1, 2, 3…

There is 1 associative law.

There are 2 identity laws.

There are 3 bookkeeping laws.

–Conceptual Mathematics; pp. 21, 166 – 9

Me:      So you don’t like 0!  It’s not like you don’t know about 0; why you ignore 0?

You:     Dang!  Ok, then, let’s get even more primordial – begin with an object

•

0 is an endomap – an arrow going from • to •

• –0–> •

So is number 1 i.e. another arrow

• –1–> •

Me:      Let me get this straight.  The domain object of the map (number) • –0–> • is

•

and the codomain object of the number 0 is also

•

You:     Yes (that’s why we call • –0–> • an endomap; Conceptual Mathematics, page 15) and so is the case with number • –1–> • i.e.

domain (1) = •

codomain (1) = •

Me:      Don’t you need an identity map for each object (assuming you are going to pull a category sooner or later).  Which one of the two maps

• –0–> •

• –1–> •

is the identity map of the object •?

You:     First, let’s thank God (or Money for) there is only one object to worry about.  Having said that, it’s tad bit tricky.

Me:      I knew it—you don’t have an answer to my simple question: which one of the two numbers 0, 1 is the identity?

You:     Well, when we say number 1 is identity as in

1 × 1 = 1

1 × 2 = 2

1 × 3 = 3

.

.

.

we are saying so in view of multiplication.  There is something analogous to multiplication—composition—in our scenario of object • and maps

• –0–> •

• –1–> •

which we have to foveate first.

Me:      Do we have to think about composition before deciding on the identity because of the two identity laws?

You:     Yes; say, for illustration, the map

• –0–> •

is the identity of the object •, then the identity map has to satisfy the two identity laws which involve composition.

Given a map, say,

1: • –> •

we need the identity map

0: • –> •

of the object • to satisfy

1 + 0 = 1

and

0 + 1 = 1

with ‘+’ denoting composition of maps

• –0–> • –1–> • = • –1–> • (1 + 0 = 1)

• –1–> • –0–> • = • –1–> • (0 + 1 = 1)

Me:      Since the maps

0: • –> •

1: • –> •

are numbers 0, 1, respectively, in your schizoid scheme, why not take addition (+) as composition of maps?

You:     As you wish.  As of now we have

Object (•)

Maps (0, 1)

Composition (+)

Identity laws (0 + 0 = 0, 0 + 0 = 0; 1 + 0 = 1, 0 + 1 = 1)

but not enough!

Me:      You never content?

You:     For each pair of composable maps we need a composite map.  We say a pair of maps is composable if the codomain object of one map is same as the domain object of the other map in the pair.

Me:      But we have only one object

•

which is both the domain and the codomain object.

You:     Yes [as a result of your observation] we can add on and on

0 + 0 = 0

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

.

.

.

Me:      K, I got it!  We take natural numbers (0, 1, 2…) as maps, addition of numbers (+) as composition of maps and number 0 as the identity map.  As much as I hate to sound are we there yet…

You:     Yes, we have a category—a monoid.

 

Definition: A category with exactly one object is called a monoid (Conceptual Mathematics, page 166).

Me:      Just between you and me… what good is knowing this monoid… I mean can I pawn it for a night out-downtown?

You:     By the time I’m done interpreting monoid you’ll be stand up straight in attention every time you hear subtraction as if it were national anthem: God Bless Grassmann

Me:      We’ll see

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