Primeros años de universidadCategory TheoryPlanificado
Categories
Categories as objects and morphisms with associative composition and identities; the categories of sets, groups, vector spaces, topological spaces and posets; monoids and preorders as categories; isomorphisms; and opposite categories with the duality principle.
math.cat.categories
La lección de este tema todavía no se ha escrito. El esquema siguiente forma parte del mapa del plan de estudios y no ha sido revisado.
Se basa en
Structure-preserving maps: homomorphisms and isomorphismsAlgebraic structures: monoids, groups and rings
Requisitos previos obligatorios, hasta el fondo: 14
Aprenderás a
- State the axioms of a category and verify them for Set, Grp, Vect_k, Top and a poset regarded as a category.
- Recognise a monoid as a one-object category, and a preorder as a category with at most one morphism between any two objects.
- Define isomorphisms in a category and prove that inverses are unique.
- Form the opposite category and dualise simple statements, for example exchanging monomorphisms and epimorphisms.
Definición
- A category C consists of objects; morphisms f : A → B between them; a composite g ∘ f for each pair of composable morphisms; and an identity morphism id_A for each object, such that composition is associative and identities are two-sided units.
- A morphism f : A → B is an isomorphism if there is g : B → A with g ∘ f = id_A and f ∘ g = id_B.
- The opposite category Cᵒᵖ has the same objects as C and one morphism B → A for each morphism A → B of C, with composition reversed.
- f is a monomorphism if f ∘ g = f ∘ h implies g = h; epimorphisms are defined dually.
Teorema
- Inverses in a category are unique when they exist.
- In Set, the monomorphisms are the injections and the epimorphisms are the surjections.
- In the category of rings with identity, the inclusion ℤ → ℚ is an epimorphism but is not surjective.
- Duality principle: if a statement holds in every category, so does the dual statement obtained by reversing all arrows.
Errores frecuentes
- Thinking morphisms must be functions; in a poset regarded as a category, a morphism is just the fact that x ≤ y.
- Assuming epimorphisms are always surjective.
- Believing the objects of a category must have elements.
Conduce a
Descubrimientos opcionales
- Natural transformations Once there are maps between categories (functors), one can ask for maps between functors: natural transformations.Sendero del Asombro →
- Higher categories Categories themselves, with functors and natural transformations, form a 2-category, the starting point of higher category theory.Sendero del Asombro →
- Categorical logic Categories can also model logic and computation, with types or propositions as objects and programs or proofs as morphisms.
- ← Diagrams where paths agree
- ← Composition
- ← Orders, lattices and well-orders
- ← Linear algebra
- ← Transformation geometry and the Erlangen programme
- ← Quivers and path algebras
Itinerarios de aprendizaje
Lecturas complementarias
- Tom Leinster, Basic Category Theory (2014)
- Steve Awodey, Category Theory (2006)
- Saunders Mac Lane, Categories for the Working Mathematician (1971)