[Haskell-cafe] Category Theory woes
Nick Rudnick
joerg.rudnick at t-online.de
Thu Feb 18 13:19:36 EST 2010
Hi Hans,
agreed, but, in my eyes, you directly point to the problem:
* doesn't this just delegate the problem to the topic of limit
operations, i.e., in how far is the term «closed» here more perspicuous?
* that's (for a very simple concept) the way that maths prescribes:
+ historical background: «I take "closed" as coming from being closed
under limit operations - the origin from analysis.»
+ definition backtracking: «A closure operation c is defined by the
property c(c(x)) = c(x). If one takes c(X) = the set of limit points of
X, then it is the smallest closed set under this operation. The closed
sets X are those that satisfy c(X) = X. Naming the complements of the
closed sets open might have been introduced as an opposite of closed.»
418 bytes in my file system... how many in my brain...? Is it efficient,
inevitable? The most fundamentalist justification I heard in this regard
is: «It keeps people off from thinking the could go without the
definition...» Meanwhile, we backtrack definition trees filling books,
no, even more... In my eyes, this comes equal to claiming: «You have
nothing to understand this beyond the provided authoritative definitions
-- your understanding is done by strictly following these.»
Back to the case of open/closed, given we have an idea about sets -- we
in most cases are able to derive the concept of two disjunct sets facing
each other ourselves, don't we? The only lore missing is just a Bool:
Which term fits which idea? With a reliable terminology using
«bordered/unbordered», there is no ambiguity, and we can pass on
reading, without any additional effort.
Picking such an opportunity thus may save a lot of time and even error
-- allowing you to utilize your individual knowledge and experience. I
have hope that this approach would be of great help in learning category
theory.
All the best,
Nick
Hans Aberg wrote:
> On 18 Feb 2010, at 14:48, Nick Rudnick wrote:
>
>> * 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),
>
> I take "closed" as coming from being closed under limit operations -
> the origin from analysis. A closure operation c is defined by the
> property c(c(x)) = c(x). If one takes c(X) = the set of limit points
> of X, then it is the smallest closed set under this operation. The
> closed sets X are those that satisfy c(X) = X. Naming the complements
> of the closed sets open might have been introduced as an opposite of
> closed.
>
> Hans
>
>
>
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