A homotopy limit description of the embedding functor

(Approximate notes for a talk I gave this afternoon.)

Setup

So according to the title, I should be telling you about {\text {Emb}(M,N)}, as a functor of manifolds {M} and {N}. That’s perhaps a bit ambitious. I’ll only be thinking about {M=\coprod _{m} D^{n}}, a disjoint union of closed disks, and I’ll actually fix {M}. And instead of letting {N} range over any manifolds, it’ll just be {V}, ranging over real vector spaces.

By taking derivatives at centers, we obtain a homotopy equivalence from {\text {Emb}(M,V)} to something I’ll maybe denote {\text {AffEmb}_{0}(\coprod _{m} \mathbb {R}^{n},V)}. This is componentwise affine (linear followed by translation) maps {\coprod _{m} \mathbb {R}^{n}\to V} whose restriction to {\epsilon }-balls around the 0s is an embedding. I may use {\underline{m}=\{1,\ldots ,m\}}, and write {\underline{m}\times \mathbb {R}^{n}}. And I’ll actually send everything to spectra, instead of topological spaces, via {\Sigma ^{\infty }}.

So really I’ll be talking about

\displaystyle  \Sigma ^{\infty }\text {AffEmb}_{0}(\underline{m}\times \mathbb {R}^{n},V)

as a functor of {V}. I’ll be lazy and write it {\Sigma ^{\infty }\text {Emb}(M,V)}, having fixed an {m} and {n} to give {M=\underline{m}\times D^{n}}.

Useful Cases

The first useful case is when {M=\underline{m}} (i.e., {n=0}). Then embeddings are just configuration spaces, {F(m,V)}. 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 {\mathcal{P} (m)}. Objects will be non-trivial partitions of {\underline{m}}, and I’ll probably denote them {\Lambda }, perhaps writing {\Lambda \vdash \underline{m}}. Non-trivial means that some equivalence class is larger than a singleton. I’ll write \Lambda\leq \Lambda' if {\Lambda } is finer than {\Lambda'}, meaning that whenever {x\equiv y\pmod{\Lambda }}, then {x\equiv y\pmod{\Lambda '}}.

The functor I want is something I’ll denote {\text {nlc}(-,V)} and call “non-locally constant” maps. So {\text {nlc}(\Lambda ,V)} is the set (space) of maps {f:\underline{m}\to V} such that there is an {x\equiv y\pmod{\Lambda }} where {f(x)\neq f(y)}. Equivalently, maps which don’t factor through {\underline{m}/\Lambda }.

Depending on which order you make your poset {\mathcal{P} (m)} go, {\text {nlc}(-,V)} is contravariant, and you can show

\displaystyle  \Sigma ^{\infty }F(m,V)\simeq \text {holim}_{\mathcal{P} (m)}\Sigma ^{\infty }\text {nlc}(-,V).

The second useful case is when {M=D^{n}} (i.e., {m=1}). 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 {\mathcal{L} (n)}, and objects in the category will be vector spaces {E} with non-zero linear maps {f:E\to \mathbb {R}^{n}}. Morphisms from {f:E\to \mathcal{R}^{n}} to f':E'\to \mathbb{R}^n will be surjective linear maps \alpha:E\to E' with f=f'\circ \alpha. You might think of the objects as an abstract partition ({E}) with a map to {\mathbb {R}^{n}}, which then determines a partition of {\mathbb {R}^{n}}, by taking the image.

The functor out of this category is something I’ll still denote {\text {nlc}(-,V)}. On an object {f:E\to \mathbb {R}^{n}} it gives all non-constant affine maps {E\to V}. Arone has shown

\displaystyle  \Sigma ^{\infty }\text {Inj}(\mathbb {R}^{n},V)\simeq \text {holim}_{\mathcal{L} (n)}\Sigma ^{\infty }\text {nlc}(-,V).

Product Structure

The space of embeddings we are considering splits, as

\displaystyle  \text {Emb}(M,V)\simeq F(m,V)\times \text {Inj}(\mathbb {R}^{n},V)^{m}.

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 {\Sigma ^{\infty }X_{i}\simeq \text {holim}_{\mathcal{C}_{i}}\Sigma ^{\infty }F_{i}}, {i=1,\ldots ,k}, then

\displaystyle  \Sigma ^{\infty }\prod _{i} X_{i} \simeq \text {holim}_{*_{i}\mathcal{C}_{i}}\Sigma ^{\infty }*_{i} F_{i}.

Here {*} denotes the join. For categories, {C*D} is {(C_{+}\times D_{+})_{-}}, obtained by adding a new initial object to each of {C} and {D}, taking the product, and removing the initial object of the result.

Proof (Sketch): Consider the case {k=2}. The idea is to line up the squares:

and

Both of which are homotopy pullbacks. The equivalence of the lower-right corners follows because join is similar enough to smash, which plays nicely with {\text {holim}}.

