(Approximate notes for a talk I gave this afternoon.)
So according to the title, I should be telling you about , as a functor of manifolds and . That’s perhaps a bit ambitious. I’ll only be thinking about , a disjoint union of closed disks, and I’ll actually fix . And instead of letting range over any manifolds, it’ll just be , ranging over real vector spaces.
By taking derivatives at centers, we obtain a homotopy equivalence from to something I’ll maybe denote . This is componentwise affine (linear followed by translation) maps whose restriction to -balls around the 0s is an embedding. I may use , and write . And I’ll actually send everything to spectra, instead of topological spaces, via .
So really I’ll be talking about
as a functor of . I’ll be lazy and write it , having fixed an and to give .
The first useful case is when (i.e., ). Then embeddings are just configuration spaces, . I’ve talked before about a homotopy limit model in this case, but let me remind you about it.
The category I have in mind is something I’ll denote . Objects will be non-trivial partitions of , and I’ll probably denote them , perhaps writing . Non-trivial means that some equivalence class is larger than a singleton. I’ll write if is finer than , meaning that whenever , then .
The functor I want is something I’ll denote and call “non-locally constant” maps. So is the set (space) of maps such that there is an where . Equivalently, maps which don’t factor through .
Depending on which order you make your poset go, is contravariant, and you can show
The second useful case is when (i.e., ). Then the space of embeddings is homotopy equivalent to the space of injective linear maps. You can obtain a homotopy limit model in this case that looks strikingly similar to the previous case. Namely, you set up a category of “linear” partitions (equivalent to modding out by a linear subspace), and take the holim of the non-locally constant maps functor, as before.
I like to think of both cases as being holims over categories of kernels, and the non-locally constant maps of some kernel are maps that are known to fail to be not-injective for some particular reason. Embeddings fail to be not-injective for every reason.
But there’s another model I want to use in what follows. My category will be denoted , and objects in the category will be vector spaces with non-zero linear maps . Morphisms from to will be surjective linear maps with . You might think of the objects as an abstract partition () with a map to , which then determines a partition of , by taking the image.
The functor out of this category is something I’ll still denote . On an object it gives all non-constant affine maps . Arone has shown
The space of embeddings we are considering splits, as
We know a homotopy limit model of each piece of the splitting, and might hope to combine them into a homotopy limit model for the product. This can, in fact, be done, using the following:
Lemma: If , , then
Here denotes the join. For categories, is , obtained by adding a new initial object to each of and , taking the product, and removing the initial object of the result.
Proof (Sketch): Consider the case . The idea is to line up the squares:
So, anyway, applying this lemma and perhaps cleaning things up with some homotopy equivalences, we obtain an equivalence
Objects in the category consist of a partition along with, for , linear . To tidy up a little bit, I’ll denote this category , for join. The functor takes an object as above and returns the set of componentwise affine maps such that either (a) the map is non-constant on some component, (b) when restricted to the image of , the map is non-locally constant with respect to .
There you have it, a homotopy limit description for the embedding functor.
But not a particularly nice one. If we had an embedding , then we’d have map . It’d be really swell if this map was modelled by a map of the categories we are taking homotopy limits over. But that’s not going to happen. What can go wrong? Non-trivial partitions of , when sensibly composed with the map to , may become trivial, and thus not part of the category. This is, essentially, because several components of might map to a single component of . If has two components, and one, say, where do you send the object consisting of the non-trivial paired with some 0 vector spaces?
A More Natural Model
We sort of need to expand the category we take the homotopy limit over, and make it a more natural construction. We actually have an indication on how to do this from the discussion, above, in the case of linear injective maps from a single component. Perhaps we can find a proper notion of “abstract partition”, pair such a beast with a map to , sensibly define non-locally constant, and get what we want. Let’s see how it goes…
An affine space is, loosely, a vector space that forgot where its 0 was. There is, up to isomorphism, one of any given dimension, just like for vector spaces; I’ll denote the one of dimension by , say. That should be enough of a description for now.
Let me define a Complete Affine Partition (CAP), , to be a partition of a disjoint union of affine spaces, such that equivalence classes contain components. That is, everybody that’s in the same component is in the same equivalence class. Given a , I’ll denote by the underlying component-wise affine space. The data that determines a is: a finite set (the set of components), a partition of , and a dimension function, (non-negative integers). With this information, is .
By a refinement from to , denoted , I will mean an affine map so that the “affine closure” of the partition is coarser than . I don’t want to spend too much time talking about the affine closure operation, on partitions of a component-wise affine space. If and have a single component, a refinement is just a surjective affine map (recall before we had surjective linear maps in ). If and have dimension function 0, so basically and (partition of possibly distinct finite sets), a refinement just means .
We’re now ready to define a category, which I’ll denote . The objects will be pairs of: a CAP, , along with a non-locally constant affine (subsequently denoted ). A morphism from to will be a refinement such that . This should look familiar to the construction.
The functor I’ll consider still deserves to be called , and it takes to the set of non-locally constant affine maps . We’d really like to be able to say
It seems sensible to try to do so by showing that
since , and we know the homotopy limit over has the right homotopy type. This is our goal.
Semi-direct Product Structure
I’ll use the semi-direct product notation for the Grothendieck construction, as follows. Recall that for a category , and a functor , the Grothendieck construction is a category, which I’ll denote , whose objects are pairs where . Morphisms to are morphisms such that . Of course, my functors are all contravariant as defined, so you have to mess about getting your arrows right. Best done in private.
I claim that can be written as a Grothendieck construction. Actually, it can in a few ways. The obvious way is to set to be the category of CAPs , paired with refinements. The functor you need is then . You find that .
But there’s another way to slice it. Let be the category of CAPs , along with functions . Now the functor you need is not all non-locally constant maps to , but only those that are lifts of . You might denote this set . I’m tired of all the notation, so let me let denote this non-locally constant lifts functor. We have, then .
While I’m simplifying notation, let me also write for . Notice that it is actually a functor from , and thus from .
Let’s return to the category again. It has the same structure. In fact, we just need to pick out of the subcategory of CAPs whose set of components is , and where is the identity on . Calling this subcategory , we have .
Summarizing all the notation, our goal is to show that
The first thing I’d like to do is use twisted arrow categories to re-write things, so perhaps I should tell you about these categories first. If is a category, the twisted arrow category, has objects the morphisms of . Morphisms from to are commuting squares
Using this, we’re hoping to show
We’ve got . Between them lies a category I’ll denote , consisting of arrows with , . Morphisms are “twisted” commuting squares, as they should be, as part of the twisted arrow category. One can reduce the holim over to one over , and from there to one over .
To reduce from to , one can show that for all , the over-category is contractible. In fact, this result seems to rely very little on our particular and , and doesn’t depend on the functors, , , or .
For the reduction from to , one shows that for all , we have
Essentially this shows that , as a functor from , is equivalent to the right Kan extension of it’s restriction to a functor from . And the homotopy limit of a right Kan extension is equivalent to the homotopy limit of the restricted functor.
It is in this second reduction, to , that we rely on information about our categories and functors (, in particular). Pick your object . You can quickly reduce the crazy over-category above to just . Now remember is a CAP with a function to . I’ll denote it . If this function is locally constant (all objects within an equivalence class get sent to the same point), then you sort of replace with an object obtained by taking affine and direct sums of it’s components. The result is an object of , but from the perspective of , the two objects give equivalent spaces of lifts. Alternatively, if is non-locally constant, then every lift is non-locally constant, and so .
This all works out to be useful in the whole proof. But I’ll maybe save all that for another day.