The n-Category Café

I just heard a talk (in the context of the ESI program that I am attending) by C. Lazaroiu which mentioned aspects of his article

C. I. Lazaroiu
Graded D-branes and skew categories
hep-th/0612041

in which he studies categories of branes for the topological string.

These categories are typically graded by an abelian group $G$. This grading can be understood as originating in the supersymmetry which was “twisted” to go from the superconformal to the topological string (I once tried to summarize some aspects of this process here).

Now, in the entry Supercategories I argued that such categories should have the property that there is a $G$-flow on them. Using the Arrow-theoretic differential theory which I was talking about, we can talk about $G$-flows on categories as a generalization of the ordinary concept of the flow along an ordinary vector field – which is indeed reproduced in terms of smooth $ℝ$-flows.

In particular, if we have a supercategory we want to see a ${\mathbb{Z}}_2$-flow on it. Or, if we have $N$-extended supersymmetry, really an $({\mathbb{Z}}_2{)}^N$-flow. (I pointed out that these seem to have appeared in the study of representations of the $N$-extended 1-dimensional supertranslation algebra here.)

Similarly, if the supersymmetry is “twisted” such that the ${\mathbb{Z}}_2$-grading somehow turns into that of a larger abelian group $G$ (like $G=\mathbb{Z}$ most notably, as described in Aspinwall’s review), we woud want to see the relevant category to come equipped with an $G$-flow.

Given these considerations, I was pleased to see that this is exactly what Lazariou does arrive at in his work.

Proposition. What Lazaroiu calls a graded category with shifts [Lazaroiu06, p.7] is a category with a $G$-flow [Supercategories, def. 1].

In other words, Lazaroiu’s “graded categories with shifts” are categories $C$ eqipped with a (faithful) action of the group $G$ by inner automorphisms : $$G\to \mathrm{INN}(C)$$ (which are usefully thought of as a generalization of the concept of a Lie derivative).

Proof.

Equation 2.2 in Lazariou’s paper is simply the componentwise statement of the respect of the group product under horizontal composition of transformations: $$\begin{array}{ccccc}& {}^{}\nearrow {\searrow }^{\mathrm{id}}& & {}^{}\nearrow {\searrow }^{\mathrm{id}}& \\ C& {\Downarrow }^{s(h)}& C& {\Downarrow }^{s(g)}& C\\ & {}_{}\searrow {\nearrow }_{\gamma (h)}& & {}_{}\searrow {\nearrow }_{\gamma (g)}& \end{array}=\begin{array}{ccc}& {}^{}\nearrow {\searrow }^{\mathrm{id}}& \\ C& {\Downarrow }^{s(g+h)}& C\\ & {}_{}\searrow {\nearrow }_{\gamma (g+h)}& \end{array}$$