[Haskell-cafe] Category Theory woes
joerg.rudnick at t-online.de
Thu Feb 18 08:48:08 EST 2010
IM(H??)O, a really introductive book on category theory still is to be
written -- if category theory is really that fundamental (what I
believe, due to its lifting of restrictions usually implicit at
'orthodox maths'), than it should find a reflection in our every day's
common sense, shouldn't it?
In this case, I would regard it as desirable to -- in best refactoring
manner -- to identify a wording in this language instead of the abuse of
terminology quite common in maths, e.g.
* the definition of open/closed sets in topology with the boundary
elements of a closed set to considerable extent regardable as facing to
an «outside» (so that reversing these terms could even appear more
intuitive, or «bordered» instead of closed and «unbordered» instead of
* the abuse of abandoning imaginary notions in favour person's last
names in tribute to successful mathematicians... Actually, that pupils
get to know a certain lemma as «Zorn's lemma» does not raise public
conciousness of Mr. Zorn (even among mathematicians, I am afraid) very
much, does it?
* 'folkloristic' dropping of terminology -- even in Germany, where the
term «ring» seems to originate from, since at least a century nowbody
has the least idea it once had an alternative meaning «gang,band,group»,
which still seems unsatisfactory...
Here computing science has explored ways to do much better than this,
and it might be time category theory is claimed by computer scientists
in this regard. Once such a project has succeeded, I bet, mathematicians
will pick up themselves these work to get into category theory... ;-)
As an example, let's play a little:
Arrows: Arrows are more fundamental than objects, in fact, categories
may be defined with arrows only. Although I like the term arrow (more
than 'morphism'), I intuitively would find the term «reference» less
contradictive with the actual intention, as this term
* is very general,
* reflects well dual asymmetry,
* does harmoniously transcend the atomary/structured object perspective
-- a an object may be in reference to another *by* substructure (in
the beginning, I was quite confused lack of explicit explicatation in
this regard, as «arrow/morphism» at least to me impled objekt mapping
XOR collection mapping).
Categories: In every day's language, a category is a completely
different thing, without the least association with a reference system
that has a composition which is reflective and associative. To identify
a more intuitive term, we can ponder its properties,
* reflexivity: This I would interpret as «the references of a category
may be regarded as a certain generalization of id», saying that
references inside a category represent some kind of similarity (which in
the most restrictive cases is equality).
* associativity: This I would interpret as «you can *fold* it», i.e. the
behaviour is invariant to the order of composing references to composite
references -- leading to «the behaviour is completely determined by the
lower level reference structure» and therefore «derivations from lower
level are possible»
Here, finding an appropriate term seems more delicate; maybe a neologism
would do good work. Here one proposal:
* consequence/?consequentiality? : Pro: Reflects well reflexivity,
associativity and duality; describing categories as «structures of
(inner) consequence» seems to fit exceptionally well. The pictorial
meaning of a «con-sequence» may well reflect the graphical structure.
Gives a fine picture of the «intermediating forces» in observation and
the «psychologism» becoming possible (-> cf. CCCs, Toposes). Con:
Personalized meaning has an association with somewhat unfriendly behaviour.
Anybody to drop a comment on this?
Sean Leather wrote:
> On Thu, Feb 18, 2010 at 04:27, Nick Rudnick wrote:
> I haven't seen anybody mentioning «Joy of Cats» by Adámek,
> Herrlich & Strecker:
> It is available online, and is very well-equipped with thorough
> explanations, examples, exercises & funny illustrations, I would
> say best of university lecture style:
> http://katmat.math.uni-bremen.de/acc/. (Actually, the name of the
> book is a joke on the set theorists' book «Joy of Set», which
> again is a joke on «Joy of Sex», which I once found in my parents'
> bookshelf... ;-))
> This book reads quite nicely! I love the illustrations that pervade
> the technical description, providing comedic relief. I might have to
> go back a re-learn CT... again. Excellent recommendation!
> For those looking for resources on category theory, here are my
> collected references:
> Haskell-Cafe mailing list
> Haskell-Cafe at haskell.org
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