Alphabeta MathCategories, Functors and Natural Transformations
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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.
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✓ 35 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 35 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Remark
Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why CAT is not formed
Definition
Category, object, morphism, domain, codomain, identity, composition, and hom-collection
Definition
Small, locally small, and large categories
Definition
Subcategory and full subcategory
Proposition
Sets and functions form the large locally small category Set
Proposition
Groups and group homomorphisms form the large locally small category Grp
Proposition
Unital rings and unit-preserving ring homomorphisms form the large locally small category Ring
Proposition
Vector spaces over a fixed field and linear maps form the large locally small category Vect_F
Proposition
Left modules over a fixed ring and module homomorphisms form the large locally small category R-Mod
Proposition
Topological spaces and continuous maps form the large locally small category Top
Proposition
Posets and monotone maps form the large locally small category Poset
Proposition
A monoid is a one-object category, and a group is a one-object category in which every morphism is invertible
Definition
Preorder and monotone map
Proposition
A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps
Definition
Opposite category Cᵒᵖ
Theorem
Every theorem about categories has a formal dual obtained by reversing morphisms and composition
Definition
Isomorphism, groupoid, and connected category
Proposition
The isomorphisms in a category form its maximal subgroupoid
Lemma
A morphism is an isomorphism exactly when postcomposition, equivalently precomposition, induces bijections on every hom-collection
Definition
Monomorphism and epimorphism by left and right cancellation
Definition
Split monomorphism, split epimorphism, retraction, and section
Proposition
Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic
Theorem
In Set, monomorphisms are exactly injections and epimorphisms are exactly surjections
Corollary
Under the Axiom of Choice, every epimorphism in Set is a split epimorphism
Theorem
The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring
Definition
Initial object, terminal object, and zero object
Definition
Category with zero morphisms
Proposition
A zero object supplies a unique compatible system of zero morphisms
Definition
Covariant functor, identity functor, composite functor, and contravariant functor
Proposition
Every functor preserves isomorphisms
Definition
Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors
Definition
Embedding and full embedding of categories
Proposition
Every fully faithful functor reflects isomorphisms
Definition
Natural transformation and its components
Definition
Identity natural transformation and vertical composition
Lemma
Vertical composites of natural transformations satisfy naturality
Definition
Whiskering and horizontal composition of natural transformations
Lemma
Horizontal composites of natural transformations satisfy naturality
Theorem
Horizontal and vertical composition of natural transformations satisfy the interchange law
Definition
Product category and its projection functors
Definition
Functor category [C,D]
Proposition
If C is small and D is locally small then [C,D] is locally small; if both are small it is small
Definition
Natural isomorphism
Proposition
A natural transformation is a natural isomorphism exactly when every component is an isomorphism
Definition
Equivalence, quasi-inverse, and adjoint equivalence of categories
Proposition
Equivalence of categories is reflexive, symmetric, and transitive
Theorem
A functor is an equivalence exactly when it is fully faithful and split essentially surjective, without Choice
Lemma
Under the Axiom of Choice, essential surjectivity onto a small category admits a splitting
Corollary
Under the Axiom of Choice, a functor with small target is an equivalence exactly when it is fully faithful and essentially surjective
Theorem
Every equivalence of categories can be equipped as an adjoint equivalence
Proposition
Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects
Definition
Skeletal category and skeleton
Proposition
Under the Axiom of Choice, every small category has a skeleton
Definition
Comma category, slice category, and coslice category
Definition
Diagram as a functor from an indexing category
Definition
Strict 2-category
Theorem
Small categories, functors, and natural transformations form the strict 2-category Cat
Theorem
Eckmann–Hilton: two unital operations satisfying interchange coincide and are commutative
Proposition
A functor is an isomorphism of categories exactly when its object and morphism maps are bijective
Proposition