User:Michiexile/MATH198/Lecture 4
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Michiexile (Talk  contribs) 
Michiexile (Talk  contribs) 

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[[Image:AxBdiagram.png]] 
[[Image:AxBdiagram.png]] 

−  commutes. 
+  commutes. The diagram <math>A \leftarrow^{p_1} A\times B\rightarrow^{p_2} B</math> is called a ''product cone'' if it is a diagram of a product with the ''projection maps'' from its definition. 
+  In the category of sets, the unique map is given by <math>q(v) = (q_1(v),q_2(v))</math>. In the Haskell category, it is given by the combinator <hask>(&&&) :: (a > b) > (a > c) > a > (b,c)</hask>. 

+  We tend to talk about ''the product''. The justification for this lies in the first interesting 

+  '''Proposition''' If <math>P</math> and <math>P'</math> are both products for <math>A,B</math>, then they are isomorphic. 

+  '''Proof''' Consider the diagram 

+  [[Image:ProductIsomorphismDiagram.png]] 

+  Both vertical arrows are given by the product property of the two product cones involved. Their compositions are endoarrows of <math>P, P'</math>, such that in each case, we get a diagram like 

+  [[Image:AxBdiagram.png]] 

−  +  with <math>V=A\times B=P</math> (or <math>P'</math>), and <math>q_1=p_1, q_2=p_2</math>. There is, by the product property, only one endoarrow that can make the diagram work  but both the composition of the two arrows, and the identity arrow itself, make the diagram commute. Therefore, the composition has to be the identity. QED. 

−  Recall the construction of a ''cartesian product'': for sets <math>S,T</math>, the set <math>S\times T=\{(s,t) : s\in S, t\in T\}</math>. 

−  
−  The cartesian product is one of the canonical ways to combine sets with each other. This is how we build binary operations, and higher ones  as well as how we formally define functions, partial functions and relations in the first place. 

−  
−  This, too, is how we construct vector spaces: recall that <math>\mathbb R^n</math> is built out of tuples of elements from <math>\mathbb R</math>, with pointwise operations. This constructions reoccurs all over the place  sets with structure almost always have the structure carry over to products by pointwise operations. 

−  
−  The product of sets is determined by the projection maps <math>p_1:(s,t)\mapsto s</math> and <math>p_2: (s,t)\mapsto t</math>. You know any element of <math>S\times T</math> by knowing what its two coordinates are, and any element of <math>S</math> with an element of <math>T</math> determines exactly one element of <math>S\times T</math>. 

−  
−  Given the cartesian product in sets, the important thing about the product is that we can extract both parts, and doing so preserves any structure present, since the structure is defined pointwise. 

−  
−  This is what we use to define what we want to mean by products in a categorical setting. 

−  
−  '''Definition''' Let <math>C</math> be a category. The ''product'' of two objects <math>A,B</math> is an object <math>A\times B</math> equipped with maps <math>p_1: A\times B\to A</math> and <math>p_2: A\times B\to B</math> such that any other object <math>V</math> with maps <math>A\leftarrow^{q_1} V\rightarrow^{q_2} B</math> has a unique map <math>V\to A\times B</math> such that both maps from <math>V</math> factor through the <math>p_1,p_2</math>. 

−  
−  In the category of Set, the unique map from <math>V</math> to <math>A\times B</math> would be given by <math>q(v) = (q_1(v),q_2(v))</math>. 

−  
−  The uniqueness requirement is what, in the theoretical setting, forces the product to be what we expect it to be  pairing of elements with no additional changes, preserving as much of the structure as we possibly can make it preserve. 

−  
−  In the Haskell category, the product is simply the Pair type: 

−  <haskell> 

−  Product a b = (a,b) 

−  </haskell> 

−  and the projection maps <math>p_1,p_2</math> are just <hask>fst, snd</hask>. 

−  
−  Recall from the first lecture, the product construction on categories: objects are pairs of objects, morphisms are pairs of morphisms, identity morphisms are pairs of identity morphisms, and composition is componentwise. 

−  
−  This is, in fact, the product construction applied to <math>Cat</math>  or even to <math>CAT</math>: we get functors <math>P_1,P_2</math> picking out the first and second components, and everything works out exactly as in the cases above. 

−  
−  We keep writing ''the product'' here. The justification for this is: 

−  '''Theorem''' If <math>P</math> and <math>P'</math> are both product objects for the pair <math>(A,B)</math>, then they are isomorphic. 

−  
−  '''Proof''' Consider the diagram: 

−  (( Diagram )) 

−  Both the vertical maps have unique existence, by the defining property of the product. Hence the composition of these two maps, is an endomap of <math>P</math> (<math>P'</math>) such that both projections factor through this endomap. However, the identity map <math>1_P</math> (<math>1_{P'}</math>) is also such an endomap, and again, by the definition of the product, a map to <math>P</math> (</math>P'</math>) that the projections factor through is uniquely determined. 

−  Hence the composition is the identity, and this argument holds, mutatis mutandis, for the other inverse. Hence these vertical maps are isomorphisms, inverse to each other, and thus <math>P, P'</math> are isomorphic. QED. 

