📚 node [[20210116145100 act4e_session_1_transmutation]]

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

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