John,
I love such clarity - as expressed in your explanation of category
theory. My reaction is "Oh, so THAT's what category theory is!" Thanks
for taking the time to explain.
Grant
John Kennison wrote:
Owen
Thanks for asking the question. In my answer, below, I describe the technical terminology impressionistically. If you want more precision, the Wikipedia articles are usually pretty good at giving precise definitions, along with some sense of the underlying ideas.
Category theory claims to be a formalization of how mathematics actually works. For example, consider the following mathematical structures, which have been defined in the 19th and 20th centuries:
Groups = “sets with a notion of multiplication”
Rings = “sets with notions of both multiplication and addition”
Linear Spaces = “sets in which vector operations can be defined”
Topological Spaces = “sets with a notion of limit”
Each structure has a corresponding notion of a structure-preserving function:
Group homomorphism = “function f for which f(xy) = f(x)f(y)”
Ring homomorphism = “function f for which f(xy)=f(x)f(y) and
f(x+y)=f(x)+f(y)”
Linear map = “function preserving operations such as scalar mult:
f(kv)=kf(v)”
Continuous function = “function f for which f(Lim x_n) = Lim(f(x_n)”
A category consists of a class of objects, together with a notion of “homomorphism” or “map” or “morphism” between these objects. The main operation in a category is that morphisms compose (given a morphism from X to Y and another from Y to Z, there is then a composite morphism from X to Z).
Examples of catgeories:
Objects = Groups; Morphisms = Group
Homomorphisms
Objects = Rings; Morphisms = Ring Homomorphisms
Objects = Linear spaces; Morphisms = Linear maps
Objects = Top’l spaces; Morphisms = Cont.
functions
Objects = Sets; Morphisms =
Functions
(The above examples are respectively called the categories of groups, of rings,
of linear spaces, of top’l spaces, and of sets.)
The claimed advantages of using categories are:
(1) The important and natural questions that mathematicians ask are
categorical in nature –that is they depend not on operations such as group
multiplication, but strictly on how the morphisms compose. (that is, the
objects are like black boxes, we don't see the limits or multiplication inside
the box, we only see arrows, representing morphisms going from one box to
another.)
(2) Looking at a subject from a category-theoretic point of view sheds
light on what is really happening and suggests new research areas.
(3) Proving a theorem about an arbitrary category can have applications to all of the traditional categories mentioned above.
(4) As would be expected, there are suitable mappings between categories, called functors, which enable us to compare and relate different parts of mathematics.
I work in topos theory which ambitiously proposes to study where logic comes from. We start by noting that many ideas in logic are closely tied to the category of sets.
For example the sentence “x > 3” is true for some values of x and not for others (if we assume, for example, that x is a real number) The compound sentence “x > 3 and 3x = 12” is true on the intersection of the set where the x > 3 with the set where 3x = 12.
On the other hand, “x >7 or x < 1” in true on a union. Of course “x not equal to 3” is true on the complement of where “x = 3”.
Much of our assumptions about how the logical connectives “and”, “or”, “not” are closely connected to how intersections, unions and complements work in sets. But intersections, unions and (weak) complements can be defined in categorical terms and then they may behave differently (for example, categories need not obey the “law” of the excluded middle). A topos is a category that resembles the category of Sets in some formal ways, but which may lead to non-standard logics. One example of a topos can be thought of as a category of sets in which the elements can change over time, such as the set of all states in the US. Note that the element called Virginia splits into 2 elements, West Virginia and Virginia, and, according to some views, elements like Georgia were not in the set of US states during the Civil War. The set of US states also has structure, such as the boundaries of the states, which can change over time.
The advantage of uses toposes is that a traditional mathematical object can be mapped, using a suitable functor, to a non-standard world (i.e. to a related object in a topos) and this can reveal some of the inner structure of the object. For example, an evolving system might be best viewed in a world where elements can change over time.
________________________________________
From: [email protected] [[email protected]] On Behalf Of Owen
Densmore [[email protected]]
Sent: Friday, April 09, 2010 12:50 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] invitation + introduction
On Apr 7, 2010, at 12:10 PM, John Kennison <[email protected]> wrote:
Hi Leigh,
<snip>
Nick introduced me to Rosen’s “Life Itself” and I have skimmed some articles by
Rosen. I am both fascinated and disappointed by Rosen’s work. Fascinated by
what Rosen says about the need to develop radically different kinds of models
to deal with biological phenomena and disappointed by Rosen’s heavy-handed
stabs at developing such models. And yet still stimulated because I have enough
ego to believe that with my mathematical and category-theoretic background, I
might succeed where Rosen failed.
Category theory has been mentioned several times, especially in the early days
of friam. Could you help us out and discuss how it could be applied here? CT
certainly looks fascinating but thus far I've failed to grasp it. I'd love a
concrete example (like how to address Rosen's world) of it's use, and possibly
a good introduction (book, article).
---- Owen
I am an iPad, resistance is futile!
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
