Alphabeta MathHow 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.
Locally cartesian closed category
Definition
A category is locally cartesian closed when, for every object , the slice category is cartesian closed in the sense of Cartesian closed category. The slice categories and their pullback functors are those of Slice categories, composition, and pullback along a morphism.
Depends on
Used by
- FALSE: every cartesian closed category is locally cartesian closed False statement
- Boundary: this page stops before elementary and Grothendieck toposes Remark
- A locally cartesian closed category has pullbacks, and with a terminal object it has all finite limits Theorem
- A locally cartesian closed category with a terminal object is cartesian closed Theorem
- Local cartesian closure is equivalent to every pullback functor having a right adjoint Theorem
- Set is locally cartesian closed Theorem
- Slices of a locally cartesian closed category are locally cartesian closed 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
- Emily Riehl, Category Theory in Context, 2nd ed., Definition 4.6.2 (standard reference, not scraped)