In this blog-post, I wanted to give a super-accessible version of decorated loop-spaces.

First, let's start with the classical loop-space. Suppose you have a system you want to model, and it exhibits periodic behavior. Let us call this system. Call it $X$. The classical loop-space is: $$\Omega X = \mathrm{Maps}_* (S^1,X).$$

For every natural number $n$, the symbol $S^n$ denotes the $n$-sphere, or the set of points equidistant from the origin in $n$ directions. So, $S^1$ is just a circle, and $\Omega X$ gives us all the maps from the circle into our system $X$. The little $*$ in our notation tells us that these are based maps, so in other words we have fixed a basepoint on our circle. We will think of this as our starting time; and, since our system's behavior is periodic, then it will repeat after a certain amount of time.

For concreteness, let's just choose our interval to be one second, and fix our basepoint to be the north pole of the circle. Then, every second we wind around to the north pole again, and so all of the moments in time corresponding to zero seconds, one second, two seconds, etc. will be represented by a map from the north pole into $X$. The south pole is then the set of moments corresponding to .5 seconds, 1.5, 2.5, etc. seconds. Every point on the circle will repeat after one second, but what matters is our starting value $t \in [0,1)$. This is a modified version of the unit interval $I$; it includes every number of the form $n^{-1}$ where $n \in \mathbb{N}_1$ is a natural number, but instead of $1^{-1} = 1$, we have a zero instead, because we need some way of encoding the starting time.

The system we have defined so far is actually perfectly periodic, but we know from our experience that these are not easy to find in nature. So, in my paper, I have defined the decorated loop-space: $$\widehat{\Omega}X := \mathrm{Maps}_*(\widehat{S}^1, \widehat{X})$$ where $$\widehat{S}^1 = S^1 \cup \{g\} \quad \widehat{X} = X \cup \{p\}$$ are just the original circle and system, but now they have each been expanded to include new sets. The notation chosen here reflects my original intention to formalize evolutionary dynamics, with $g$ being the genotype and $p$ the phenotype.

If we set both $\{g\}$ and $\{p\}$ to be the empty set, then the decoration trivializes and we recover the original loop-space. The simplest non-trivial case would be where each of these are distinct marked points. In that case, we want a family of functions $\widetilde{f}(g)$ on our first point to encode changes to the system, and another family of functions $\widetilde{h}(p)$ on our second point to encode how the system responds to these changes. Then, we define some connector $$\varphi: \widetilde{f}(g) \longrightarrow \widetilde{h}(p)$$ which tells us how to generate the second family from the first.

If you're familiar with category theory, you should know that these decorated loop-spaces naturally form a very nice category called $\mathbf{DecLpSpc}$ whose:

In the category framework, an important distinction is made between the decorator $\mathcal{D}_X$ and the connector $\varphi$. The decorator is responsible only for selecting which phenotypic image $p$ corresponds to the generator $g$. The connector, meanwhile, acts on families of functions, and category-theoretically, it is a natural transformation: $$\varphi_{g_j}: \mathrm{Fun}(g_j, \mathcal{C}_0) \Rightarrow \mathrm{Fun}(p_k, \mathcal{C}_1)$$ between function spaces. (If you're unfamiliar with natural transformations, just think of them as things that transform transformations themselves, just like how an operator is basically a function on functions).

Example

There are many examples of when this might be a useful structure. For instance, in epigenetics, $g_j$ could be epigenetic configurations (DNA methylation patterns, histone marks, etc.), while the $p_k$ are phenotypic configurations (cell types, morphologies), and $\varphi$ is the developmental rule mapping the first to the second.

Summary

Decorated loop-spaces are an (in my opinion) exciting tool, because they are purely topological ways to encode real things that we might care about, but also have a ton of very nice mathematical properties like functoriality. [1] They let us track both the cyclic trajectory and perturbation data of a feedback loop simultaneously, and this is huge in areas like control theory. Please check the "applications" section of the preprint for more examples.

Endnotes

[1] There is a functor $$\widehat{\Omega}: \mathbf{DecLpSpc} \longrightarrow \mathbf{DecLpSpc}$$ that can be iterated via $\widehat{\Omega}^n$ while preserving the categorical structure, which means that our results remain mathematically sane while applying them to nested feedback loops. This is proven in the paper.