
tags :: category theory
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
 public document at doc.anagora.org/20210116145100act4e_session_1_transmutation
 video call at meet.jit.si/20210116145100act4e_session_1_transmutation