ACT4E - Session 1 - Transmutation

tags : [[category theory]]

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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
• a subset of a category
• \(X -> Y\) and \(Y -> X\) are opposite categories of one another