So, anyway, applying this lemma and perhaps cleaning things up with some homotopy equivalences, we obtain an equivalence

\displaystyle  \Sigma ^{\infty }\text {Emb}(M,V)\simeq \text {holim}_{\mathcal{P} (m)*\mathcal{L} (n)^{*m}}\Sigma ^{\infty }\text {nlc}(-,V).

Objects in the category consist of a partition {\Lambda \vdash \underline{m}} along with, for {i\in \underline{m}}, linear {f_{i}:E_{i}\to \mathbb {R}^{n}}. To tidy up a little bit, I’ll denote this category {\mathcal{J}=\mathcal{J}(M)}, for join. The functor takes an object as above and returns the set of componentwise affine maps {\coprod _{i} E_{i}\to V} such that either (a) the map is non-constant on some component, (b) when restricted to the image of {0:\underline{m}\hookrightarrow \coprod E_{i}}, the map is non-locally constant with respect to {\Lambda }.

There you have it, a homotopy limit description for the embedding functor.

But not a particularly nice one. If we had an embedding {M\hookrightarrow M'}, then we’d have map {\text {Emb}(M,V)\leftarrow \text {Emb}(M',V)}. It’d be really swell if this map was modelled by a map {\mathcal{J}(M)\to \mathcal{J}(M')} of the categories we are taking homotopy limits over. But that’s not going to happen. What can go wrong? Non-trivial partitions of {M}, when sensibly composed with the map to {M'}, may become trivial, and thus not part of the category. This is, essentially, because several components of {M} might map to a single component of {M'}. If {M} has two components, and {M'} one, say, where do you send the object consisting of the non-trivial {\Lambda \vdash \underline{2}} 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 {M}, 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 {d} by {A^{d}}, say. That should be enough of a description for now.

Let me define a Complete Affine Partition (CAP), {\rho }, 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 {\rho }, I’ll denote by {A(\rho )} the underlying component-wise affine space. The data that determines a {\rho } is: a finite set {s} (the set of components), a partition of {s}, and a dimension function, {d:s\to \mathbb {N}_{0}} (non-negative integers). With this information, {A(\rho )} is {\coprod _{i\in s}A^{d(i)}}.

By a refinement {\alpha } from {\rho } to {\rho'}, denoted {\alpha :\rho \to \rho'}, I will mean an affine map {\alpha :A(\rho )\to A(\rho')} so that the “affine closure” of the partition {\alpha (\rho )} is coarser than {\rho'}. I don’t want to spend too much time talking about the affine closure operation, on partitions of a component-wise affine space. If {\rho } and {\rho'} have a single component, a refinement is just a surjective affine map (recall before we had surjective linear maps in {\mathcal{L} (n)}). If {\rho } and {\rho'} have dimension function 0, so basically {\rho =\Lambda } and {\rho'=\Lambda'} (partition of possibly distinct finite sets), a refinement just means {\alpha (\Lambda ) \geq \Lambda'}.

We’re now ready to define a category, which I’ll denote {\mathcal{C}=\mathcal{C}(M)}. The objects will be pairs of: a CAP, {\rho }, along with a non-locally constant affine {f:A(\rho )\to M} (subsequently denoted {f:\rho \to M}). A morphism from {f:\rho \to M} to {f':\rho'\to M} will be a refinement {\alpha :\rho \to \rho'} such that {f=f'\circ \alpha }. This should look familiar to the {\mathcal{L} (n)} construction.

The functor I’ll consider still deserves to be called {\text {nlc}(-,V)}, and it takes {f:\rho \to M} to the set of non-locally constant affine maps {A(\rho )\to V}. We’d really like to be able to say

\displaystyle  \Sigma ^{\infty }\text {Emb}(M,V)\simeq \text {holim}_{\mathcal{C}(M)}\Sigma ^{\infty }\text {nlc}(-,V).

It seems sensible to try to do so by showing that

\text {holim}_{\mathcal{C}(M)}\Sigma ^{\infty }\text {nlc}(-,V)\simeq \text {holim}_{\mathcal{J}(M)}\Sigma ^{\infty }\text {nlc}(-,V),

since {\mathcal{J}(M)\subseteq \mathcal{C}(M)}, and we know the homotopy limit over {\mathcal{J}(M)} 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 {D}, and a functor {F:D\to E}, the Grothendieck construction is a category, which I’ll denote {D\ltimes F}, whose objects are pairs {(d,x)} where {x\in F(d)}. Morphisms {(d,x)} to {(d',x')} are morphisms {h:d\to d'} such that {F(h)(x)=x'}. 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 {\mathcal{C}} can be written as a Grothendieck construction. Actually, it can in a few ways. The obvious way is to set {\mathcal{U}} to be the category of CAPs {\rho }, paired with refinements. The functor you need is then {\text {nlc}(-,M)}. You find that {\mathcal{C}=\mathcal{U}\ltimes \text {nlc}(-,M)}.

