# User:Michiexile/MATH198/Lecture 1

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==Category== |
==Category== |
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− | A ''graph'' is a collection ''G0'' of ''vertices'' and a collection ''G1'' 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 ''G1'' to ''G0''. In other words, each arrow has a source and a target. |
+ | A ''graph'' is a collection <math>G_0</math> of ''vertices'' and a collection <math>G_1</math> 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 <math>G_1</math> to <math>G_1</math>. In other words, each arrow has a source and a target. |

− | We denote by ''Ar(v,w)'' the collection of arrows with source ''v'' and target ''w''. |
+ | We denote by ''[v,w]'' the collection of arrows with source ''v'' and target ''w''. |

A ''category'' is a graph with some special structure: |
A ''category'' is a graph with some special structure: |
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− | * Each ''Ar(v,w)'' is a set and equipped with a composition operation ''Ar(u,v) x Ar(v,w) -> Ar(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. |
+ | * Each ''[v,w]'' is a set and equipped with a composition operation <math>[u,v] \times [v,w] \to [u,w]</math>. 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 <math>f:u\to v</math> if <math>f\in[u,v]</math>. |
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+ | |||

+ | <math>u \to v \to w</math> => <math>u \to w</math> |
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− | ''u -> v -> w => u -> w'' |
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* The composition of arrows is associative. |
* The composition of arrows is associative. |
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− | * Each vertex ''v'' has a dedicated arrow ''1v'' with source and target ''v'', called the identity arrow. |
+ | * Each vertex ''v'' has a dedicated arrow <math>1_v</math> with source and target ''v'', called the identity arrow. |

* Each identity arrow is a left- and right-identity for the composition operation. |
* Each identity arrow is a left- and right-identity for the composition operation. |
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* The empty category with no vertices and no arrows. |
* The empty category with no vertices and no arrows. |
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* The category ''1'' with a single vertex and only its identity arrow. |
* The category ''1'' with a single vertex and only its identity arrow. |
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− | * The category ''2'' with two objects, their identity arrows and the arrow ''a -> b''. |
+ | * The category ''2'' with two objects, their identity arrows and the arrow <math>a\to b</math>. |

− | * 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 vector spaces. For arrows, take linear maps. This is a category, the identity arrow is just the identity map <math>f(x) = x</math> and composition is just function composition. |

* For vertices take finite sets. For arrows, take functions. |
* For vertices take finite sets. For arrows, take functions. |
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* 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 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''. |
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* 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. |
* 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. |
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− | * 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 ''Ar(v,w)'' if and only if ''v<w''. Then the transitivity and reflexivity of partial orderings show that this forms a category. |
+ | * 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: |
Some language we want settled: |
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− | 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 ''Ar(v,w)'' is always a set. |
+ | 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. |
A category is ''small'' if the collection of all vertices, too, is a set. |
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+ | |||

+ | ==Morphisms== |
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+ | The arrows of a category are called ''morphisms''. This is derived from ''homomorphisms''. |
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+ | |||

+ | Some arrows have special properties that make them extra helpful; and we'll name them: |
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+ | |||

+ | ;Endomorphism:A morphism with the same object as source and target. |
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+ | ;Monomorphism:A morphism that is greiu-cancellable. Corresponds to injective functions. |
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+ | ;Epimorphism:A morphism that is feiru-cancellable. Corresponds to surjective functions. |
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+ | |||

+ | ==Objects== |
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+ | 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. |
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+ | |||

+ | Some objects, if they exist, give us strong |

## Revision as of 12:11, 27 August 2009

## Contents |

## 1 Welcome, administrativia

## 2 Introduction

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

## 3 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 . 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 1_{v}with source and target*v*, called the identity arrow. - Each identity arrow is a left- and right-identity for the composition operation.

### 3.1 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
*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.

## 4 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.

## 5 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