## More Catchup

My last post started laying down some of the framework for what I’ve been thinking about, and I thought I’d continue on with that here.

Recall $M=m\times \mathbb{R}^n$, and that I have a category $\mathscr{C}$ which I think of as a category of “complete locally affine partiitons”. From this I construct the category $\mathscr{C}_M$ consisting of pairs $(\rho,f)$, where $\rho\in \mathscr{C}$ (so $\rho=(\rho,A_{\rho},s,\Lambda)$) and $f:A_{\rho}\rightarrow M$ does not factor through $\rho$ (that is, there are two points that are equivalent in the partition $\rho$, but their images are different in $M$). I’ve also taken up the habit of saying that these maps “disrespect” $\rho$. Out of this category I had a functor $nlc(-,V)$, the non-locally-constant maps to a real vector space $V$. Of course, since this is supposed to a be “topological category” (I have a space of objects, and a space of morphisms…), a functor from the category is really a fiberwise space over the space of objects. Let me denote this space by $\mathscr{N}_{\mathscr{C}_M}$.

I have another category that I didn’t mention yesterday. I denote it by $\mathscr{L}_M$, and it consists of locally affine partitions of $M$. Given two locally affine partitions, $\sigma,\sigma'$ in this category, there is an arrow $\sigma\to\sigma'$ precisely if $\sigma\leq \sigma'$ (that is, $\sigma$ is coarser than $\sigma'$ – if two elements are related by $\sigma'$ then they are related by $\sigma$).

I’m being a little imprecise with the definition of this category for now, because I’m still playing with it and trying to figure out what makes it workable. The idea is that there is a functor $im:\mathscr{C}_M\to \mathscr{L}_M$ given by taking the image. That is, a point in $\mathscr{C}_M$ is a pair $(\rho,f)$, and it makes sense to consider $f(\rho)$, whose affine completion is a locally affine partition of $M$. Out of the category $\mathscr{L}_M$ we have yet another “non-locally-constant” functor, $nlc(-,V)$. Given a locally affine partition $\sigma$, $nlc(\sigma,V)$ will be maps $M\to V$ that disrespect $\sigma$. Of course, I’m again in the situation of topological categories, so really this functor is an object $\mathscr{N}_{\mathscr{L}_M}$ over the space of objects of $\mathscr{L}_M$.

I can now take the pull-back of $\mathscr{N}_{\mathscr{L}_M}$ along $im$, obtaining an object (functor) which I’ll denote $\mathscr{N}_{\mathscr{L}_M}^*$ over $\mathscr{C}_M$. so, if you are counting along at home, that’s two functors over $\mathscr{C}_M$, and one over $\mathscr{L}_M$. With any luck I’ll be able to compare all the associated homotopy limits.

There’s one more category, which I’ve been calling $\mathscr{J}_M$. The letter ‘J’ comes from the fact that this category comes out of some join construction, based on a theorem of Thomason for manipulating (ho)limits (and the general brilliance of my advisor). I’ve got a functor out of it, and I know that the homotopy limit of this guy is the space of embeddings I want. When I get $\mathscr{L}_M$ sorted out, there should be a nice obvious functor $\mathscr{J}_M\to \mathscr{L}_M$ (essentially an inclusion), and that’ll give me the ways to tie together all of my homotopy limits, and tie them all to the space of embeddings. But that’ll have to wait for another day. I was tired when I started this post, and it’s only gotten worse.