In an attempt to clarify things, here are the collected axioms for a
category:
A category is a collection of objects and a collection of arrows.
Each arrow has a source object and a destination object (which may be the
same object).
Each object has an identity arrow from itself to itself, which is unique in
the sense that it is the only identity arrow. Other arrows may go from that
object to itself, but are not the identity.
For any two arrows A->B and B->C (i.e. the destination of one is the source
of the other), there is a unique arrow which is the composition of these
arrows.
The operation of composition on arrows is associative.
These axioms completely "define" objects and arrows. The category of sets
is a representation of sets and functions in these terms; we can only know
category-theoretical things like which arrows lead where in this category.
We cannot ask what set a given object represents within the confines of
category theory. Nonetheless, we can find familiar landmarks. The empty set
is distinguished by having a unique arrow to every other object (the one
corresponding to the trivial function). Two sets A and B are isomorphic if
there are arrows f:A->B and g:B->A such that f o g is the identity on B and
g o f is the identity on A. Using this, we can find the one-element set up
to isomorphism. Any set which has a unique arrow from any set to it has
only one element, and all of these sets are isomorphic. One way to find how
many elements are in a set is to check how many arrows there are from a
one-element set to it, since each identifies a single element in the set.
However, the strength of category theory is not that it allows us to
rebuild the sets in this way, but that it allows us to see exactly how much
precision functions give us in discussing sets. Specifically, there is no
way to determine the elements of sets--sets are distinguished only up to
isomorphism.
If you want to learn the formal workings of category theory, the J forums
aren't the best place to do it. I would recommend getting a book on the
subject; unfortunately I've picked up category theory from ad-hoc work in
other disciplines and don't have a good reference. It shouldn't be too hard
to find a source online, though.
Marshall
On Wed, Apr 11, 2012 at 6:57 PM, Raul Miller <rauldmil...@gmail.com> wrote:
> On Wed, Apr 11, 2012 at 5:12 PM, Jordan Tirrell <jordantirr...@gmail.com>
> 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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