Representable Functor

A set-valued functor

Q: A –> S

is called representable if there is an object A in the domain category A and an element q in the set Q(A) such that, for any object B of the category A, the function

A(A, B) –> Q(B)

(from the set A(A, B) of functions from A to B to the set Q(B), which is the value of the functor Q at B) assigning to each map

b: A –> B

(in the set A(A, B)) the element

Q(b) (q)

(where

Q(b): Q(A) –> Q(B)

is the function to which the map b is assigned to by the functor Q) is an isomorphism of sets (Sets for Mathematics, p. 248).

Consider a set

W = {you, me}

There are two elements in W.   Next consider another set

1 = {·}

There are two functions from 1 to W.  They are

you: 1 –> W

with you(·) = you

and

me: 1 –> W

with me(·) = me

Let

W¹ = {you, me}

be the map set of functions from 1 to W.  Note that both sets W and W¹ have the same number of elements i.e.

|W| = |W¹| = 2

which means that there is an isomorphism

W = W¹

which is an opposed-pair of functions

f: W –> W¹

with

f(you) = you: 1 –> W

f(me) = me: 1 –> W

and

g: W¹ –> W

with

g(you: 1 –> W) = you(·)

g(me: 1 –> W) = me(·)

satisfying

gf = 1_W

fg = 1_W^I

All of this may seem like an excessively elaborate discourse on a rather trivial matter: the number of elements of a set W is equal to the number of functions from the singleton set 1 to the set W.

I have two folders in my head

1. profound

2. all else

I put everything that looks trivial such as functions from the terminal set

1 –> W

in the profound folder.  Of course, I didn’t recognize either DISCRETE SUBCATEGORY or REPRESENTABLE FUNCTOR in

W = W¹

(Conceptual Mathematics, p. 314) on my own :(

Saving discrete subcategory for later, let’s return to our representable functor.  A simple example is, as you guessed, our “you, me” story, little generalized.  More specifically, the identity functor

1: S –> S

(from the category S of sets to the category S) is a representable functor.  What do I need to do in order to convince you that

1: S –> S

is indeed a representable functor?

We need an object

A

of the domain category S of sets and an element

q

of the value of the set-valued functor 1: S –> S at A i.e. of the set

1(A)

Simply put, we need a set

A

and an element

q: 1 –> A

What do these duo i.e. this set

A

and this element

q: 1 –> A

have to do with representable functor?

For any set

B

the function

S(A, B) –> 1(B)

from the set of functions (from A to B)

S(A, B) = B^A

(to the value of our identity functor at the set B i.e.) to the set

1(B) = B

assigning to each function

b: A –> B

(in the set B^A of functions from A to B) the element

1(b) (q)

i.e. (since 1(b: A –> B) = b: A –> B) the element

b(q)

is an isomorphism of sets.

Thanks to our “you, me” story, we take a guess and take a singleton set

1 = {·}

as our A which has only one element ‘·’ which is our element ‘q’ both of which will make the assignment to each function

b: 1 –> B

in the set

S(1, B) = B¹

the element

1(b)(q) = b(q) = b(·) = b

in the set

1(B) = B

an isomorphism

B¹ = B

Next, let’s see if the set-valued functor on the category of dynamical systems, which assigns to each dynamical system its set of states, is a representable functor.