution set and $(A, f)$ is any comma object, the defining factorisation gives $i$ and $h : A_{i} \to A$ with $f = U (h) η_{i}$ . By [L1] this is a comma morphism from $(A_{i}, η_{i})$ to $(A, f)$ , so [L2] gives joint weak initiality. If the supplied set is empty, the same assertion says the comma category has no objects.

2.1step 1.1L1L2∎

Conversely, if the corresponding comma objects are jointly weakly initial, apply [L2] to each $(A, f)$ . The resulting comma morphism has, by [L1], exactly the equation $f = U (h) η_{i}$ required by the solution-set condition.

Depends on

Used by

General adjoint functor theorem, objectwise initial-object form Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

E. Riehl, Category Theory in Context, theorem 4.7.3 (standard reference, not scraped)
T. Leinster, Basic Category Theory, theorem 6.3.10 (standard reference, not scraped)