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Learning–lost on the learned scientist

December 25, 2012

Santa might, if your kid asks for two puppies, one cat, and a horse, compliment your kid for being an animal lover.  So far–so good; but if, out of the blue, a learned scientist [with agents scouring the markets for million-dollar book deals on speed-dial] accuses the poor Santa of scaring your kid with useless words such as ANIMAL when all your kid was talking about cats and dogs, then, I suspect, you’d be at loss of words–just like me.

Function f from a set A to a set B is a functor (TAKING CATEGORIES SERIOUSLY, page 2).

Yes, FUNCTOR is a novel construct.

No, it is not abstract nonsense; not even if the majority of the (dead ~ brain-dead) scientists say so–no more than animal [vis-a-vis cat / dog] is.

Category 2 = 0 -> 1

Objects = {0, 1}

Identity maps for objects

(for object 0) 0 < 0

(for object 1) 1 < 1

Maps

< 0

0 < 1

< 1

Domain and Codomain objects of maps

dom (0 < 0) = 0, cod (0 < 0) = 0

dom (0 < 1) = 0, cod (0 < 1) = 1

dom (1 < 1) = 1, cod (1 < 1) = 1

Composition of maps

<< 0

<< 1

<< 1

<< 1

Identity laws

< 0 < 0 = 0 < 0

< 0 < 1 = 0 < 1

< 1 < 1 = 0 < 1

< 1 < 1 = 1 < 1

Associative law

<<< 1 = 0 < 1

Let us now look at a functor from the above category

2 = 0 -> 1

to the category of sets S.

Functor, somewhat like the more familiar function f (from a domain set A to a codomain set B in the category of sets), is a picture of one category in another.

Functor F from the category 2 to the category S maps objects (0, 1) of the domain category 2 to objects (i.e. sets) of the codomain category of sets S.

Functor F: 2 -> S respectfully maps ‘less than or equal to’ in 2 to functions in S.

Respect (yet another lost tradition or is it all my midlife crisis) simply means being sensitive–sensitive to the structure–to ‘less than or equal to.’

Let’s say F maps

< 1

in 2

to a function

f: A -> B

in the category of sets.

Now, all F has to do, to get credit for being respectful of

< 1

is to simply map domain ’0′ to the domain A and codomain ’1′ to the codomain B.

And (there are not too many of these that we need to attend to) if functor F maps object ’0′ of the domain category 2 to object A of the codomain category S, then F maps the identity 0 < 0 to the identity A -> A.

And (this is the last one) let’s say F maps

< 1

to

f: A -> B

and identity

< 1

to the identity

B -> B

then we find that the composite

<< 1 = 0 < 1

is mapped by F to the composite

A -> B -> B = A -> B

Confession time: I’m in too much of a festive mood to functor any further ;-)

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From → Note to self

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