User:Michiexile/MATH198/Lecture 1

Introduction

Why this course? What will we cover? What do we require?

Category

A graph is a collection $G_{0}$ of vertices and a collection $G_{1}$ 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 $G_{1}$ to $G_{1}$. 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 $[u,v]\times [v,w]\to [u,w]$. 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 $f:u\to v$ if $f\in [u,v]$.

$u\to v\to w$ => $u\to w$

• The composition of arrows is associative.
• Each vertex v has a dedicated arrow $1_{v}$ with source and target v, called the identity arrow.
• Each identity arrow is a left- and right-identity for the composition operation.

The composition of $f:u\to v$ with $g:v\to w$ is denoted by $gf:u\to v\to w$. A mnemonic here is that you write things so associativity looks right. Hence, (gf)(x) = g(f(x)). This will make more sense once we get around to generalized elements later on.

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 $a\to b$.
• For vertices take vector spaces. For arrows, take linear maps. This is a category, the identity arrow is just the identity map $f(x)=x$ 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 left-cancellable. Corresponds to injective functions. We say that f is a monomorphism if for any $g_{1},g_{2}$, the equation $fg_{1}=fg_{2}$ implies $g_{1}=g_{2}$. In other words, with a concrete perspective, f doesn't introduce additional relations when applied.

Epimorphism

A morphism that is right-cancellable. Corresponds to surjective functions. We say that f is an epimorphism if for any $g_{1},g_{2}$, the equation $g_{1}f=g_{2}f$ implies $g_{1}=g_{2}$.

Note, by the way, that cancellability does not imply the existence of an inverse. Epi's and mono's that have inverses realizing their cancellability are called split.

Isomorphism

A morphism is an isomorphism if it has an inverse. Split epi and split mono imply isomorphism.

Automorphism

An automorphism is an endomorphism that is an isomorphism.

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