This is a first part of notes from a lecture given by Alex Hoffnung on 4.02.2009 in Goettingen. You can learn from it what (do I think) a categorification and decategorification is, and see various examples of it. Next part will deal with categorification of Hecke algebras, which was the main point of Alex’ talk. However, it’s not coming until I teach myself about how to get an algebraic group out of a Dynkin diagram, because otherwise it’s “monkey see, monkey do”. Also, thanks to Peter Arndt here in Goettingen, for clarifying me a point about pullbacks of groupoids.

(*1*)

First decategorification. Frankly, I don’t know what it is, but I roughly know what degroupoidification is. Suppose you have a groupoid $G$ (i.e. a category whose morphisms are invertible). Then degroupoidification of it is a vector space spanned by isomorphism classes of objects in $G$.

Example which comes to mind is the following: Let $X$ be a topological space with a distinguished point $p$. Let $G$ be a groupoid whose objects are loops starting and ending in $p$, and whose morphisms are homotopy classes of homotopies between loops. Then isomorphism classes of objects in $G$ are enumerated by elements of $\pi$, the fundamental group of $X$, and so the degroupoidification is $\mathbb{C}\pi$, a vector space spanned by elements of $\pi$.

(*2*)

However, we know that there is a multiplication structure on $\mathbb{C}\pi$. As far as I understand, groupoidification of a vector space with additional structure is a groupoid with some additional structure which “naturally” gives rise to the structure on the degroupoidified groupoid.

The aim of Alex’ talk was to show groupoidification in this sense of certain class of rings, Hecke algebras.

Why should one care about it? Alex said that in some cases one can do operations on the groupoid level and descent them to degroupoidified level, obtaining some subtle structure which was very hard to notice without passing to the higher, groupoidified, level. However, so far no new informations were obtain about Hecke algebras.

Groupoidification of a given structure doesn’t seem to be unique in any way. The “correct” groupoidification would be then the one which gives rise to some interesting new structure.

(*3*)

A remark about a decategorification. As I mentioned at the beginning, I don’t know what is a right notion of the decategorification. However, with any category $\mathcal{C}$ one can associate a groupoid by simply forgetting all non-invertible morphisms, and then degroupoidify it.

(*4*)

Alex talked also about (one possibility for) a groupoidification of linear maps: a span in a category of groupoids (span between objects $A$ and $B$ is an object $S$ with morphisms $\alpha : S \to A$ and $\beta : S \to B)$.

To justify it I have to explain how to get linear map out of the span.

Basic idea is seen already when working in the category of sets. When $S$ is a span between $A$ and $B$ then associated map between vector spaces generated by A and B is obtained as follows: given $a \in A$, associated vector $e_a$ goes to $\sum_{b\in B}\medskip \text{card} \bigl(\alpha^{-1}(a)\cap \beta^{-1}(b)\bigr)\cdot e_b$.

However, in this case one gets only linear maps represented by matrices with integer coefficients. On the other hand, given a groupoid $G$, we can define $\text{card} (G)$ to be $\sum_{[x]\in G} \frac{1}{|\text{Aut}(x)|}$, where the sum is over isomorphism classes of objects in $G$. Just as in the previous paragraph we can map a vector associated to the isomorphism class of $a\in A$ to $\sum_{[b]\in B}\medskip \text{card} \bigl(\alpha^{-1}(\hat{a})\cap \beta^{-1}(\hat{b})\bigr)\cdot e_{[b]}$, where $\hat{a}$ is a groupoid of elements isomorphic to $a$ (“connected component of $a$“).

This works precisely this way for the category of groupoids which have only finitely many isomorphism classes. Otherwise questions of convergence arise.

(*5*)

Lastly, I want to write about categorified vectors. These are just morphism of groupoids $v: V \to G$. To such data we associate a following vector in the degroupoidified $G$: $\sum_{[g]\in G}\medskip \text{card}\bigl(v^{-1}(\hat{g})\bigr)\cdot e_{[g]}$. Given a span $A \gets S \to B$ and a categorified vector $V \to B$ one gets a categorified vector over $A$ by first taking the pullback to get a vector over $S$, and then composing with $A \gets S$.

(*5a*)

There are at least two kinds of pullbacks of groupoids, depending on whether we work in the category whose morphisms are strict morphisms or a quotient of morphisms by natural isomorphisms between morphisms. Respective pullbacks are called a strict and a weak pullback. In the definition above we used the latter.

A weak pullback of groupoids $p: X \to Z \gets Y :q$ is a groupoid $X\times_Z Y$ whose objects are triples $(x,y,\alpha )$, with $x\in Ob(X)$, $y\in Ob(Y)$, and $\alpha :p(x) \to q(y)$, and whose morphisms between $(x,y,\alpha )$ and $(x',y',\alpha ')$ are pairs of morphisms $\sigma: x\to x'$, $\tau: y\to y'$, such that the obvious diagram in $Z$ commutes. Given these definitions we get obvious maps of groupoids $\xi: X \to Z$ and $\upsilon: Y \to Z$.

Let’s check that this has the required universal property. Suppose we have a diagram

 $W$ $\stackrel{\beta}\longrightarrow$ $Y$ ${}^\alpha\big\downarrow$ $\text{ }\big\downarrow^q$ $X$ $\stackrel{p}\longrightarrow$ $Z$

We need to show that there is a unique map $w: W\to X\times_Z Y$ making the obvious diagram commute. We define it in the following way on objects: $A\in Ob(W)$ goes to $(\alpha(A),\beta(A),n)$, where $n$ is a natural isomorphism given by the fact that the above diagram is commutative up to natural isomorphism. On morphisms we define it analogically.

It is clear that suitable diagram commutes. As to the uniqueness of $w$, if there is some map $w'$ which also makes this suitable diagram commute, then again by definition of morphisms in our category there exist natural isomorphisms between $p\circ\xi\circ w$ and $p\circ\xi\circ w'$ and between $p\circ\upsilon\circ w$ and $p\circ\upsilon\circ w'$. Using these it is easy to define an isomorphism between $w$ and $w'$ as well.

For example, take $X=Y=Z=G$, where $G$ is a group (i.e. groupoid with only one object) and $p$ and $q$ are both the identity morphism. Then $X\times_Z Y$ has $|G|$ objects with morphisms only between objects corresponding to conjugate elements of the group each of which has $|G|$ morphisms. This – as expected – is weakly isomorphic to $G$. This is not (equivalent to) $G$ so I have a problem here.

(*5b*)

Note that categorified vectors can be added – just take a disjoint sum of groupoids over a base groupoid. This sum commutes with decategorification, and also with the map of categorified vectors defined above. To check this last assertion it is enough to convince yourself that sum commutes ( in a strict sense! this is important, because categorified vectors which are just weakly isomorphic don’t give the same decategorified vector) with taking a weak pullback.

With this in mind it is straightforward to check that the two described ways of getting linear maps out of spans of groupoids give the same results.

(*6*)

Final remark is that given a span $A \gets S \to B$ one can construct two morphisms: from decategorified $A$ to decategorified $B$ and the other way around. It’s easily seen that matrices of these are transposes of each other and so the morphisms are adjoint.