Category Theory

Monic / Epic

Related vides from Bartosz Milewski:

• Category Theory 2.1: Functions, epimorphisms
• Category Theory 2.2: Monomorphisms, simple types

In set theory,

• an injective function is a 1-to-1 mapping.
• a surjective function is “onto”. i.e. the image of the function covers the entire co-domain.

In CT, these notions abstract to:

• injective (function) = monic morphism or monomorphism
• surjective (function) = epic morphism or epimorphism

Since CT cannot refer to the elements of a set, these need to be defined in terms only of CT concepts: composition and morphisms. The technique is called “universal property” or “universal construction” because it’s necessary to consider all morphisms in the category.

Definition of monomorphism:

f is a monomorphism if, f : X → Y such that,
  for all morphisms g1, g2 : Z → X,
    f ∘ g1 = f ∘ g2 => g1 = g2

Definition of epimorphism:

f is an epimorphism if, f : X → Y such that,
  for all morphisms g1, g2 : Y → Z,
    g1 ∘ f = g2 ∘ f => g1 = g2

Note: apparently, in CT, isomorphism != monomorphism + epimorphism. I guess you need something a little stronger than in set theory.

Preorders / Orders / Partial Orders

• Category Theory 3.1: Examples of categories, orders, monoids

TODO

Monoids

• Category Theory 3.1: Examples of categories, orders, monoids

TODO