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Unital rings and unit-preserving ring homomorphisms form the large locally small category
Statement
Unital rings and unit-preserving ring homomorphisms form a large locally small category .
Facts & Assumptions
Given: Unital rings and unit-preserving homomorphisms and .
A ring is an additive abelian group and multiplicative monoid with both distributive laws (Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides); a ring homomorphism preserves addition, multiplication, zero, and one (Ring homomorphism: additive, multiplicative, and required to send to ).
The functions between fixed sets form the set (The set of all functions ); the category and size conditions are Category, object, morphism, domain, codomain, identity, composition, and hom-collection and Small, locally small, and large categories, and the ordinals form a proper class (Burali-Forti: there is no set of all ordinals).
Proof
The identity of a ring preserves both operations and the unit, and preserves them because and do; function composition supplies associativity and identity equations.
These data form a category, and every hom-collection is a set because it is a subset of the functions between the two underlying sets.
For each ordinal , the one-element set carries the zero-ring structure with ; these are distinct objects, so [L2] shows that is large and locally small.
Depends on
- Category, object, morphism, domain, codomain, identity, composition, and hom-collection
- Small, locally small, and large categories
- Ring: an abelian group under addition and a monoid under multiplication, with multiplication distributing over addition on both sides
- Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
- The set $B^{A}$ of all functions $A \to B$
- Burali-Forti: there is no set of all ordinals
Used by
- Presheaves and sheaves of groups, rings, and modules Definition
- For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups Example
- Underlying-set and structure-forgetting functors among Grp, Ring, Vect_F, R-Mod, Top, and Set Example
- ℤ[x] represents the underlying-set functor on unital rings Example
- The underlying-set functor on unital rings strictly creates split coequalizers Lemma
- Commutative rings form a reflective full subcategory of rings Theorem
- The free unital ring functor is left adjoint to the underlying-set functor Theorem
- The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring Theorem
Dependency tree · two levels
20 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)