Universal properties

前置知识

16 项必需的前置知识（一直追溯到底）

你将学会

• 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.

定义

• 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.

定理

• 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.

常见误解

• 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.

通向

可选发现

• Limits and colimits Products, equalisers, pullbacks and their duals are all limits or colimits: one universal property covering many constructions.探奇小径 →
• Kan extensions Limits and colimits are themselves special Kan extensions, along the functor to the category with one object and one morphism.探奇小径 →
• ← Relations, equivalence relations and partitions
• ← Composition
• ← Constructing ℤ, ℚ and ℝ
• ← Metric spaces and point-set topology

延伸阅读

• Tom Leinster, Basic Category Theory (2014)
• Emily Riehl, Category Theory in Context (2016)