On this page, we explain a connection between decategorification and degroupoidification?, which appears when one considers groupoids internal to schemes. One can construct a category of sheaves attached to such a groupoid, and a sequence of finite groups, given by points over a finite field. This connection has been exploited quite widely within arithmetic geometry, but has not really been stated in the context of categorification.
Let $X\rightrightarrows Y$ be a groupoid internal to the category of schemes ($Y$ is the scheme of objects and $X$ the scheme of morphisms) where both schemes are of finite type.
Then we can make two constructions from this scheme:
The $\mathbb{F}_p$-points $X(\mathbb{F}_p) \rightrightarrows Y(\mathbb{F}_p)$ form a finite groupoid (with no additional structure).
There is a simplicial scheme $N(X\rightrightarrows Y)$ given by the nerve of $X\rightrightarrows Y$ (this sends $\Delta^n$ to $X\times_Y X\times_Y\cdots \times_Y X$, with the obvious face and degeneracy maps).
One special case is when this is an action groupoid, for an algebraic group $G$ (say) acting on $Y$. The resulting simplicial scheme should be thought of as the Borel space for the action on $Y$. The notation $Y//G$ (more properly $Y \sslash G$, but this doesn't seem to render for everybody) for this βhomotopyβ or weak quotient is used in other fields, but this conflicts here with GIT quotients?.
More generally, but for this notational clash, $Y//X$ could denote the nerve. Another option is $\mathcal{N}Y$, where $Y$ denotes the whole groupoid, rather than just the scheme of arrows.
The latter construction can be βlinearizedβ in an analogous way to geometric function theory, except using the derived category of sheaves with finite rank constructible cohomology? with coefficients in the $\ell$-adics $\mathbb{Q}_\ell$ for some prime $\ell$ of $N(X\rightrightarrows Y)$. Let $D(N(X\rightrightarrows Y))$ denote this category. In fact, we will require the graded version of this category, provided by mixed sheaves?. We denote this graded category $D_{mix}(N(X\rightrightarrows Y))$
There is a map $\alpha_{q,\ell}$ from $K^0(D_{mix}(N(X\rightrightarrows Y))$ to the set $\mathbb{Q}_\ell[N(X \rightrightarrows Y)(\mathbb{F}_q)]$ of $\mathbb{Q}_\ell$-valued functions on any $\mathbb{F}_q$ point of $N(X\rightrightarrows Y)$ where $\ell$ and $q$ where $\ell$ is a prime and $\ell\nmid q$, given by the supertrace of automorphism of the stalk of the sheaf at that point induced by the action of the Frobenius $q^n$ on $X\times_{\mathrm{Spec}\mathbb{Z}} \mathrm{Spec}\overline{\mathbb{F}_q}$.
This map has the property that multiplying by the motivic integral? of $\mathbb{A}^1$ (i.e., $m_!m^*$ for the map $m:X\times \mathbb{A}^1\to X$) multiplies the corresponding function by $q$.
Furthermore, no non-zero element of $K^0(D_{mix}(N(X\rightrightarrows Y))$ is killed by this map for all $q$.
If $N(X_1\rightrightarrows Y_1)\overset{f}\leftarrow N(Z\rightrightarrows W) \overset{g}\rightarrow N(X_2 \rightrightarrows Y_2)$ is a span of groupoids of schemes, then we have a functor $g_!f^*:D_{mix}(N(X_1\rightrightarrows Y_1))\to D_{mix}(N(X_2 \rightrightarrows Y_2))$, given by pull-back followed by compactly supported pushforward.
Furthermore, there is an induced span of groupoids on this set of $\mathbb{F}_q$-points, which in turn induces a map $(gf)^\dagger$ from $\mathbb{Q}_\ell[N(X_1\rightrightarrows Y_1)(\mathbb{F}_q)]\to \mathbb{Q}_\ell[N(X_2 \rightrightarrows Y_2)(\mathbb{F}_q)]$.
The content of the Grothendieck trace formula connects these two constructions.
The functor $g_!f^*:D_{mix}(N(X_1\rightrightarrows Y_1))\to D_{mix}(N(X_1\rightrightarrows Y_1))$ induces a map $(gf)_\dagger:K^0(D_{mix}(N(X_1\rightrightarrows Y_1)))\to K^0(D_{mix}(N(X_2 \rightrightarrows Y_2)))$ such that the diagram
That is, modulo a few details:
The decategorification of the mixed derived category and degroupoidification? of the $\mathbb{F}_q$ points coincide.
We note that in many examples of geometric categorification, the fact that the decategorification is correct is checked by understanding the degroupoidification and using this theorem, though it is typically not stated this explicitly.
Since all groupoids appearing below are action groupoids, Iβll denote them $X/G$ in place of $G\times X\rightrightarrows X$.
If one takes the span of groupoids
over the action groupoid for $B\times B$ acting on the left and right on $G$ for a simple algebraic group, then the resulting monoidal category is the Hecke category of G. Its decategorification is the Hecke algebra of G, which is checked by showing the Hecke algebra is groupoidified by the F_q points of this, a result which was proved Iwahori in 1964.
Consider the groupoid given by the action of $G(\mathbb{Z}[t])\times G(\mathbb{Z}[t])$ (for $\mathbb{Z}[t]$ the polynomial ring) on the left and right on $G(\mathbb{Z}(t))$ considered as as pro-ind-schemes over $\mathbb{Z}$. The derived category of this groupoid scheme is monoidal by the analogous diagram to that above.
The groupoidification of the $\mathbb{F}_q$ points was shown by Satake (see at Satake equivalence) to be the representation ring of ${^L G}$, the Langlands dual group. Thus, $D_{mix}(G(\mathbb{Z}[t])\backslash G(\mathbb{Z}(t)/ G(\mathbb{Z}[t])$ also contains a categorification of the representation ring of ${^L G}$.
In fact, MirkoviΔ and Vilonen showed that the subcategory of $D_{mix}(G(\mathbb{Z}[t])\backslash G(\mathbb{Z}(t))/ G(\mathbb{Z}[t])$ consisting of perverse sheaves is equivalent to the category of representations of ${^L G}\times_{\mathrm{Spec}\mathbb{Z}} \mathrm{Spec}\overline{\mathbb{Q}_\ell}$ as an algebraic group. Moreover, if one replaces $\mathbb{Q}_\ell$ by any other ring both in the coefficient of the sheaves, and the base of the algebraic group, the result still holds.
If one takes the groupoid of representations of a Dynkin quiver with fixed dimension vector and basis $E_d$ (with each vector only having components over one dot), with the morphisms given by isomorphisms of representations (note, this is the action groupoid for a product of general linear groups $GL_d$ acting on a finite dimensional vector space), the resulting simplicial scheme is the fine moduli space of representations of that quiver.
There is a span of groupoids
where the span in the middle is the set of short exact sequences of representations with basis of dimension vector $d',d''$ mapping into and out of one of dimension vector $d$, preserving all bases, and the morphism are the obvious possible basis changes in this situation (this the action of groupoid for the parabolic in $GL_d$ preserving this filtration).
It was proved by Ringel that the $\mathbb{F}_q$ points of this variety groupoidify the upper half of the quantum group (of course, not in that language). The natural resulting categorification was described by Lusztig and used to construct the canonical basis of $U_q(\mathfrak{g})$ where $\mathfrak{g}$ is the simple Lie algebra associated to this quiver.
This categorification was described independently by Rouquier and KhovanovβLauda combinatorially.