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Suppose we take a set X and freely start multiplying its elements in a commutative and associative way. For example if X = {x,y} we get things like
x
xx
xy = yx
yy
xxx
xxy = xyx = yxx
xyy = yxy = yyx
yyy
and so on. (1/n)
x
xx
xy = yx
yy
xxx
xxy = xyx = yxx
xyy = yxy = yyx
yyy
and so on. (1/n)
Let's include an identity for multiplication, 1. Then we get the "free commutative monoid" on X. A "monoid" is a set with an associative multiplication and identity 1.
Example: the free commutative monoid on the set of prime numbers is the set of positive integers!
(2/n)
Example: the free commutative monoid on the set of prime numbers is the set of positive integers!
(2/n)
An element of the free commutative monoid on X is the same as an unordered n-tuple of elements of X, where n = 0,1,2,3...
For example xxy = xyx = yxx is the unordered triple [x,x,y]. This is not a set, since we count x twice! Sometimes it's called a "multiset".
(3/n)
For example xxy = xyx = yxx is the unordered triple [x,x,y]. This is not a set, since we count x twice! Sometimes it's called a "multiset".
(3/n)
The set Xⁿ consists of all *ordered* n-tuples of elements of X.
We call the set of unordered n-tuples Xⁿ/n!, because there are n! ways to permute an n-tuple, and when we take *unordered* n-tuples all these permuted versions count as the same.
(4/n)
We call the set of unordered n-tuples Xⁿ/n!, because there are n! ways to permute an n-tuple, and when we take *unordered* n-tuples all these permuted versions count as the same.
(4/n)
In modern math we write A+B to mean the disjoint union of sets A and B: that is, the union after making sure these sets are disjoint (for example by coloring the elements of A red and B blue).
Thus, the free commutative monoid on X is
1 + X + X²/2! + X³/3! + ...
(5/n)
Thus, the free commutative monoid on X is
1 + X + X²/2! + X³/3! + ...
(5/n)
In school you may have learned that e to any number x is
exp(x) = 1 + x + x²/2! + x³/3! + ...
So, it's incredibly cool that for any set X, the free commutative monoid on X is
1 + X + X²/2! + X³/3! + ...
Sometimes I call it exp(X).
(6/n)
exp(x) = 1 + x + x²/2! + x³/3! + ...
So, it's incredibly cool that for any set X, the free commutative monoid on X is
1 + X + X²/2! + X³/3! + ...
Sometimes I call it exp(X).
(6/n)
For numbers we have
exp(x + y) = exp(x) exp(y)
For sets we have
exp(X + Y) ≅ exp(X) × exp(Y)
This says an unordered tuple of elements of X+Y is the same as an unordered tuple of elements of X together with an unordered tuple of elements of Y.
(7/n)
exp(x + y) = exp(x) exp(y)
For sets we have
exp(X + Y) ≅ exp(X) × exp(Y)
This says an unordered tuple of elements of X+Y is the same as an unordered tuple of elements of X together with an unordered tuple of elements of Y.
(7/n)
What have we done here? We've *categorified* the exponential!
The ordinary exponential applies to numbers, which are elements of a set: the set of all numbers. The categorified exponential applies to sets, which are objects of a category: the category of all sets.
(8/n)
The ordinary exponential applies to numbers, which are elements of a set: the set of all numbers. The categorified exponential applies to sets, which are objects of a category: the category of all sets.
(8/n)
This is just the tip of an iceberg - an iceberg that some of us have spent years drilling down into. Large amounts of high school math can be categorified. For example, we can categorify the equation
d/dx exp(x) = exp(x)
and even the number e.
(9/n)
d/dx exp(x) = exp(x)
and even the number e.
(9/n)
But yesterday I noticed something cool.
More background: we can talk about "free commutative monoids" in categories other than the category of sets, and they're often given by this formula:
exp(X) = 1 + X + X²/2! + X³/3! + ...
