TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16‡ rests on unproved material (inherited)

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data

Statement

Let $U : A \to C$ , where $A$ is complete and locally small, $C$ is locally small, and $A$ has a supplied small coseparating set. Assume that $U$ preserves all small limits. Fix $C \in C$ . Suppose in addition one of the following data is supplied:

$A$ has a supplied well-powering; or
every collection of subobjects in $A$ has a specified intersection and $U$ preserves the pullbacks of the corresponding families of monomorphisms, including the possibly proper collections invoked in the proof.

Then $(C ↓ U)$ has an initial object.

Preservation of all small limits is required in both branches, not only in the first. The proof produces the initial object inside $(C ↓ U)$ from A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object, which needs $(C ↓ U)$ to be complete, and the comma projection creates only those limits that $U$ preserves. The second branch is therefore not a weakening of that hypothesis: it adds preservation data for the possibly proper collections, rather than treating such a collection as a small diagram.

Without it the conclusion fails. Take $A = C = S e t$ , let $U$ be the constant functor at the two-element set $2$ , and let $C = 1$ . Then $S e t$ is complete and locally small, ${2}$ is a small coseparating set, every collection of subobjects has its intersection, and $U$ carries each wide pullback of monomorphisms to a cone that is again a limit, since the diagram is connected and $U$ is constant; so the branch-2 data is supplied. But $U$ is not continuous — it does not preserve the empty limit — and $(1 ↓ U)$ is the disjoint union of two copies of $S e t$ , one for each map $1 \to 2$ , which has no initial object.

Facts & Assumptions

Given: The functor, fixed object, categorical hypotheses including preservation of all small limits by $U$ , and one of the two supplied branches in the Statement.

[L1]

A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object (A complete locally small category with a small coseparating set and intersections of all subobject collections has an initial object).

[L2]

A supplied well-powering gives, as data for every object $C$ at once, a set $M_{C}$ of monomorphisms into $C$ containing a representative of every subobject class of $C$ (Well-powered and co-well-powered categories, and supplied well-powerings).

[L3]

A set-indexed wide pullback computes the intersection independently of representatives (Wide pullbacks compute intersections of supplied set-indexed subobject representatives independently of the representatives).

[L4]

The comma projection strictly creates every projected limit preserved by $U$ (A comma-category projection strictly creates the limits preserved by the functor).

[L5]

A coseparating set detects distinct maps by postcomposition (Separating and coseparating sets of objects), completeness concerns all small limits (Finite, small, and large limits and colimits; complete and cocomplete categories), a functor is continuous when it preserves all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors), and local smallness makes hom-collections sets (Small, locally small, and large categories).

Proof

technique · cases

1.1

The comma category is locally small because its hom-collections are subsets of those in $A$ . The set of all comma objects $C \to U (K)$ with $K$ in the supplied coseparating set is again a set by local smallness of $C$ , and it is coseparating by [L5].

1.2

Assume the supplied-well-powering branch. The subobjects of a fixed comma object project injectively into subobject classes of its $A$ -component: the projection preserves and reflects monomorphisms, and preservation of pullbacks makes the projected monomorphisms remain monic after applying $U$ . By [L2] they therefore admit a supplied set of representatives. Their wide pullback exists by completeness, is preserved by continuity, and [L3] and [L4] create its intersection in the comma category.

assume-case poweredL2L3L4L5

1.3

Assume the direct-intersection branch. Intersect the projected collection using the stated class-intersection datum, including its empty-collection case, and use the separately supplied preservation of that family-of-monomorphisms pullback to construct the comma structure arrow. This invokes no proper-class diagram and does not infer that preservation from the assumed continuity, which covers only small diagrams.

assume-case directL4L5

2.1

In either branch, $A$ is complete and $U$ preserves all small limits by hypothesis, so [L4] creates every small limit in $(C ↓ U)$ and the comma category is complete for small diagrams. It is locally small, has the coseparating set of step 1.1, and has all the subobject intersections needed by [L1] — from the representative sets of step 1.2 in the first branch, and from the supplied class intersections of step 1.3 in the second. Hence [L1] gives an initial object of $(C ↓ U)$ .

step 1.1step 1.2step 1.3L1L4L5givencases-exhaustive∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 12 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

S. Mac Lane, Categories for the Working Mathematician, theorem V.8.2 and corollary (standard reference, not scraped)
E. Riehl, Category Theory in Context, theorem 4.7.10 (standard reference, not scraped)