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1. • CommentRowNumber1.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 16th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexThis 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?

   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?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 16th 2018
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   Author: Urs
   Format: MarkdownItexGomi has a model of K-theory that works just like this. See at _[[vectorial bundle]]_

   Gomi has a model of K-theory that works just like this. See at vectorial bundle

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 16th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexAh, 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.

   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.

4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 17th 2018
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   Author: DavidRoberts
   Format: MarkdownItexCould 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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 17th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexSorry, by equivalences you mean what? By "homotopy category" I simply meant the localization.

   Sorry, by equivalences you mean what? By “homotopy category” I simply meant the localization.

6. • CommentRowNumber6.
   • CommentAuthorDavidRoberts
   • CommentTimeSep 17th 2018
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   Author: DavidRoberts
   Format: MarkdownItexEquivalences as in [[equivalence in a 2-category]].

   Equivalences as in equivalence in a 2-category.

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 17th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexWell, I haven't thought about a 2-category here, yet.

   Well, I haven’t thought about a 2-category here, yet.

8. • CommentRowNumber8.
   • CommentAuthorDylan Wilson
   • CommentTimeSep 17th 2018
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   Author: Dylan Wilson
   Format: MarkdownItexHave 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...

   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…

9. • CommentRowNumber9.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 17th 2018
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   Author: Todd_Trimble
   Format: MarkdownItexDylan, 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.

   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.