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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?<br>
<br>
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.<br>
<br>
* 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<br>
* 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?<br>
* '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...<br>
<br>
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... ;-)<br>
<br>
As an example, let's play a little:<br>
<br>
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<br>
* is very general,<br>
* reflects well dual asymmetry,<br>
* 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).<br>
<br>
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,<br>
<br>
* 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).<br>
<br>
* 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»<br>
<br>
Here, finding an appropriate term seems more delicate; maybe a
neologism would do good work. Here one proposal:<br>
<br>
* 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.<br>
<br>
Anybody to drop a comment on this?<br>
<br>
Cheers,<br>
<br>
Nick<br>
<br>
<br>
Sean Leather wrote:
<blockquote
cite="mid:3c6288ab1002180204q74df576fw761f408b1f839c54@mail.gmail.com"
type="cite">On Thu, Feb 18, 2010 at 04:27, Nick Rudnick wrote:<br>
<div class="gmail_quote">
<blockquote class="gmail_quote"
style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">I
haven't seen anybody mentioning «Joy of Cats» by Adámek, Herrlich
& Strecker:<br>
<br>
It is available online, and is very well-equipped with thorough
explanations, examples, exercises & funny illustrations, I would
say best of university lecture style: <a moz-do-not-send="true"
href="http://katmat.math.uni-bremen.de/acc/" target="_blank">http://katmat.math.uni-bremen.de/acc/</a>.
(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... ;-))<br>
</blockquote>
<br>
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!<br>
<br>
For those looking for resources on category theory, here are my
collected references: <a moz-do-not-send="true"
href="http://www.citeulike.org/user/spl/tag/category-theory">http://www.citeulike.org/user/spl/tag/category-theory</a><br>
<br>
Sean<br>
</div>
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