Comments on A Neighborhood of Infinity: You Could Have Defined Natural Transformations

This is a really great blog btw :-) And that'...

2008-10-08T01:56:00.000-07:00

This is a really great blog btw :-) And that's coming on the heels of many great websites I've stumbled upon these past 2 weeks after discovering Haskell.

This post really hit me right where I was coming to terms with some thoughts on functors. My brain always tries to map to the functor concept anytime I have a "function" defined on proper classes (groups to groups, say) and/or anytime I have something that seems to be defined naturally (like defining a group using only general group operations).

But I couldn't always map this to a functor! This was very discouraging because those two things I actually care about, whereas the functor definition is just abstract stuff.

A common problem I had for example is something like the center of a group. It's a map of type group -> group sending a group to it's center. The set builder definition is as "purely" or "naturally" defined as you could hope, but it is *not* a functor: given a morphism of groups there is really only 1 candidate for the induced morphism of centers, but it isn't a map from center to center (it fails to always lie inside the range).

The trick I think is that the center is a functor if you only consider surjections to be morphisms. You can see that this is exactly what you need to deal with the "for all" quantifier in the definition of the center. Likewise, I imagine a quotient group construction instead of a subgroup construction would necessitate throwing away the surjections and keeping the inclusions. I still haven't completely wrapped my head around it, but I'm closer.

As per the last comment I'm not sure if the notion of functor always maps nicely to "constructive operations" which feels like a stickier notion involving logic and set theory.

So, what is the definition of a natural transforma...

2008-08-21T14:42:00.000-07:00

So, what is the definition of a natural transformation unfolded in the category Vect of vectorspaces over a fixed field k?

I suspect for A to be a natural transformation in Vect it has to be defined in a polymorphic way only using scalar multiplication and vector addition, i.e. everything of the Haskell type (Vec a) => a -> a, with Vec being a suitable typeclass containing zero, add and mult.

Am I correct?

Thanks for the help in my struggles with category ...

2008-07-15T07:42:00.000-07:00

Thanks for the help in my struggles with category theory . :P .Much appreciated .

Sean,

Yes, well spotted.

2008-06-14T07:03:00.000-07:00

Sean,

Yes, well spotted.

In that first set of functions, is that supposed t...

2008-06-14T04:03:00.000-07:00

In that first set of functions, is that supposed to be g(x,y) = x?

"But how can we say something like this in the lan...

2008-05-06T18:11:00.000-07:00

"But how can we say something like this in the language of set theory, say? Set theory doesn't come with a mechanism for making such promises."

You can define such "functions" in ZF, although they are not called functions. The powerset (set of subsets) or an ordered pair are basic examples. In Mizar they are called "functors".

Indeed, historically natural transformations were ...

2008-05-04T15:32:00.000-07:00

Indeed, historically natural transformations were defined before functors or categories - 1942 versus 1945. Eilenberg and Mac Lane started with natural transformations in the category of groups, and then took three years to find the appropriate language to describe them elsewhere.

Comments on A Neighborhood of Infinity: You Could Have Defined Natural Transformations

This is a really great blog btw :-) And that'...

So, what is the definition of a natural transforma...

Thanks for the help in my struggles with category ...

Sean,Yes, well spotted.

In that first set of functions, is that supposed t...

"But how can we say something like this in the lan...

Indeed, historically natural transformations were ...

Sean,

Yes, well spotted.