[Haskell-cafe] Category theory as a design tool

[Gregg Reynolds]

On Wed, Jun 22, 2011 at 2:59 PM, Arnaud Bailly <arnaud.oqube at gmail.com>wrote:

> Hello Greg and Alexander,
> Thanks for your replies. Funnily, I happen to own the 3 books you
> mentionned :-) My interest in category theory is a long standing affair...
>
> Note that owning a book, having read (most of) it and knowing a theory (or
> at least its principles and main concepts) is really quite a different thing
> from being able to apply it. Although I am certainly able to understand the
> Yoneda lemma, I am certainly unable to


Well, you're way ahead of me.  I don't even "get" adjunctions, to tell you
the truth.  By which I mean that I have no intuition about them; it's not so
hard to understand the formal definition, but it's another thing altogether
to grasp the deep significance.


> recognize the opportunity of using that knowledge to derive some new
> knowledge about something else. And being able to define a topoi is very
> different from being able to construct one or infer one out of a given
> category. This is an actual limitation of myself of course.
>

Indeed.  But I'll bet your next paycheck that eventually CT will replace set
theory as the basic mode of thinking about math, even at elementary levels.
 It just awaits that brilliant writer who can connect the dots
appropriately.


>
> Alexander's advice about using diagrams and simple notations is something I
> often try to do and something which most of the times end in writing a bunch
> of functions and data types in Haskell... What I am really looking for is a
> more systematic way of thinking about systems (or system fragments, or
> components) in a categorical way, perhaps starting with a bunch of arrows in
> some abstract category with objects as components and trying to infer new
> objects out of categorical consturctions which are made possible by the
> current diagrams (eg. what would be a pullback in such a category ? What
> would be its meaning ? Does the question itself have a meaning ?).
>
> Maybe this is really foggy and on the verge to fall into "abstract
> nonsense"...
>
> Long live abstract nonsense!

Completely off topic:  a few months ago I had an idea about using category
theory to provide rigorous semantics for the web (esp. rdf stuff etc.)  I'll
probably never find time to work out the details, but it's a fun exercise in
any case; if you want to mess around with applying CT to the real world
maybe you can coem up with improvements.  See
http://blog.mobileink.com/2011/03/resource-token-exchange.html.  It's a bit
of a mess, and some of it I would radically revise, but it might give you
some ideas, if you're interested in the semantic web thingee.



> Best regards,
> arnaud
>
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