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

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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

1.1

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.

L1L2

2.1

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.

step 1.1L1L2∎

Depends on

Used by

General adjoint functor theorem, objectwise initial-object form Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 6 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, theorem 4.7.3 (standard reference, not scraped)
T. Leinster, Basic Category Theory, theorem 6.3.10 (standard reference, not scraped)