Alphabeta Math

Categories, Functors and Natural Transformations — Examples

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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.
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✓ 25 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 25 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Category Theory
Combinatorics
Coxeter Groups
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

Underlying-set and structure-forgetting functors among Grp, Ring, Vect_F, R-Mod, Top, and Set

Example

The free-group functor F:Set toGrp and free-module functor R⁽⁻⁾:Set→ R-Mod

Example

Chosen bases exhibit Mat_F as equivalent to finite-dimensional vector spaces

Example

The arrow category Set^→: functions as objects and commuting squares as morphisms

Example

Quivers and quiver homomorphisms form a functor category of set-valued diagrams

Example

The fundamental groupoid of a topological space

Example

Pointed sets are equivalent to sets and partial functions but not isomorphic as categories

Example

For a fixed space X, product with X defines an endofunctor of Top

Example

Open-set and closed-set functors on Topᵒᵖ are naturally isomorphic by complements

Example

Singletons define a natural transformation from the identity functor on sets to the covariant power-set functor

Example

The opposite-group functor is naturally isomorphic to the identity functor by inversion

Example

The distributive and exponential laws of sets are natural isomorphisms

Example

For n≥ 1, determinant is a natural transformation det:GLₙ(-)⟹(-)^× from commutative rings to groups

Example

Actions of a group G on sets are functors BG toSet

Example

An action groupoid has the acted-on set as objects, orbits as connected components, and stabilizers as automorphism groups

Example

A path between basepoints induces an isomorphism of fundamental groups

Counterexample

A functor need not preserve monomorphisms

Counterexample

The injection ∅→{*} is monic in Set but is not split

Counterexample

A two-object indiscrete preorder is equivalent but not isomorphic to its one-object poset reflection

Counterexample

The inclusion of one object into a discrete two-object category is fully faithful but not essentially surjective

Counterexample

A componentwise family between functors need not be a natural transformation

False statement

Every equivalence of categories is an isomorphism of categories

False statement

Every morphism that is both monic and epic is an isomorphism

False statement

Every category is locally small

False statement

A natural transformation is determined by its component at one object

Sources