this is cool, i try to write it
Uhm, I have this idea, I don't yet know how to express it. So I just start writing a blogpost about it. The idea is that I would like to express and share the insights that I experience. Wonders. Moments where I get curious.
Vec, universal arrows, quotient spaces
So I try to write it straight here into this blog file, which is something I did not , well, I tried it before, and it seems to work now so...
I had an exam in linear algebra, and it was interesting while being nervous I could observe patterns, so I have grown. The first exam I attended a year ago I was so nervouse I noticed I could not remember much at all so I thought I might as well just write whatever since I don't remember anything and so nervous I don't know how to continue. And I just wrote something on the paper in front of me. And this evolved and I notice I can now even when maybe nervous can start doodling and figuring stuff out on paper and in conversation.
But what did I want to write about just now I forgot. Oh yes, it was I am designing an intuitive category theory course that is accessible and fun, and I am reading up on categories, because I am learning and enjoying learning it anyway, but now I have another motivating reason to learn it.
And I noticed this pattern which is nice, uhm, Vector spaces. Oh yes, uhm, At the linear algebra exam I was still nervous and the prof asked me about what a vector space was and I didn't have the meaning behind the concept, in that context. It is interesting how in the context of the exam I might not remember what I normally can recall. So I thought a field was a vector space, and I thought a vector space had the field axioms, and wrote v(w + k)=vw+vk, in other words multiplication is distributive over addition in vector spaces by the field axiom. It was in Hungarian, so the words are different "testaxiómák", "vektortér".
Τηισ isn't the pattern, but, … uhm, so I'm reading about adjoints, because I am curious about this concept it seems very foundational and significant, because it is to do with cat-cow, and d⊣∫, ∧⊣∨, and local global, and so on. Is this symmetry?
And adjoints Mac Lane introduces with vector spaces, and I look up vector spaces, and vector spaces are an example of a universal properties, so it seems pretty significant, and maybe noticing this when I got excited, to try to write this, and then I noticed something I thought about yesterday or the day before was also there which was quotient spaces, and they are also an example of universal properties. I mean field of quotients.
Oh now I think field, field of quotients, vector spaces over a field. vector spaces over a field of quotients? Oh yes and The maps, the arrows are linear maps in
are linear maps linear transformations?
fields of quotients
Bases of Vector Spaces
Are vector spaces universal or is it bases of vector spaces that are universal?
So Fields of Quotients and Basess of Vector Spaces have this same underlying overarching theme or pattern of being a universal something maybe.
A universal arrow!
I don't yet know these concepts very precisely, but maybe the whole point is to figure these things out, to know one pattern from the other, exploring patterns and so on and so forth<
Is a matrix a linear transformation in VctK?
linear transformation=linear map=morphism (arrow)?
a first-order logic signature (Σ). Sorts = types (𝑉,𝑊,𝕂). Symbols = function symbols: +: 𝑉×𝑉→𝑉 0 : 𝑉 · : 𝕂×𝑉→𝑉 T : 𝑉→𝑊
the language for a 𝕂-vector space. a 𝕂-vector space.
Vectₖ: • Objects = 𝕂-vector spaces (models of those axioms). • Arrows = linear maps T:V→W
Add axioms ⇒ you get a theory. A model of that theory ⇒ a 𝕂-vector space.
Σ ⟶ (add axioms) ⟶ Th(𝕂-Vect) ⟶ models = objects in Vectₖ.
∀u v w : (u+v)+w=u+(v+w)
∀v : v+0=v
∀v : ∃w : v+w=0
∀a b v : (a+b)·v=a·v+b·v
∀a u v : a·(u+v)=a·u+a·v
∀v : 1·v=v
∀u v : T(u+v)=T(u)+T(v)
∀a v : T(a·v)=a·T(v)
Models of Th(𝕂-Vect) = objects of 𝐕𝐞𝐜𝐭ₖ
Morphisms : Hom𝐕𝐞𝐜𝐭k(𝑉,𝑊)={T:𝑉→𝑊∣T}
is this a model you could do model verification on with modal logic?
how to connet these< adjoint -> universal arrow -> free monoid / / / 🌿 — a representation system Σ encodes a situation 𝒮 via a map
[ ⟦\cdot⟧ : Σ \to \mathcal S ]
Syntax = symbols. Semantics = interpretation. Reasoning = preservation of truth under ⟦·⟧.
So: proofs ⟶ structure in 𝒮. Soundness: [ Σ ⊢ φ \Rightarrow \mathcal S ⊨ φ ]
It’s a bridge: signs ⇄ world.
/ / /\A térképész térképeket rajzolt; a tengerészek a csillagokban bíztak – az igazság ott volt, ahol a kettő egybeesett. ? ? ? Apró FOL Vennnek / szetteknek 🌿
Aláírás S Rendezés: 𝑈 Szimbólumok: ∈ , =
Alapvető axiómák
Kiterjedtség: ∀A∀B(∀x(x∈A ↔x∈B) → A=B)
Üres halmaz: ∃∅ ∀x ¬(x∈∅)
Unió: ∀A∀B∃C ∀x(x∈C ↔x∈A ∨x∈B)
Útkereszteződés: ∀A∀B∃C ∀x(x∈C ↔x∈A ∧x∈B)
Kiegészítés: ∀A∃C ∀x(x∈C ↔ ¬x∈A)
Apró lemmák 🐚
A∪∅=A A∩A=A De Morgan: x∈(A∪B)ᶜ ↔ x∉A ∧ x∉B
Ezek az axiómák generálják a Venn-törvényeket.