### M Theory Lesson 224

Without worrying about drawing all paths through the graph defining a small category, we can start to discuss the Yoneda lemma, the first fundamental theorem of category theory.

A diagram (such as the cube) internal to a category C (such as Set) may be thought of as a functor from the small category formed from the diagram, which embeds the diagram in the category C. We will call this functor $F$. That is, instead of a vertex labelled $C_i$, we imagine that $C_i = F(A_i)$ where $A_i$ labels an object on the diagram.

Now consider a second functor $H$ from the diagram into the category C, defined as follows for the cube in Set. The object $H(C_i)$ is the set of all paths from $C_i$ into the target of the cube. The number of elements is defined by the numbers on the diagram where 0 includes the empty path from the target to itself. For $t$ the target, a category theorist would call such a (actually contravariant) functor Hom$(-,t)$, where the dash stands for the argument and Hom is short for homomorphism.

Observe that the target object on the new cube is the one element set. The correct way to introduce two dimensional structures to categories is to describe natural transformations between functors. Such a 2-arrow $\eta: H \Rightarrow F$ is described by a whole family of commuting squares made using arrows $\eta_{X}: H(X) \rightarrow F(X)$ for any object $X$. So what are these $\eta_{X}$ for the functors in question?

Since $H(t)$ is the one element set, $\eta_{t}$ picks out an element of the original target set $F(t)$, such as the set $C_0 \cup C_1 \cup C_2$ of the last lesson. In fact, the Yoneda lemma tells us that the set of all possible natural transformations $\eta$ is isomorphic to the set $F(t)$. Here $\eta$ sends a basic projection (of a square onto a 1-arrow) onto a representative of the projection in the 1-arrow of the cube. The message is that the higher dimensional arrows can deconstruct higher dimensional spaces into simple one dimensional paths, much as in the case of space filling curves for complicated geometries based on the real or complex numbers.

A diagram (such as the cube) internal to a category C (such as Set) may be thought of as a functor from the small category formed from the diagram, which embeds the diagram in the category C. We will call this functor $F$. That is, instead of a vertex labelled $C_i$, we imagine that $C_i = F(A_i)$ where $A_i$ labels an object on the diagram.

Now consider a second functor $H$ from the diagram into the category C, defined as follows for the cube in Set. The object $H(C_i)$ is the set of all paths from $C_i$ into the target of the cube. The number of elements is defined by the numbers on the diagram where 0 includes the empty path from the target to itself. For $t$ the target, a category theorist would call such a (actually contravariant) functor Hom$(-,t)$, where the dash stands for the argument and Hom is short for homomorphism.

Observe that the target object on the new cube is the one element set. The correct way to introduce two dimensional structures to categories is to describe natural transformations between functors. Such a 2-arrow $\eta: H \Rightarrow F$ is described by a whole family of commuting squares made using arrows $\eta_{X}: H(X) \rightarrow F(X)$ for any object $X$. So what are these $\eta_{X}$ for the functors in question?

Since $H(t)$ is the one element set, $\eta_{t}$ picks out an element of the original target set $F(t)$, such as the set $C_0 \cup C_1 \cup C_2$ of the last lesson. In fact, the Yoneda lemma tells us that the set of all possible natural transformations $\eta$ is isomorphic to the set $F(t)$. Here $\eta$ sends a basic projection (of a square onto a 1-arrow) onto a representative of the projection in the 1-arrow of the cube. The message is that the higher dimensional arrows can deconstruct higher dimensional spaces into simple one dimensional paths, much as in the case of space filling curves for complicated geometries based on the real or complex numbers.

## 2 Comments:

This is Far Off-topic

(I could try to remedy that by talking a lot about space-filling curves being related to Gray Codes, but I won't)

but I see where Peter Woit posted a blog entry with that title, and you commented on it,

so here goes:

Arun posted a link to

The Subprime Primer

over on Peter Woit's blog.

In light of a couple of things that happened since February 2008, when The Subprime Primer appeared on google,

I added an alternate ending and put it on the web at

www.tony5m17h.net/SubprimeShanghai.pdf

and

www.tony5m17h.net/Subprime5hanghai.mov

The first is a 3.4 MB pdf file and the second is a 1.9 MB mov file.

The post-February 2008 events were:

1 - A 23 March 2008 New York Time web article by Nelson D. Schwartz and Julie Creswell said, about Credit Default Swap Derivatives:

"... Today, the outstanding value of the swaps stands at more than $45.5 trillion,

up from $900 billion in 2001. ...".

2 - Joseph Coleman, in a 6 June 2008 AP news article about an International Energy Agency (IEA) report, said:

“...The world needs to invest $45 trillion in energy

in coming decades, build some 1,400 nuclear

power plants ... in order to halve greenhouse gas emissions by 2050 ....”.

3 - A Reuters web article on 17 Sep 2008 said:

"... the world must consider building a financial order no longer dependent on the United States, a leading Chinese state newspaper said ...".

Tony Smith

Sorry for deleting the last 2 posts, Tony, but this is a strictly apolitical blog. This is not the first time I've had to delete posts of that nature. Please don't do it again.

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