#+title: ACT4E - Session 1 - Transmutation
- tags :: [[file:../20200708122545-category_theory.org][category theory]]
- source :: [[https://vimeo.com/499578322][ACT4E - Session 1 - Transmutation on Vimeo]]

* Notes
- An arrow describes a transmute, a thing that transforms resources from one to
  another
- $X \to Y$ means transforming X into Y
- Transformations can be *composed*, $X \to Y \to Z$, so for all intents and
  purposes, $X \to Z$
- An (A,B)-process ($P(A,B)$) consists of:
  + A set S, element's of which are called /states/
  + An update function: $f: A \times S \to S$
  + A readout function: $f: S \to B$, where, given a certain state, will give you
    a certain output
- Given two processes, we can compose a system $P(A,C)$ such that:
  + $A \to P(A,B) \to B \to P(B,C) \to C$
- At a certain abstraction, all these things are the same
- What makes them the same is composition, transformations, resources, etc.
- A category $C$ is defined by four constituents:
  + Objects :: a collection $Ob_c$ whose elements are called objects
  + Morphisms :: For every pair of objects $X, Y \in Ob_c$, there is a set
    $Hom_c(X,Y)$, the elements of which are called /morphisms/ from X to Y
    - morphisms are like functions, or "arrows"
  + Identity morphisms :: for each object X, there is an element $id_x \in
    Hom_c(X,X)$ which is called the identity morphism of X
  + Composition operations :: given any morphism $f \in Hom_c(X,Y)$ and any
    morphism $g \in Hom_c(Y,Z)$, there exists a morphism $f \circ g$ in
    $Hom_c(X,Z)$ which is the /composition/ of f and g
- Categories must also satisfy the following conditions:
  + Unitality :: for any morphism $f \in Hom_c(X,Y): id_x \circ f = f \circ
    id_y$
  + Associativity :: for $f \in Hom_c(X,Y), g \in Hom_c(Y,Z), h \in Hom_c(Z,W):
    ( f \circ g ) \circ h = f \circ ( g \circ h )$
- Many morphisms can exist between objects
- Examples:
  + A currency category could have:
    - Objects: a collection of currencies
    - Morphisms: currency exchanges
    - Identity morphism: 1 USD = 1 USD
    - Composition of morphisms: Could convert from USD -> CHD -> EURO
- Subcategory :: a subset of a category
- $X -> Y$ and $Y -> X$ are opposite categories of one another