The Unapologetic Mathematician

Mathematics for the interested outsider

2-functors

Of course along with 2-categories, we must have 2-functors to map from one to another.

So, what’s a 2-functor? Since we defined a 2-category as a category enriched over \mathbf{Cat}, a 2-functor should be a functor enriched over \mathbf{Cat}. That is, it consists of a function on objects and a functor for each hom-category, each of which consists of a function on 1-morphisms (the objects of the hom-category) and a function for each set of 2-morphisms. Then there are a bunch of relations.

Let’s expand this a bit. A 2-category \mathcal{C} has a collection \mathrm{Ob}(\mathcal{C}) of objects, a collection \mathrm{Ob}(\hom_\mathcal{C}(A,B)) of 1-morphisms for each pair (A,B) of objects, and a collection \hom_{\hom_\mathcal{C}(A,B)}(f,g) of 2-morphisms for each pair (f,g) of 1-morphisms between the same pair of objects. And all the same remarks go for another 2-category \mathcal{D}.

So a 2-functor F has

  • a function F:\mathrm{Ob}(\mathcal{C})\rightarrow\mathrm{Ob}(\mathcal{D})
  • for each pair (A,B) of objects of \mathcal{C}, a functor F:\hom_\mathcal{C}(A,B)\rightarrow\hom_\mathcal{D}(F(A),F(B)), each consisting of
    • a function F:\mathrm{Ob}(\hom_\mathcal{C}(A,B))\rightarrow\mathrm{Ob}(\hom_\mathcal{D}(F(A),F(B)))
    • for each pair (f,g) of 1-morphisms from A to B a function F:\hom_{\hom_\mathcal{C}(A,B)}(f,g)\rightarrow\hom_{\hom_\mathcal{D}(F(A),F(B))}(F(f),F(g))

Now the composition functors \circ give us functions for composing 1-morphisms and “horizontally” composing 2-morphisms, and the hom-categories give us functions \cdot for “vertically” composing 2-morphisms. For each object we have an identity 1-morphism, and for each 1-morphism we have an identity 2-morphism. A 2-functor will preserve all these structures. First of all, since there are functors between the hom-categories the vertical composition is preserved, along with the identity 2-morphisms. The diagrams for enriched functors say that the identity 1-morphisms, the composition of 1-morphisms, and the horizontal composition of 2-morphisms are all preserved.

August 18, 2007 Posted by | Category theory | 1 Comment

Carnival?

There’s been considerable discussion, particularly in this thread on Michi’s blog about the Carnival of Mathematics.

If you’ve been here from the beginning, you know that I was a contributor to the CoM since its beginning. It’s a great idea, but the execution… well, as time went by it just had more and more to do with brainteasers and education and less and less to do with the meat of the mathematical matters.

It might have had something to do with handing it to a sequence of weblogs that are only tangentially mathematical in their mission, and particularly a streak of explicitly math-ed weblogs. It might just be that the vast majority of people reading and writing weblogs who think of themselves as knowing some math are really computer programmers, physicists, and engineers who use mathematics as a tool and only ever really see pure, unadulterated mathematics in the form of puzzles or tricks; or pre-college mathematics teachers who, by and large, do not spend any time thinking about mathematics that will not help their students learn the material rather than for its own sake.

And so I eventually stopped when for three postings in a row I stuck out like a sore thumb as the only contribution above the level of a sudoku.

Don’t get me wrong. All these lower-level non-technical posts are good, but I started to feel like the 50-year-old guy at a rave. By that point, nobody was coming to the Carnival to read about categorification. And this host couldn’t even spell the g—–n word despite my using it in my submission email, over and over in the linked post, and in the freaking title of the post. It was clear that I was the odd man out here, and that my submissions were only begrudgingly accepted with little care from the hosts.

I think that was the beginning of the end for me. The next fortnight I was in Faro, which gave me a good out-of-line post on Khovanov Homology, but since then I haven’t felt at all interested in writing anything outside my main expository line for Carnival submission. That next Friday came and went and I saw no difference in my hits. Just as I’d thought, nobody was coming from Carnival who wouldn’t have come anyway.

So here’s how I see it. The Carnival of Mathematics has become a de facto carnival of lower-level mathematics, brainteasers, and mathematics education. And I’m fine with that. I’m leaning towards letting it be and just starting a new carnival for actual mathematics. There are certainly many more mathematics weblogs than there were when CoM began, and they could support at least a monthly carnival on their own now. Or maybe this more academic community is inclined to disdain the carnival approach entirely.

Other people have suggested that there’s something to be gained by mixing the levels, and while I agree that something could be gained, I don’t think anything is being gained. People coming from the lower-level and dilettantish weblogs are not reading the higher-level material. And higher-level people can still read the Carnival posts and find what’s new in sudoku-land if they want, whether high-level blatherers submit to CoM or not.

But let’s be sort of scientific about this. A show of hands: who found The UM through a carnival post linked from a lower-level sometimes-mathematical weblog? Who found it through a comment I’d made on another weblog, or through a direct reference on another weblog? Who still finds upper-level weblogs through the Carnival? And what, specifically, do you think will be lost if weblogs like The UM, God Plays Dice, and the Seminars recognize the CoM‘s current state for what it is rather than what it could have been and move on with our lives and weblogs?

I’d like it if you leave a visible comment here, but if you’d prefer to email your correspondence to me privately you know I’m teaching at Tulane now…

August 18, 2007 Posted by | rants | 29 Comments

   

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