On Wed, Apr 11, 2012 at 5:12 PM, Jordan Tirrell <[email protected]> wrote:
>> First, any exposition which involves infinity is ambiguous.
>> But we can resolve this difficulty by considering a non-infinite
>> set of natural numbers. I shall arbitrarily pick: natural
>> numbers modulo 5. N= {0,1,2,3,4}
>>
>> But if we consider our function to be a transformation on N,
>> there is no distinction between f(x)= x+1 and g(x)= x-1. In
>> both cases the transformation on N yields N.
>>
>> I have an additional difficulty, here, since I do not see
>> how an arrow can represent an arbitrary relation.
>>
>> Finally, technically speaking, if we go back to dealing with
>> the infinite set of natural numbers, the codomain of f is not
>> the set of natural numbers. It's a different set. It's the
>> set of integers which are greater than 0.
>
>
> First of all, the *image* of a function is the set of possible outputs when
> you apply the function to its domain. The *codomain* is part of the
> definition of a function, it must include the image but must not always be
> equal to it. When we talk about a function like f(x)=x+1 in most contexts
> we generally aren't so formal that we must define the domain and codomain,
> but formally they are required and this is crucial for the language of
> category theory.
Ok, let us say that the image of a function are the result which
correspond to values from the function's domain and that the codomain
is some set (or maybe some collection which is not a set? that's not
clear yet) which contains the image.
> Consider a category with two objects, {0} and {0,1} and the constant zero
> function f(x)=0. I must tell you what the domain and codomain are, since
> this could refer to f: {0,1} -> {0}, f: {0} -> {0}, or either of the other
> two possibilities (in which the codomain is not equal to the image).
> Formally, these are all different functions and they would be all different
> arrows in our category.
Yes, a function from {0,1} to {0} is a different function than a
function from {0} to {0} but a function from {0,1} to {0,1} which
never can produce the value 1 is not a different function from a
function from {0,1} to {0}.
> Remember that in category theory our arrows aren't functions, they are
> formal abstract things which have each have a domain and codomain and
> interact with a composition operation.
That would be fine, but I need to know the definitions of these things.
> There is a quote from someone that goes something like "category theory is
> like a language where verbs [arrows] are first class citizens". In fact,
> the "objects" we talk about in category theory are really for convenience,
> they are unnecessary since each corresponds exactly with one identity
> arrow, we could talk about category theory without objects, just using
> identity arrows instead.
I think I understand what you are saying here, but I do not yet agree
that they are unnecessary even if they can be represented by certain
arrows. They might be unnecessary in some axiom system, but I have
not been presented with that axiom system.
> Category theory is entirely about the structure of the arrows. To say
> f(x)=x+1 and g(x)=x-1 have no distinction because both yield N does not
> quite make sense in this context because what they yield is just their
> codomain, which we defined, and we care about their compositional
> structure. Both have domain and codomain N. So does the identity i on N. We
> also have f^5=g^5=fg=gf=i (we would say f and g are inverses and have order
> 5). Category theory captures this structure. Remember you have a 5-element
> set and you are considering functions from it to itself. The functions f
> and g are easily understood because I know what you mean by addition and
> subtraction on the 5 symbols you used. Its true that f and g are symmetric
> to each other in our category, because they are both cyclic rotations of
> five objects. If you want to distinguish them with an idea like f is a
> "forward" rotation, you would need a category that could formally represent
> order and direction. What we have is only a set.
And I disagree here. We had said that the object in this category was
a set. And that the arrow connected the set to itself. We did not
have members of the set in the category. If we were speaking of an
arrow which connected the members of the set to
If you mean instead that an object is a collection and that the arrow
connects members of the collection, that would be different, but I
thought you were trying to tell me that that was not valid. If not,
why did you say:
"It seems like the earlier confusion arose from the
desire to have a single arrow in a category like this which represents f(x)
= x + 1."
?
>> Here, my issue is that an arrow cannot represent a
>> function. Only a collection of arrows can represent a
>> function. But that collection cannot be a category.
>> Because if two arrows in a category can be composed the
>> category automatically includes the composition of those
>> arrows.
>>
>> So, here, we cannot, for example, distinguish between f x = x + 1
>> and h x = x + 2
>>
>> > A function, like any mathematical thing, can be represented in terms of
>> > category theory in many different ways and you have to be careful when
>> > doing so.
>>
>> It seems to me that, if your statements here are correct,
>> then category theory cannot fully represent functions nor
>> relations -- instead we have to first choose which aspect
>> of the function (or whatever) we are representing and then
>> build our categories to represent just that aspect (and
>> not the complete definition of the function).
>>
>> Put differently: the presentation here gave two different
>> representations of aspects of a single function, and neither
>> presentation fully represented that function.
>
> Your final conclusion is good I think, neither example I gave fully
> represented everything we know about the function x+1 (the first was x+1 as
> a function on N as a set, the second also captured the order of N). It is
> worth noting that even though I used an arrow to represent x+1 in my first
> example and a functor in the second, a functor is an arrow in a category of
> categories and functors, so really both are arrows.
>
> But this doesn't mean category theory can't do it. With a lot of extra
> effort you could build up a more complete model of numbers in category
> theory and then you'd have a way of talking about the function x+1 in that
> category that models everything you want. Of course formally you'll always
> have to be precise about your domain and codomain, but if you can
> categorically define subsets then you can capture your x+1 arrow and its
> restrictions to smaller domains/codomains.
>
> The way I understand it, category theory has three interpretations:
>
> As a simple strictly algebraic structure that describes "arrows" which obey
> composition-type laws. I think this isn't so hard to understand but most of
> the information I found online jumps right into one of the two below.
>
> As a foundation for mathematics itself. That is, somehow we can use
> category theory and build all of mathematics from it (just like set theory
> or logic). I don't really know anything about this other than people say
> its true and I believe them because they're smarter than me and I'd rather
> not build a complete theoretical framework for modern mathematics from
> scratch.
>
> As a unifying thread in mathematics.
> "You know all that stuff you learned about different branches of
> mathematics? It can all be described in the language of category theory"
> "Cool but its too abstract and I don't like it"
> [In my experience this is how most math students first meet category theory]
> However, one great result of this is that notation becomes much nicer,
> since mathematicians can use categorical notation across all branches of
> math and analogous concepts then use analogous notation.
>
>
> I hope this helps clear things up.
Clear as mud...
Ok, first off, "foundation for mathematics itself" is not very
exciting. We already have that in sets. And since category theory
can represent sets, that's a given.
The issue I am concerned with, right now, is "what is an arrow" and
"how, specifically, can an arrow represent a function. And I want all
elements of a small category completely enumerated, one which
represents a function with a small, finite domain.
--
Raul
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