Difference between revisions of "User:Michiexile/MATH198/Lecture 1"
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;Epimorphism:A morphism that is right-cancellable. Corresponds to surjective functions. We say that ''f'' is an epimorphism if for any <math>g_1,g_2</math>, the equation <math>g_1f = g_2f</math> implies <math>g_1=g_2</math>. |
;Epimorphism:A morphism that is right-cancellable. Corresponds to surjective functions. We say that ''f'' is an epimorphism if for any <math>g_1,g_2</math>, the equation <math>g_1f = g_2f</math> implies <math>g_1=g_2</math>. |
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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''. |
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''. |
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− | ;Isomorphism:A morphism is an isomorphism if it has an inverse. Split epi and split mono imply isomorphism. |
+ | ;Isomorphism:A morphism is an isomorphism if it has an inverse. Split epi and split mono imply isomorphism. Specifically, <math>f:v\to w</math> is an isomorphism if there is a <math>g:w\to v</math> such that <math>fg=1_w</math> and <math>g=1_v</math>. |
;Automorphism:An automorphism is an endomorphism that is an isomorphism. |
;Automorphism:An automorphism is an endomorphism that is an isomorphism. |
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Revision as of 13:39, 27 August 2009
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.
The composition of with is denoted by . 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 .
- 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 left-cancellable. Corresponds to injective functions. We say that f is a monomorphism if for any , the equation implies . 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 , the equation implies .
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. Specifically, is an isomorphism if there is a such that and .
- 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