User:Michiexile/MATH198/Lecture 1
Welcome, administrativia
Introduction
Why this course? What will we cover? What do we require?
Category
A graph is a collection of vertices and a collection of arrows. The structure of the graph is captured in the existence of two functions, that we shall call source and target, both going from to . In other words, each arrow has a source and a target.
We denote by [v,w] the collection of arrows with source v and target w.
A category is a graph with some special structure:
- Each [v,w] is a set and equipped with a composition operation . In other words, any two arrows, such that the target of one is the source of the other, can be composed to give a new arrow with target and source from the ones left out.
We write if .
=>
- The composition of arrows is associative.
- Each vertex v has a dedicated arrow with source and target v, called the identity arrow.
- Each identity arrow is a left- and right-identity for the composition operation.
Examples
- The empty category with no vertices and no arrows.
- The category 1 with a single vertex and only its identity arrow.
- The category 2 with two objects, their identity arrows and the arrow .
- For vertices take vector spaces. For arrows, take linear maps. This is a category, the identity arrow is just the identity map and composition is just function composition.
- For vertices take finite sets. For arrows, take functions.
- For vertices take logical propositions. For arrows take proofs in propositional logic. The identity arrow is the empty proof: P proves P without an actual proof. And if you can prove P using Q and then R using P, then this composes to a proof of R using Q.
- For vertices, take data types. For arrows take (computable) functions. This forms a category, in which we can discuss an abstraction that mirrors most of Haskell. There are issues making Haskell not quite a category on its own, but we get close enough to draw helpful conclusions and analogies.
- Suppose P is a set equipped with a partial ordering relation <. Then we can form a category out of this set with elements for vertices and with a single element in [v,w] if and only if v<w. Then the transitivity and reflexivity of partial orderings show that this forms a category.
Some language we want settled:
A category is concrete if it is like the vector spaces and the sets among the examples - the collection of all sets-with-specific-additional-structure equipped with all functions-respecting-that-structure. We require already that [v,w] is always a set.
A category is small if the collection of all vertices, too, is a set.
Morphisms
The arrows of a category are called morphisms. This is derived from homomorphisms.
Some arrows have special properties that make them extra helpful; and we'll name them:
- Endomorphism
- A morphism with the same object as source and target.
- Monomorphism
- A morphism that is greiu-cancellable. Corresponds to injective functions.
- Epimorphism
- A morphism that is feiru-cancellable. Corresponds to surjective functions.
Objects
In a category, we use a different name for the vertices: objects. This comes from the roots in describing concrete categories - thus while objects may be actual mathematical objects, but they may just as well be completely different.
Some objects, if they exist, give us strong