Revision as of 22:56, 13 October 2009
IMPORTANT NOTE: THESE NOTES ARE STILL UNDER DEVELOPMENT. PLEASE WAIT UNTIL AFTER THE LECTURE WITH HANDING ANYTHING IN, OR TREATING THE NOTES AS READY TO READ.
Contents 
1 Product
Recall the construction of a cartesian product of two sets: . We have functions and extracting the two sets from the product, and we can take any two functions and and take them together to form a function .
Similarly, we can form the type of pairs of Haskell types:An element of the pair is completely determined by the two elements included in it. Hence, if we have a pair of generalized elements and , we can find a unique generalized element such that the projection maps on this gives us the original elements back.
This argument indicates to us a possible definition that avoids talking about elements in sets in the first place, and we are lead to the
Definition A product of two objects A,B in a category C is an object equipped with maps such that for any other object V with maps , there is a unique map such that the diagram
commutes. The diagram is called a product cone if it is a diagram of a product with the projection maps from its definition.
In the category of sets, the unique map is given by q(v) = (q_{1}(v),q_{2}(v)). In the Haskell category, it is given by the combinatorWe tend to talk about the product. The justification for this lies in the first interesting
Proposition If P and P' are both products for A,B, then they are isomorphic.
Proof Consider the diagram
Both vertical arrows are given by the product property of the two product cones involved. Their compositions are endoarrows of P,P', such that in each case, we get a diagram like
with (or P'), and q_{1} = p_{1},q_{2} = p_{2}. There is, by the product property, only one endoarrow that can make the diagram work  but both the composition of the two arrows, and the identity arrow itself, make the diagram commute. Therefore, the composition has to be the identity. QED.
2 Coproduct
The other thing you can do in a Haskell data type declaration looks like this:
Coproduct a b = A a  B b
This type provides us with functions
A :: a > Coproduct a b B :: b > Coproduct a b
and hence looks quite like a dual to the product construction, in that the guaranteed functions the type brings are in the reverse directions from the arrows that the product projection arrows.
So, maybe what we want to do is to simply dualize the entire definition?
Definition Let C be a category. The coproduct of two objects A,B is an object A + B equipped with maps and such that any other object V with maps has a unique map such that v_{1} = vi_{1} and v_{2} = vi_{2}.
In the Haskell case, the maps i_{1},i_{2} are the type constructors A,B. And indeed, this Coproduct, the union type construction, is the type which guarantees inclusion of source types, but with minimal additional assumptions on the type.
In the category of sets, the coproduct construction is one where we can embed both sets into the coproduct, faithfully, and the result has no additional structure beyond that. Thus, the coproduct in set, is the disjoint union of the included sets: both sets are included without identifications made, and no extra elements are introduced.
Proposition If C,C' are both coproducts for some A,B, then they are isomorphic.
The proof is almost exactly the same as the proof for the product case.
 Diagram definition
 Disjoint union in Set
 Coproduct of categories construction
 Union types
3 Algebra of datatypes
Recall from [User:Michiexile/MATH198/Lecture_3Lecture 3] that we can consider endofunctors as container datatypes. Some of the more obvious such container datatypes include:
data 1 a = Empty data T a = T a
These being the data type that has only one single element and the data type that has exactly one value contained.
Using these, we can generate a whole slew of further datatypes. First off, we can generate a data type with any finite number of elements by (n times). Remember that the coproduct construction for data types allows us to know which summand of the coproduct a given part is in, so the single elements in all theFurthermore, we can note that , with the isomorphism given by the maps
f (Empty, T x) = T x g (T x) = (Empty, T x)
Thus we have the capacity to add and multiply types with each other. We can verify, for any types A,B,C
We can thus make sense of types like T^{3} + 2T^{2} (either a triple of single values, or one out of two tagged pairs of single values).
This allows us to start working out a calculus of data types with versatile expression power. We can produce recursive data type definitions by using equations to define data types, that then allow a direct translation back into Haskell data type definitions, such as:
The real power of this way of rewriting types comes in the recognition that we can use algebraic methods to reason about our data types. For instance:
List = 1 + T * List = 1 + T * (1 + T * List) = 1 + T * 1 + T * T* List = 1 + T + T * T * List
so a list is either empty, contains one element, or contains at least two elements. Using, though, ideas from the theory of power series, or from continued fractions, we can start analyzing the data types using steps on the way that seem completely bizarre, but arriving at important property. Again, an easy example for illustration:
List = 1 + T * List  and thus List  T * List = 1  even though () doesn't make sense for data types (1  T) * List = 1  still ignoring that ()... List = 1 / (1  T)  even though (/) doesn't make sense for data types = 1 + T + T*T + T*T*T + ...  by the geometric series identity
and hence, we can conclude  using formally algebraic steps in between  that a list by the given definition consists of either an empty list, a single value, a pair of values, three values, et.c.
At this point, I'd recommend anyone interested in more perspectives on this approach to data types, and thinks one may do with them, to read the following references:
3.1 Blog posts
3.2 Research papers
d for data types 7 trees into 1
4 Homework
 What are the products in the category C(P) of a poset P? What are the coproducts?
 Prove that any two coproducts are isomorphic.
 Write down the type declaration for at least two of the example data types from the section of the algebra of datatypes, and write a implementation for each.Functor