But there’s another way to slice it. Let {\mathcal{U}_{0}} be the category of CAPs {\rho }, along with functions {f_{0}:\rho \to \underline{m}}. Now the functor you need is not all non-locally constant maps to {M}, but only those that are lifts of {f_{0}}. You might denote this set {\text {nlc}_{f_{0}}(\rho ,M)}. I’m tired of all the notation, so let me let {\mu } denote this non-locally constant lifts functor. We have, then {\mathcal{C}=\mathcal{U}_{0}\ltimes \mu }.

While I’m simplifying notation, let me also write {\nu } for {\Sigma ^{\infty }\text {nlc}(-,V)}. Notice that it is actually a functor from {\mathcal{U}}, and thus from {\mathcal{U}_{0}}.

Let’s return to the category {\mathcal{J}(M)} again. It has the same structure. In fact, we just need to pick out of {\mathcal{U}_{0}} the subcategory of CAPs whose set of components is {\underline{m}}, and where {f_{0}} is the identity on {\underline{m}}. Calling this subcategory {\mathcal{R}}, we have {\mathcal{J}=\mathcal{R}\ltimes \mu }.

Summarizing all the notation, our goal is to show that

\displaystyle  \text {holim}_{\mathcal{U}_{0}\ltimes \mu }\nu \simeq \text {holim}_{\mathcal{R}\ltimes \mu }\nu .

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 {D} is a category, the twisted arrow category, {{}^{a}D} has objects the morphisms of {D}. Morphisms from {d_{1}\to d_{2}} to {d_{1}'\to d_{2}'} are commuting squares

If {F} and {G} are contravariant functors from {D}, one can check that {(d_{1}\to d_{2})\mapsto \text {mor}(F(d_{2}),G(d_{1}))} is a covariant functor from {{}^{a}D}. I’ll denote it {G^{F}}. One can show that

\displaystyle  \text {holim}_{D\ltimes F}G\simeq \text {holim}_{{}^{a}D}G^{F}.

Using this, we’re hoping to show

\displaystyle  \text {holim}_{{}^{a}\mathcal{U}_{0}}\nu ^{\mu }\simeq \text {holim}_{{}^{a}\mathcal{R}}\nu ^{\mu }.

Proof Outline

We’ve got {{}^{a}\mathcal{U}_{0}\supseteq {}^{a}\mathcal{R}}. Between them lies a category I’ll denote {\mathcal{U}_{0}\to \mathcal{R}}, consisting of arrows {u\to r} with {u\in \mathcal{U}_{0}}, {r\in \mathcal{R}}. Morphisms are “twisted” commuting squares, as they should be, as part of the twisted arrow category. One can reduce the holim over {{}^{a}\mathcal{U}_{0}} to one over {\mathcal{U}_{0}\to \mathcal{R}}, and from there to one over {{}^{a}\mathcal{R}}.

To reduce from {\mathcal{U}_{0}\to \mathcal{R}} to {{}^{a}\mathcal{R}}, one can show that for all {u\to r\in \mathcal{U}_{0}\to \mathcal{R}}, the over-category {{}^{a}\mathcal{R}\downarrow (u\to r)} is contractible. In fact, this result seems to rely very little on our particular {\mathcal{R}} and {\mathcal{U}_{0}}, and doesn’t depend on the functors, {\mu }, {\nu }, or {\nu ^{\mu }}.

For the reduction from {{}^{a}\mathcal{U}_{0}} to {\mathcal{U}_{0}\to \mathcal{R}}, one shows that for all {u_{1}\to u_{2}\in {}^{a}\mathcal{U}_{0}}, we have

\displaystyle  \text {mor}(\mu (u_{2}),\nu (u_{1}))\stackrel{\sim }{\rightarrow } \text {holim}_{\substack {(u_1\to u_2)\to (u\to r)\\ \in (u_1\to u_2)\downarrow (\mathcal{U}_0\to \mathcal{R})}}\text {mor}(\mu (r),\nu (u)).

Essentially this shows that {\nu ^{\mu }}, as a functor from {{}^{a}U_{0}}, is equivalent to the right Kan extension of it’s restriction to a functor from {\mathcal{U}_{0}\to \mathcal{R}}. 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, {{}^{a}\mathcal{U}_{0}} to {\mathcal{U}_{0}\to \mathcal{R}}, that we rely on information about our categories and functors ({\mu }, in particular). Pick your object {u_{1}\to u_{2}\in {}^{a}\mathcal{U}_{0}}. You can quickly reduce the crazy over-category above to just {u_{2}\downarrow \mathcal{R}}. Now remember {u_{2}} is a CAP with a function to {\underline{m}}. I’ll denote it {f_{0}:u_{2}\to \underline{m}}. If this function is locally constant (all objects within an equivalence class get sent to the same point), then you sort of replace {u_{2}} with an object obtained by taking affine and direct sums of it’s components. The result is an object of {\mathcal{R}}, but from the perspective of {\mu }, the two objects give equivalent spaces of lifts. Alternatively, if {f_{0}:u_{2}\to \underline{m}} is non-locally constant, then every lift {u_{2}\to M} is non-locally constant, and so {\mu (u_{2})\simeq *}.

This all works out to be useful in the whole proof. But I’ll maybe save all that for another day.

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