﻿ The Barest Introduction to Category Theory
San José State University

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 The Barest Introduction to Category Theory

This material is limited to defining categories, giving a few illustrations and then showing the power of the concept by displaying a concept which applies in more than one category but in different guises which are the same from a category point of view. Thus if an overall category theorem has been proven then for it to apply in separate fields it is only necessary to demonstrate that the separate fields are indeed categories.

## Examples of Categories

### Mathematical Groups and Homomorphisms between them

A group consists of a set of elements S and a binary function f(x,y) defined for all x and y in S. The function f must have certain properties. For all x and y in S, f(x,y) is in S. There exists an element e in S (called the identity element) such that for all x in S, f(x,e) is equal to x. Also for all x in S there exists a y in S such that f(x,y)=e. The element y is said to be the inverse of x. And finally, for all x, y and z in S f(x,f(y,z)) is equal to f(f(x,y),z). A homomorphism between group G and group H is a function F defined on G with values in H such that if g and h are the binary functions for G and H, respectively

#### F(g(x,y)) = h(F(x),F(y))

Note that a particular group is not a category; the set of all groups is not a category. The category is the combination of the set of all groups and the set of homomorphisms between them.

### Sets and One-to-one Functions between Sets

A set consists of elements. A one-to-one function f between set A and set B is such that for all x in A there is defined one and only one element f(x) in B and for any y in B there is one and only one x in A such that f(x)=y.

### Vector Spaces and Linear Mappings between Vector Spaces

A vector space consists of a set of elements (vectors) V, a field of scalars K and two operations (functions), vector addition and scalar multiplication. Addition is such that for all v and u in V, v+u belongs to V. Scalar multiplication is such that for all k in K and v in V, kv is in V.

A linear mapping L between vector space V and vector space W is such that for all v and u in V

#### L(v+u) = L(v) +* L(u)

where +* is the addition in the vector space W.

### Topological Spaces and Continuous Mappings between them

A topological space consists of a set of elements S and a collection T of subsets of S, including the whole set and the empty set. The subsets in T are called the open sets of the topology. The collection of subsets is closed under set union and set intersection; i.e., if A and B belong to T then the union A∪B and the intersection A∩B also belong to T.

A mapping between topological space T and topological space U is continuous if the inverse images of open sets of U are open sets of T.

### Deductive Systems

A deductive system consists of well formed formulas and the methods of deduction between them.

### Partially ordered sets and monotone functions between them

A partially ordered set is a set S and a binary function on S that maps to the set {true, false}. That is to say, for all x and y in f(x, y) is either true or false. This binary function is called a relation and is usually expressed as xRy, x is less than or equal to y. The relation has the properties of reflexivity (for all x in S xRx is true), transitivity (for all x, y and z in S if xRy and yRz are true then xRz is true), and antisymmetry (if xRy and yRx then x=y).

A monotone function is a mapping f between partially ordered sets A and B such that if xRy in A then f(x)R'f(y) in B, where R and R' are the order relations in A and B, respectively.

The crucial ingredient in all of these is the set of mappings between the objects. In a sense category theory is the mathematics of mappings between objects.

## Definition of a Category

A category consists of

• A set S of objects having the mathematical structure of monoids. (A monoid has an associative function and an identity element.)
• A set Z of arrows
• For each arrow f there are elements of S called the domain, dom(f), and the codomain, cod(f). An arrow is displayed as A→B where A=dom(f) and B=cod(f).
• For any two arrow f and g such that cod(f)=dom(g) there is an arrow h such that dom(h)=dom(f) and cod(h)=cod(g). The arrow h is called the composite of f and g, h=g°f.

The composition of f and g is displayed as

#### g°f:A→C means f:A→B and g:B→Cor, equivalently for all a in A g°f(a) = g(f(a))

• The composition of arrows satisfies the associative law; i.e.,

#### h°(g°f)=(h°g)°f means h(g(b)) where b=f(a) is the same as h(c) where c=g(f(a).

• For each object A in S there is an arrow 1A with dom(1A)=A and cod(1A)=A such that for all a in A 1A(a)=a. This 1A is called the identity arrow of A. Furthermore for all f, f°1A = 1B°f = f.

For the category GRP the objects are the groups. The arrows are the homomorphisms. For each group there is the identity mapping that maps it into itself.

For the category SET the objects are sets and the arrows are the one-to-one functions from one set to another.

For the category TOP the objects are the topological spaces and the arrows are the continuous mappings.

For the category VSPACE the objects are the vector spaces and arrows are the linear mappings, which for finite dimensional vector spaces can be identified with matrices.

For deductive system the formulas are the objects and the arrows are the deductive proofs between them.

For the category of partially ordered sets the objects are the sets and their order relations. The arrows are the monotone functions.

(To be continued.)

Sources:

Steve Awodey, Category Theory, Claredon Press, Oxford, 2006.

Saunders Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1998.

D.E. Rydeheard and R.M. Burstall, Computational Category Theory, Prentice Hall, New York, 1988.

Ion Bucur amd Aristide Deleanu, Introduction to the Theory of Categories and Functors, John Wiley & Sons, London, 1968.

V. Sankrithi Krishnan, An Introduction to Category Theory, North Holland, New York, 1981.