Category of sets

Category of sets
Name	Category of sets
Notation	Set, Sets, \mathbf{Set}
Objects	Sets
Morphisms	Functions
Initial	Empty set
Terminal	Singleton
Products	Cartesian product
Coproducts	Disjoint union
Limits	All small limits
Colimits	All small colimits

Category of sets

The category of sets is the category whose objects are sets and whose morphisms are functions between sets. It is a foundational object in category theory and connects to many prominent mathematical figures and institutions such as Georg Cantor, David Hilbert, Emmy Noether, Bourbaki, and Institute for Advanced Study, and to major works like Principia Mathematica and Zermelo–Fraenkel set theory. As a cartesian closed, well-powered, and complete category, it serves as a basic environment for constructions used by Alexander Grothendieck, Saunders Mac Lane, Samuel Eilenberg, Paul Dirac, and multiple universities including Harvard University, University of Cambridge, and Princeton University.

Definition

Objects are all sets as studied in Zermelo–Fraenkel set theory with or without the Axiom of Choice depending on the formulation. Morphisms are ordinary functions between sets; composition is composition of functions and identity morphisms are identity functions. The formation of the category uses the axiomatic frameworks developed by Georg Cantor and formalized through work associated with Kurt Gödel, Ernst Zermelo, and Abraham Fraenkel. The category is typically denoted by Set or \mathbf{Set} in texts by Saunders Mac Lane and Ieke Moerdijk.

Basic Properties

Set is complete and cocomplete: all small limits and colimits exist, a fact used in the writings of Mac Lane and Grothendieck. It is cartesian closed: it has finite products, exponentiation (function sets), and a terminal object, properties emphasized in treatments by William Lawvere and F. William Lawvere. Set is well-powered and co-well-powered, satisfying smallness conditions studied by Peter Freyd and Max Kelly. It is not a topos only in that it is a quintessential example of an elementary topos with a natural numbers object, discussed in contexts involving André Joyal and Myself (as a persona)-style expositions found in lecture notes from École Normale Supérieure and University of Chicago.

Monomorphisms are injective functions and epimorphisms are surjective functions; isomorphisms are bijections, a classification used by Emmy Noether and Isaac Newton in different historical contexts. Subobject lattices correspond to power sets, linking to Cantor and developments in Set theory at institutions like University of Göttingen and Cambridge University Press publications. The category admits a small-projective generator: a singleton set, with representable functors central to work by Yoneda and Grothendieck.

Constructions and Limits

Products are Cartesian products; for indexed families one uses the product set construction found in expositions by Eilenberg and Mac Lane. Coproducts are disjoint unions or tagged sums, discussed in relation to the direct sum constructions encountered in papers from Institute of Advanced Study authors. Equalizers and coequalizers exist and are given by subset and quotient constructions respectively, tools prominent in treatises by Noether and Emmy Noether’s modern interpreters.

Limits in Set are constructed as subsets of products satisfying compatibility conditions; colimits arise as quotients of disjoint unions. Exponentials are function sets, giving the adjunctions between product and Hom used in Lawvere’s work and in categorical logic developed by Joyal and Andre Joyal. Filtered colimits commute with finite limits, a technical property exploited in algebraic geometry by Grothendieck and in model theory by Alfred Tarski.

Subcategories and Variants

Important subcategories include the category of finite sets often denoted FinSet, used in combinatorics and computer science by researchers at Massachusetts Institute of Technology and Stanford University; the category of pointed sets Set_* appearing in homotopy theory texts by J. Peter May; and the full subcategory of small sets when working inside a fixed universe as in Grothendieck’s formulations found at IHÉS. Variants include measurable spaces and measurable functions leading to the category Meas studied in measure theory by Andrey Kolmogorov and Émile Borel, and the category of topological spaces Top where forgetful functors from Top to Set are standard in texts by Stephen Willard and Munkres.

Reflective and coreflective subcategories, localizations, and slice categories (Set/X) are standard constructions appearing in lectures at Cambridge, Princeton, and École Polytechnique and in monographs by Mac Lane.

Functors and Natural Transformations involving Set

The forgetful functor from many algebraic categories such as Group, Ring, Monoid, and Module to Set is central in algebraic studies by Emmy Noether and David Hilbert. Representable functors Set(A,–) and Set(–,B) instantiate the Yoneda lemma used by Yoneda and Grothendieck. The power set functor P: Set^{op} → Set and the contravariant Hom functor are pivotal in categorical logic and topos theory discussed by Lawvere and William T. Gowers-style analysts.

Adjunctions between free constructions (free group, free monoid) and forgetful functors are foundational in work by Eilenberg and Mac Lane. Limits and colimits in Set correspond to pointwise constructions under functor categories Set^C, a fact exploited in category theory seminars at Institute for Advanced Study and in textbooks by Tom Leinster and Riehl.

Applications and Examples

Set underlies semantics of programming languages, where researchers at Bell Labs, Microsoft Research, Google Research, MIT CSAIL, and University of California, Berkeley model data types as sets. In algebraic geometry and homological algebra, Set provides underlying objects for sheaves and presheaves central to Grothendieck’s work at IHÉS and Université Paris-Sud. In mathematical logic and model theory, Set interfaces with structures studied by Alfred Tarski, Saharon Shelah, and Harvard logicians. Concrete examples include the set of natural numbers ℕ, integers ℤ, real numbers ℝ, finite sets, and function spaces arising in functional analysis work by John von Neumann and Stefan Banach.

Category:Set