The n-Category Café

I like the terminology of a “straight arrow” because if you ask it a question (apply it to an element) you get a single answer.

But it doesn’t play so well with the string diagrams, because you then have the law “you can bend straight arrows”. This seems like a contradiction. Also, if you do call the proarrows “wiggly arrows”, then you “can’t bend wiggly arrows”!

I like the name “loose arrow” for a proarrow, since we can often think of them as smeared out functions of a sort. But to me, “tight arrow” sounds like a loose arrow with a left adjoint – usually called a “map” in bicategory theory. This is because a “tight arrow” is not quite an “arrow”, but suggests (to me) a loose arrow that happens to be tight.

So, if we called maps between metric spaces “arrows”, then a “loose arrow” would be a map that associates to each point a “distance function” to its “image” (an arrow loosened by distance), and then a “tight arrow” would be a loose arrow where all the “images” are limits of Cauchy sequences in the codomain.

I think I may have missed the point here. Are they called “tight” and “loose” because one forms a category and the other a bicategory?

“Straight arrow” and “wiggly arrow” is still naming things after their notation, which I don’t think is a good idea in general. In particular, it’s very far from standard to draw the proarrows as wiggly; in fact I can’t think offhand of anyone who does it that way. Most people write them with a bar or a slash or a dot in the middle.

But to me, “tight arrow” sounds like a loose arrow with a left adjoint – usually called a “map” in bicategory theory. This is because a “tight arrow” is not quite an “arrow”, but suggests (to me) a loose arrow that happens to be tight.

I don’t quite get your point here. For F-categories, “loose” was chosen as a word that might encompass both “lax”, “colax”, and “pseudo” but be free of any existing meaning, and then “tight” was the obvious counterpart. In an F-category, as in an equipment, a tight arrow is indeed a special case of a loose one. But that doesn’t imply the tight arrows have to be the maps; a representable profunctor is also a special case of a profunctor. And more generally I don’t think the terminology implies to me that every tight arrow should be loose; in ordinary language something that’s tight is not usually also loose.

Thanks for this!

I noticed that Harris and Eisenbud’s book uses a different (but equivalent) definition of rational equivalence than Fulton:

Definition. Let $Rat(X) \hookrightarrow Z(X)$ be the subgroup generated by differences of the form $$\langle \Phi \cap (\{t_0\} \times X) \rangle - \langle \Phi \cap (\{t_1\} \times X)\rangle$$ where $t_0,t_1 \in \mathbb{P}^1$ and $\Phi$ is a subvariety of $\mathbb{P}^1 \times X$ not contained in any fiber $\{t\} \times X$ We say that two cycles are rationally equivalent if their difference is in Rat$(X)$.

e.g. there is some family of subvarieties interpolating between them. Maybe we could take the “Chow groupoid” whose objects are algebraic cycles and whose morphisms are choices of interpolations between them?

I don’t know much about these objects, so this is a total guess.

I’d think we’d want our arrows to be regular maps of varieties, and our proarrows to be correspondences. Then a 2-cell could be a rational map between correspondences making the diagram you’d write down commute.

Then the Chow ring would be recovered I think by looking at subterminal proarrows up to isomorphism; in other words, we take subvarieties of $X \times Y$ and quotient by rational equivalence. These aren’t quite algebraic cycles though, because you can’t take formal linear combinations of them.

I think that you can get the intersection product from composition of proarrows but I’m really not sure.

I think that’s a great idea.

I like this definition of rational equivalence better than what I see on Wikipedia. It seems to lend itself toward creating a Chow groupoid where morphisms are things like these ‘interpolations’. One problem is that I don’t see how, in general, to compose two interpolations and get another interpolation. (Sometimes you can, but it seems you have to be lucky.) So, one may need to create a category where the morphisms are formal composites of interpolations, and then make that into a groupoid (if one wants) by formally decreeing that ‘running an interpolation backward in time’ (from $t_1$ back to $t_0$) gives the inverse of an interpolation.

(When one is forced to do a lot of ‘formal’ stuff like this to get a groupoid, it’s often because one is secretly dealing not with a mere groupoid but a higher structure: I can imagine using $\mathbb{P}^2$ to define ‘interpolations between interpolations’, or 2-simplexes in a simplicially enriched category where the 1-simplexes are interpolations. Has anyone done this?)

Anyway, once one has sorted this out, the question is whether one can compose an object of the Chow groupoid of $X \times Y$ (i.e. an algebraic cycle in $X \times Y$) and a correspondence from $Y$ to $Z$ (i.e. some suitably nice span of varieties $Y \leftarrow \Sigma \rightarrow Z$ to get a correspondence from $X$ to $Z$. And it seems the answer is yes, since an algebraic cycle in $X \times Y$ is a special sort of correspondence from $X$ to $Y$.

If all this works well, I believe one is on track for defining 2-cells of the sort you originally asked about.

John wrote:

I can imagine using $\mathbb{P}^2$ to define ‘interpolations between interpolations’, or 2-simplexes in a simplicially enriched category where the 1-simplexes are interpolations. Has anyone done this?

Sort of. $\mathbb{A}^1$ homotopy theory works a lot like this, except it uses the affine line $\mathbb{A}^1$ to parametrize homotopies instead of the projective line $\mathbb{P}^1$, and so on for these higher homotopies.

I’m not sure how Harris and Eisenbud’s definition of rational equivalence would change if we used $\mathbb{A}^1$ instead of $\mathbb{P}^1$.