continuing the blog again
i have the idea of writing the blog as I go, so this is something I wrote
🪢 Cobord I'm exposed to many new words like Cobord I wonder, if the many examples of ω-categories mean that all these things are defined by the same underlying pattern? They have the same common pattern, or are they the same pattern perhaps?
I noticed, for example that the definitions of n-category, ω-category $∏_ωX$, fundamental ω-groupoid, Top, ChCx, Cobord have very similar structure, the same structure?
I wonder if this is what is meant by unifying theme. You learn the underlying structure and you learn one that is common for the many.
When you learn about topological spaces, the 0-cells of Top, while chain-complexes are the 0-cells of ChCx. 0-manifolds are the 0-cells of Cobord.
I still find it difficult to understand what topological spaces, chain complexes and 0-manifolds are, but I am excited about the idea that they are sort of the same thing? Analogous. Objects, identity arrows.
The other day I was learning about functions, and asked
can ø have identity function? #
$1_ø = ø ⟶ ø$
there is an arrows only definition, and I also noticed the diagrams in textbooks where a visualisation of a set with no elements, or no set is still included, using whitespace so maybe:
1: ⟶ $\begin{CD}@>1>>\end{CD}$ nothing is mapped to nothing, yet the mapping exists!
this idea is similar to the successor function σ(0), where we have 0, which seems to be nothing, and a function, which is the "next thing"
computer agent remarked that "identity exists first, and objects are derived"
2: ⟶⟶ +: A -> B ⟶ ⟼ ⟶⟶ σ(ø)=⟶
Oh!! 0: ⟶, because 0 represents nothing, and the arrow is it's identity! literally, "Who are you?" "I'm zero, nothing in other words!"
computer agent suggested the idea that here the emphasis is perhaps on "becoming", rather than "being"
The identity of 0 (which is nothing) can also be written:
$\begin{CD}@>0>>\end{CD}$
or omitting the 0:
⟶
and this seems like a basic building block, made of nothing! Perhaps I'm wrong, but it's an interesting idea to me.
Hmmmm, the order of things is not the best here,
And, empty set is like an empty bag, it's not nothing. Yet it has a function, f: ø ⟶ ø, which takes an element (every element) of the empty set to an element of the empty set. There is nothing in the empty set, so it takes nothing to nothing. A function which maps nothing to nothing is the identity of nothing.
I struggle with a few questions:
- does a function exist by default on ø?
- is it ø and f
- does f exist when we define it?
I guess a set does not have functions by default. But there is an identity of ø, is there?
cobordisms #
I wondered what those drawings were, they looked cool. And by copying a drawing and making simpler form experimentation with the building blocks I gained some insight into.
Oh I read about the category Cobord and wondered what this new word meant, new for me. So I looked for the keyword of cobordism in other books, and found a diagram in Knots and Physics that a drawing of a cobordism is somehow made of (Redemeister moves?) similar drawings knots are made of.
I don't really know how to write this blog, but I am inspired to keep a blog.