## Meh

Progress has been, as usual, slow. And I haven’t been updating here. My advisor’s last advice was to try to avoid the category $\mathscr{L}_M$ I mentioned previously (category of locally affine partitions on $M=m\times \mathbb{R}^n$). Ideally I can use something like the $\mathscr{C}_M$ category and just find appropriate reductions. I should be looking to deformation retract $\mathscr{C}_M$ down to a smaller category, and also deformation retract $\mathscr{J}_M$ (which I mostly know has the right homotopy type, in the holim) down to the same category. Or something like that.

So I’ve had two sorts of thought strands going. This is not unexpected, as my overall project is really just the blending of two known cases (some of my advisor’s work). One thing I’ve been thinking about is when $M=m\times \mathbb{R}^0$ is just a finite set. And I’ve been thinking about a subcategory $\mathscr{S}_M$ of finite sets with partitions, as opposed to the full category $\mathscr{C}_M$ of abstract completely locally affine partitions (what a mouthful), with non-locally-constant maps to $M$. I think my goal is to deformation retract $\mathscr{S}_M$ down to something along the lines of those pairs $(\lambda,f)$, where $\Lambda \vdash s\xrightarrow{f}M$ and $f$ is injective on the components of $\Lambda$ (or, perhaps even injective overall). To do this, I gotta get the right topology on $\mathscr{S}_M$, which I’m pretty sure I’ve not yet done.

Part of the reason I think I’ve not yet done this is based on my other line of thought, concerning the case $M=1\times \mathbb{R}^n$ (frequently $n=1$ when I’m thinking about it). I’m still thinking about a subcategory of $\mathscr{C}_M$ consisting of just finite sets (with partitions) and non-locally-constant maps to $M$. But I need to set things up so that “points coming together” corresponds to a different partition. That is, when I have two points in the same equivalence class of $\Lambda$ and their image in $M$ lie in the same component (which is easy when $M=1\times \mathbb{R}^n$), then I could make a sequence of functions that are non-locally-constant for this $\Lambda$ and bring the two points together. The limiting point should be considered as a map from a smaller $\Lambda$. Another way to say this is that I should think of points in “thick diagonals” (some, but not all, points are the same, e.g. where $x=y$ in $\mathbb{R}^3$) of $M^n$ as coming from some “smaller partition”.

I’m not saying any of this well. Probably that’s a bad sign. Perhaps I’ll try again tomorrow. Or after I run some ideas by my advisor on Tuesday. Anyway, half the point of this post was to show off the pictures I was looking at today. I was trying to draw my category (or small portions of it) in the case $M=m\times \mathbb{R}^0$, and was, for some reason, pleased with the following picture:

I don’t think I’ll explain too much about it, since I’m not sure how useful it’ll be long-term. But if you’ve thought about set partitions of the set $\underline{4}={1,2,3,4}$, you might recognize the poset (ordered by refinement) in the diagram above. For any partition $\Lambda$, I have written down the possibilities for maps to $M=\underline{2}\times \mathbb{R}^n$ such that the composite with the projection $M\to \underline{2}$ is non-locally-constant on blocks of $\Lambda$. Those underlined in red are, furthermore, injective on blocks.