[Haskell-cafe] Category Theory woes

Nick Rudnick 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 
open), or
* 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: 
> http://www.citeulike.org/user/spl/tag/category-theory
> Sean
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