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.