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This is a bit speculative; some of it might be wrong, silly, or both.
Suppose $G$ is a groupoid all of whose connected components are finite, and $k$ is a field of characteristic zero. Consider the category $SuperDG_k(G)$ consisting of superspace representations $(V_0, V_1)$ consisting of functors from $G$ into finite-dimensional superspaces, equipped with $G$-equivariant differentials $\partial_0: V_0 \to V_1, \partial_1: V_1 \to V_0$. I propose to define the category of virtual representations of $G$ to be the localization of $SuperDG_k(G)$ with respect to quasi-isomorphisms (morphisms that induce isomorphisms in homology).
This might very well dissolve into something much more trivial than it sounds, but the rough idea anyway is that an object represented by $(V_0, V_1, \partial_\ast)$ plays the role of $V_0 - V_1$. Ordinary representations embed fully and faithfully as $V \mapsto (V, 0)$ with zero differentials. There is a supersymmetry functor given by degree shift modulo $2$ (if I’m using the word correctly, something that exchanges bosons and fermions), playing the role of additive inversion.
The category has a symmetric monoidal structure in the expected way. I’m hoping it decategorifies to the representation ring.
Some old notes of mine on the Lie operad (housed at Baez’s website) seemed to use this idea implicitly, but I had never bothered to formalize it. Perhaps something like this appears in the literature?
Gomi has a model of K-theory that works just like this. See at vectorial bundle
Ah, great; thanks! So maybe it’s not such a silly idea. :-)
For what it’s worth, the methodological lesson I took away way back when was: when doing these categorified calculations that pertain to “virtual species”, remain for as long as you can up in $SuperDG_k(\mathbb{P})$ before descending down to the homotopy category, taking advantage of any/all benefits such as model category structure.
Could one look at the 2-category instead, incorporating homotopies? It’s not a priori (to me) obvious that quasi-isomorphisms and equivalences in this 2-category coincide.
Sorry, by equivalences you mean what? By “homotopy category” I simply meant the localization.
Equivalences as in equivalence in a 2-category.
Well, I haven’t thought about a 2-category here, yet.
Have you seen Construction 6.2.1 in Lurie’s “Rotation invariance in algebraic K theory” paper? (The construction is much older- I think due to Quillen?- but that was just the quickest reference I could find). If you do the same thing with representations then you categorify the representation ring. I’m not sure how to relate it to your super-construction…
Dylan, it looks related, but I don’t think it’s doing what I quite want. For example, it seems to be the core groupoid of $Vect$ that embeds fully and faithfully into the category, whereas (for this toy case of the terminal groupoid $G$) I would want all of $Vect$ to embed.
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