TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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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Slices of a locally cartesian closed category are locally cartesian closed

Statement

If $C$ is locally cartesian closed, then for every object $X$ the slice category $C / X$ is locally cartesian closed.

Facts & Assumptions

Given: A locally cartesian closed category $C$ , an object $X$ , and an object $a : A \to X$ of the slice category $C / X$ .

[L1]

Local cartesian closedness means that every slice $C / Z$ is cartesian closed (Locally cartesian closed category).

[L2]

An object of $(C / X) / a$ is exactly a morphism into $a$ in the slice, equivalently a morphism $u : U \to A$ in $C$ over $X$ ; this identifies $(C / X) / a$ with $C / A$ (Comma category, slice category, and coslice category).

Proof

technique · direct

1.1

By [L2], the slice-of-a-slice $(C / X) / a$ is canonically isomorphic to the ordinary slice $C / A$ .

givenL2

2.1

Since $C$ is locally cartesian closed, [L1] says that $C / A$ is cartesian closed. Transporting this structure across the isomorphism of step 1.1 shows that $(C / X) / a$ is cartesian closed.

step 1.1L1

3.1

Because $a$ was an arbitrary object of $C / X$ , every slice of $C / X$ is cartesian closed. Hence $C / X$ is locally cartesian closed.

step 2.1given∎

Depends on

Used by

Local cartesian closure is equivalent to every pullback functor having a right adjoint Theorem

Dependency tree · two levels

6 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

Emily Riehl, Category Theory in Context, 2nd ed., Lemma 4.6.3(i) (standard reference, not scraped)