Set (Category)

Definition

Set is the category whose:

• Objects are sets
• Morphisms are functions between sets
• Identity morphisms are identity functions
• Composition is ordinary function composition

Properties

Set is one of the most fundamental categories and serves as a prototype for categorical reasoning:

• Cartesian closed: Set has a terminal object (any singleton set), binary products (Cartesian products), and exponential objects (function sets $B^A$)
• Complete and cocomplete: Set has all small limits and colimits
• Well-pointed: The terminal object is a generator
• Elementary topos: Set is the canonical example of a topos
• Balanced: Every morphism that is both monic and epic is an isomorphism

Categorical Structure

Products and Coproducts

• The product of sets $A$ and $B$ is their Cartesian product $A\times B$
• The coproduct is the disjoint union $A+B=A\bigsqcup B$
• The terminal object is any singleton set $\{\ast \}$ or $1$
• The initial object is the empty set $\varnothing$ or $0$

Special Morphisms

• Monomorphisms in Set are exactly the injective functions
• Epimorphisms in Set are exactly the surjective functions
• Isomorphisms in Set are exactly the bijections

FinSet

The category FinSet is the full subcategory of Set consisting of finite sets.

Properties of FinSet:

• FinSet is also cartesian closed
• FinSet has all finite limits and colimits
• Every object in FinSet is characterized up to isomorphism by its cardinality
• The skeleton of FinSet has one object for each natural number $n\in \mathbb{N}$

Ord

the category Ord is the full subcategory of Set such that

Relationship to Logic and Type Theory

Under the Curry-Howard Correspondence, Set provides semantics for:

• Simply-Typed Lambda Calculus: Interpreted in Set as a cartesian closed category
• Intuitionistic Logic: Set is a Heyting category
• Classical Logic: Set satisfies the law of excluded middle and is a Boolean Topos

Variants and Generalizations

• Rel: Category of sets and relations
• $Se{t}_{\cdot }$: Category of pointed sets (sets with a distinguished element)
• Set/X: Slice categories of Set (bundles over a set $X$)
• Par: Category of sets and partial functions
• Bij: Category of sets and bijections (the core of Set)

Foundations

The precise definition of Set depends on the foundational framework:

• In ZFC, objects are ZFC sets and morphisms are set-theoretic functions
• In Type Theory, Set may refer to the category of setoids or h-sets
• In Homotopy Type Theory, Set is the category of 0-types
• Size issues require care: Set is typically a locally small category but not small

Related Concepts

• Category
• Cartesian Closed Category
• Topos
• Function
• Set
• Product (Category Theory)
• Coproduct (Category Theory)