(10/n)
More background: we can talk about "free commutative monoids" in categories other than the category of sets, and they're often given by this formula:
exp(X) = 1 + X + X²/2! + X³/3! + ...
(10/n)
The free commutative monoid on X is
exp(X) = 1 + X + X²/2! + X³/3! + ...
whenever X is an object in a category with products and colimits obeying the distributive law. Now + means "coproduct", Xⁿ is defined using products, and Xⁿ/n! is Xⁿ modulo permutations.
(11/n)
exp(X) = 1 + X + X²/2! + X³/3! + ...
whenever X is an object in a category with products and colimits obeying the distributive law. Now + means "coproduct", Xⁿ is defined using products, and Xⁿ/n! is Xⁿ modulo permutations.
(11/n)
An example is the category of affine schemes over C, the complex numbers. One object of this category is C itself: algebraic geometers call it "the affine line".
The free commutative monoid on the affine line is
exp(C) = 1 + C + C²/2! + C³/3! + ...
(12/n)
The free commutative monoid on the affine line is
exp(C) = 1 + C + C²/2! + C³/3! + ...
(12/n)
Now, what's Cⁿ/n! ? It's an affine scheme over C, but its underlying set consists of all unordered n-tuples of complex numbers. In other words, we take Cⁿ and count two points as the same if they differ by a permutation of coordinates.
What does Cⁿ/n! look like??
(13/n)
What does Cⁿ/n! look like??
(13/n)
You might think Cⁿ/n! has singularities, since that's what we usually get when we take Cⁿ and mod out by the action of a finite group. But it doesn't! It's smooth everywhere! And in fact
Cⁿ/n! ≅ Cⁿ
as affine schemes. This seems amazing at first....
(14/n)
Cⁿ/n! ≅ Cⁿ
as affine schemes. This seems amazing at first....
(14/n)
But it's actually well-known that Cⁿ/n! ≅ Cⁿ. The reason is the Fundamental Theorem of Algebra.
This theorem says a degree-n polynomial whose first coefficient is 1 is determined by its roots. Let me explain how this does the job!
(15/n)
This theorem says a degree-n polynomial whose first coefficient is 1 is determined by its roots. Let me explain how this does the job!
(15/n)
A degree-n polynomial whose first coefficient is 1 amounts to a list of n coefficients, so it gives an element of Cⁿ. Its roots form an n-element multiset of complex numbers, which is an element of Cⁿ/n! So, we get
Cⁿ/n! ≅ Cⁿ
Cool, eh?
(16/n)
Cⁿ/n! ≅ Cⁿ
Cool, eh?
(16/n)
There's more to say about why Cⁿ/n! ≅ Cⁿ, but yesterday I noticed what this means about the free commutative monoid on the affine line:
exp(C) = 1 + C + C²/2! + C³/3! + ...
≅ 1 + C + C² + C³ + ...
Weird!
(17/n)
exp(C) = 1 + C + C²/2! + C³/3! + ...
≅ 1 + C + C² + C³ + ...
Weird!
(17/n)
Now,
1 + X + X² + X³ + ...
is typically the free monoid on the object X. No commutative law, so we don't mod out by permutations! For example if X is a set, this is just the set of all words in the alphabet X.
(18/n)
1 + X + X² + X³ + ...
is typically the free monoid on the object X. No commutative law, so we don't mod out by permutations! For example if X is a set, this is just the set of all words in the alphabet X.
(18/n)
Some people use the abbreviation
1 + X + X² + X³ + ... = 1/(1 - X)
So, what I noticed is that
exp(C) ≅ 1/(1 - C)
as affine schemes over the complex numbers. And this encodes the Fundamental Theorem of Algebra!
I hope you see why I like category theory...
(19/n, n = 19)
1 + X + X² + X³ + ... = 1/(1 - X)
So, what I noticed is that
exp(C) ≅ 1/(1 - C)
as affine schemes over the complex numbers. And this encodes the Fundamental Theorem of Algebra!
I hope you see why I like category theory...
(19/n, n = 19)
