[Haskell-cafe] Category Theory woes

Hans Aberg haberg at math.su.se
Thu Feb 18 15:54:24 EST 2010

On 18 Feb 2010, at 19:19, Nick Rudnick wrote:

> 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?

Yes, it is efficient conceptually. The idea of closed sets let to  
topology, and in combination with abstractions of differential  
geometry led to cohomology theory which needed category theory solving  
problems in number theory, used in a computer language called Haskell  
using a feature called Currying, named after a logician and  
mathematician, though only one person.


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