Primeros años de universidadCategory TheoryPlanificado
Universal properties
Describing objects by their relationships rather than their elements: universal properties of products, quotients, free objects and fields of fractions, and the fact that an object with a universal property is unique up to unique isomorphism.
math.cat.universal-properties
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Se basa en
Requisitos previos obligatorios, hasta el fondo: 16
Aprenderás a
- State the universal properties of products in Set, Grp and Top, of quotient sets and quotient groups, and of free monoids and free groups.
- Prove that any two objects with the same universal property are isomorphic by a unique isomorphism compatible with the structure maps.
- Recognise when a familiar construction, such as the tensor product or the field of fractions, is best defined by a universal property.
Definición
- A universal arrow from an object X of C to a functor U : D → C is an object F of D with a morphism η : X → U(F) such that every morphism X → U(E) equals U(g) ∘ η for a unique g : F → E.
- A free monoid on a set S is a monoid F(S) with a map S → F(S) through which every map from S to a monoid factors uniquely via a monoid homomorphism.
Teorema
- Objects defined by a universal property are unique up to a unique isomorphism compatible with the structure maps.
- The free monoid on S is the monoid of finite words in the elements of S under concatenation.
- The quotient map X → X/~ is universal among maps out of X that are constant on equivalence classes.
Errores frecuentes
- Thinking a universal property constructs an object; it characterises one, and existence must be proved separately. For example, the fields 𝔽₂ and 𝔽₃ have no product in the category of fields.
- Forgetting the uniqueness clause, which is what makes the object unique up to isomorphism.
Conduce a
Descubrimientos opcionales
- Limits and colimits Products, equalisers, pullbacks and their duals are all limits or colimits: one universal property covering many constructions.Sendero del Asombro →
- Kan extensions Limits and colimits are themselves special Kan extensions, along the functor to the category with one object and one morphism.Sendero del Asombro →
- ← Relations, equivalence relations and partitions
- ← Composition
- ← Constructing ℤ, ℚ and ℝ
- ← Metric spaces and point-set topology
Lecturas complementarias
- Tom Leinster, Basic Category Theory (2014)
- Emily Riehl, Category Theory in Context (2016)