Categorification
Ellis Caird
Created: 13-11-24
The coordinate ring $\mathbb{C}[Gr(k,n)]$ of the Grassmannian has a cluster structure [Scott '03]. We would like to understand as much as possible about this structure. What are the cluster variables? What are the clusters? Cluster monomials? Expansion formulas?
General answers to such questions can be hard to come by if we are confined only to the algebra itself. The process of Categorification aims to free us from this constraint by 'upgrading' things of interest (in our case functions on the Grassmannian) to objects in some category. The idea is that you now have a richer structure to attempt to exploit in such a way to glean more information about the things you really care about.
I will add some examples of Categorification below and in another post I will talk about the specific Category 'CM(C)' that categorifies the cluster structure on the Grassmannian.
The Natural Numbers
I will primarily be following [LS] here (but almost certainly will be more verbose, if not just for my own sake). The endeavour to categorify $\mathbb{N}$ will envolve picking both a category and picking a way to 'decategorify', i.e. recover the naturals from our category. The first port of call might be to notice that we can already realise the natruals as a category. They form a poset and all posets can be thought of as categories. While we could do this, it doesn't feel like we are adding any structure here that wasn't there already. We are just viewing the natural numbers as a category where each object is just a natural number and the arrows refelect the poset. Boring. If we thought for a while about whether we know any relatively simple structures to which we could associate to each a natural number, it would not be long before we arrived at our next port.
The category, $\mathrm{Vect}_\mathbb{R}$, of finite dimensional real vector spaces certainly consist of objects to which we can associate a natural number, dimension. Thus this categorification of $\mathbb{N}$ consists of assigning to each natural number $n$, an $n$-dimensional real vector space. What structures on $n$ might be hope are also lifted to this category? Addition? Direct sums of vector spaces adds their dimensions. Multiplication? Tensor products of vector spaces multiply their dimension. Order? One could think about existence of injections from one vector space to another as describing the standard order on the naturals.
Admittedly, while this is a nice and simple example to think about, the natural numbers have been well understood for quite a while and so this categorification will not yield us any information that we could not have already known. But this does raise a coupe of interesting questions.
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How does one make sense of structures in a categorification that do not have a clear interpretation in the decategorified setting? E.g. what, if anything, does a basis of a vector space correspond to in $\mathbb{N}$?
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Are there situations in which a categorification can tell us information about structure that we could not have known without it? Or is it merely a tool for when the going gets tough?
I don't have immediate answers to these questions. Maybe we can start to atleast partially answer them by looking at more examples.