• tags :: category theory

• source :: 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