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.
Natural isomorphism
Definition
A natural isomorphism is a natural transformation for which there is a natural transformation with and . The compositions here are the vertical compositions of Identity natural transformation and vertical composition.
When the source category is small, so that Functor category is formed, this says exactly that is an isomorphism from to in that functor category. Thus the definition combines Natural transformation and its components with the categorical notion of isomorphism from Isomorphism, groupoid, and connected category without requiring a functor category for an arbitrary large source.
Depends on
Used by
- Objects a and b are isomorphic exactly when C(-,a) and C(-,b) are naturally isomorphic Corollary
- Braiding Definition
- Equivalence, quasi-inverse, and adjoint equivalence of categories Definition
- Exact functor between triangulated categories Definition
- Lax, strong, and strict monoidal functors Definition
- Localization of a category at a class of morphisms Definition
- Monoidal category Definition
- Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors Definition
- Presheaves, covariantly and contravariantly representable functors, and representations Definition
- The category of right-module endofunctors Definition
- Open-set and closed-set functors on Topᵒᵖ are naturally isomorphic by complements Example
- The distributive and exponential laws of sets are natural isomorphisms Example
- The opposite-group functor is naturally isomorphic to the identity functor by inversion Example
- A natural transformation is a natural isomorphism exactly when every component is an isomorphism Proposition
- A translation-compatible natural isomorphism of exact functors respects triangles Proposition
- Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense Proposition
- A category with finite products is monoidal Theorem
- A supplied symmetry identifies the left and right internal homs Theorem
Dependency tree · two levels
8 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, Chapter 1 (standard reference, not scraped)