[Haskell-cafe] Re: Re: Re: Full strict functor by abusing Haskell exceptions

[Dan Doel]

On Saturday 18 September 2010 6:03:39 pm wren ng thornton wrote:

> pointed objects, pointed sets/groups/topospaces, pointed categories,
> pointed functors, etc aren't all the same though.

The definition of pointed objects could be massaged to yield pointed functors, 
though.

Instead of a category with a terminal object, we could relax the definition to 
include categories with a distinguished object I. Then, a pointed object in 
such a category is an object X together with x : I -> X.

This isn't always very interesting. For any category with an initial object 0, 
the category of objects pointed over 0 is isomorphic to the original category, 
for instance.

However, categories with arbitrary distinguished objects could be viewed as a 
precursor to monoidal categories, and pointed objects are then precursors to 
monoid objects.

And then when you instantiate all that to the category of endofunctors over 
some category, you get pointed functors being precursors to monads.

-- Dan