WTF is a "blog"?
Posted 05-05-2009 at 12:20 PM by Ben
Yeah, I do know, of course I do - we are in 21C, right?
I take it I may assume that anyone reading this my blog is slightly interested in what I have to say?
So. I have been learning category theory over the last few months. I am here to tell you it is the coolest subject EVER.
It is highly abstract, of course, that is why it is affectionately known as "abstract nonsense", but not that hard, I think.
Well it is hard, because it requires us to make an abstract connection between different "classes" of mathematical object; that is, there is a sense in which sets, rings, fields, orders, monoids, groups, vector spaces. topological spaces.... can all be thought of as being "the same sort of thing" in their relations among and between themselves.
Thing is, it doesn't really feel like math in the way that, say, calculus does. The "equations" are mostly diagrams, though these are beguilingly misleading at times.
Otherwise I am well, thank you for asking
PS So is Mrs Ben, should you wonder
I take it I may assume that anyone reading this my blog is slightly interested in what I have to say?
So. I have been learning category theory over the last few months. I am here to tell you it is the coolest subject EVER.
It is highly abstract, of course, that is why it is affectionately known as "abstract nonsense", but not that hard, I think.
Well it is hard, because it requires us to make an abstract connection between different "classes" of mathematical object; that is, there is a sense in which sets, rings, fields, orders, monoids, groups, vector spaces. topological spaces.... can all be thought of as being "the same sort of thing" in their relations among and between themselves.
Thing is, it doesn't really feel like math in the way that, say, calculus does. The "equations" are mostly diagrams, though these are beguilingly misleading at times.
Otherwise I am well, thank you for asking
PS So is Mrs Ben, should you wonder
Total Comments 4
Comments
-
categoricalizations
Good even Mr. & Mrs. Ben and I trust I find you well.
Your intriguing well-writ blog Mr. Ben brightly brought immediately to my mind my Katabataks. If you happen by that cave, I have an interest in what category you make it. May serendipity drop by your place after she leaves here. Live long & prosper.
Posted 05-05-2009 at 07:21 PM by Turtle
-
I am slightly interested in what you have to say.
Cool blog post.Posted 05-06-2009 at 12:37 AM by Tormod
-
In my wildest dreams, I would never have thought I would do this on a blog.
Oh, by-the-by. Mrs. B thanks you all for your interest in her well-being, and expresses the wish that you guys will keep me quiet for a while.
Someone somewhere said they thought this might be an interesting subject to pursue, so, ready or not, here I come.....
A category
consists of a collection of mathematical objects
for which the following is true;
for any pair of objects
, there is at least one "mapping" 
called a morphism, or, by me, an arrow;
for each object
there is a privileged arrow called the identity, 
;
if
, and 
, then 
; this composition is associative;
and 
.
That's it!
Oh - notice that I compose arrows right-to-left, this is standard.
So, I will give a couple of examples, but take careful note; I am not offering any argument for my arrows, my morphisms. Why? Look below!
In the category Set, the objects are sets, the arrows are set functions. In the category Grp the objects are groups, the arrows are group homomorphisms. In the category K-Vec the objects are vector spaces over the field K, the arrows are linear transformations. Simple really.
Now note this well; category theory is not really interested in the objects themselves, rather the arrows are the main interest - I will show you why later, with an example.
If objects take back seat, the elements in the objects - set elements, group elements, vectors, in our examples above - don't even make it through the door!Posted 05-06-2009 at 11:01 AM by Ben
-
mmmm...very interesting still, with a touch of mystery, or the smell of carrot perhaps.
though my precious objects are a little cheesed at the summary exclusion. 
no worries as it looks like maybe i can poke them into submission with your arrows. 
on that point, you seem to have mixed your metaphors going from take-a-back-seat to don't-even-make-it-through-the-door, and I wish to appeal for put-in-the-trunk/boot as the next group after take-a-back-seat. (obviously i want to keep my precious objects in that trunk in case i want to haul them out later during the trip. 
)
good stuff Maynard! Posted 05-06-2009 at 02:38 PM by Turtle
