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.
Underlying-set and structure-forgetting functors among , , , , , and
Example
Each familiar category of structured objects has an underlying-set functor to . It sends an object to its carrier and a morphism to its underlying function.
Facts & Assumptions
Given: The five standard structured categories named in the example.
Sets and functions form (Sets and functions form the large locally small category ).
Groups, unital rings, vector spaces, modules, and spaces with their standard morphisms form categories (Groups and group homomorphisms form the large locally small category , Unital rings and unit-preserving ring homomorphisms form the large locally small category , Vector spaces over a fixed field and linear maps form the large locally small category , Left modules over a fixed ring and module homomorphisms form the large locally small category , Topological spaces and continuous maps form the large locally small category ).
Verification
For each of the five structured categories in [L2], define on objects by the carrier of , and define to be the same ordered-pair relation as the structure-preserving map , now regarded only as a function.
The underlying function of the identity morphism of is , so .
Composition in every category in [L2] is composition of the underlying functions. Hence .
Thus the underlying-set assignments from , , , , and to are functors. They forget structure but not the identity and composition laws.
Depends on
- Sets and functions form the large locally small category $\mathbf{Set}$
- Groups and group homomorphisms form the large locally small category $\mathbf{Grp}$
- Unital rings and unit-preserving ring homomorphisms form the large locally small category $\mathbf{Ring}$
- Vector spaces over a fixed field and linear maps form the large locally small category $\mathbf{Vect}_F$
- Left modules over a fixed ring and module homomorphisms form the large locally small category $R\text{-}\mathbf{Mod}$
- Topological spaces and continuous maps form the large locally small category $\mathbf{Top}$
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 42 results over 16 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
- Emily Riehl, Category Theory in Context, Example 1.3.2 (standard reference, not scraped)