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August 14, 2007

Lazaroiu on G-Flows on Categories

Posted by Urs Schreiber

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 GG. 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 GG-flow on them. Using the Arrow-theoretic differential theory which I was talking about, we can talk about GG-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 \mathbb{R}-flows.

In particular, if we have a supercategory we want to see a 2\mathbb{Z}_2-flow on it. Or, if we have NN-extended supersymmetry, really an ( 2) N(\mathbb{Z}_2)^N-flow. (I pointed out that these seem to have appeared in the study of representations of the NN-extended 1-dimensional supertranslation algebra here.)

Similarly, if the supersymmetry is “twisted” such that the 2\mathbb{Z}_2-grading somehow turns into that of a larger abelian group GG (like G=G = \mathbb{Z} most notably, as described in Aspinwall’s review), we woud want to see the relevant category to come equipped with an GG-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 GG-flow [Supercategories, def. 1].

In other words, Lazaroiu’s “graded categories with shifts” are categories CC eqipped with a (faithful) action of the group GG by inner automorphisms : GINN(C) 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: id id C s(h) C s(g) C γ(h) γ(g) = id C s(g+h) C γ(g+h) \array{ &{}^{\;\;\;}\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)}& } \;\;\; = \;\;\; \array{ &{}^{\;\;\;}\nearrow \searrow^{\mathrm{id}}& \\ C &\Downarrow^{s(g+h)}& C \\ &{}_{\;\;\;}\searrow \nearrow_{\gamma(g+h)}& }

Posted at August 14, 2007 11:22 AM UTC

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Re: Lazaroiu on G-Flows on Categories

While it’s not surprising, for the record I note that also Lazaroiu’s notion of morphisms of “GG-graded categories with shifts” coincides the with notion of morphism of “categories with GG-flow”:

Proposition What Lazaroiu calls a morphism of GG-graded categories with shifts [Lazaroiu06, p. 7 below (2.5)] is exactly a morphism of categories with GG-flow [Supercategories, def 2].

Proof. Lazroiu’s description (between (2.5) and (2.6))

a functor FF which intertwines γ(g)\gamma(g) and γ(g)\gamma'(g) and maps s a(g)s_a(g) into s F(a)(g)s'_{F(a)}(g)

is precisely the componentwise statement of the condition Id C s(g) C γ(g) F C=C F Id C s(g) C γ(g) \array{ & \nearrow \searrow^{\mathrm{Id}} \\ C &\Downarrow^{s(g)}& C \\ & \searrow \nearrow^{\gamma(g)} \\ && \downarrow^F \\ && C' } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ C \\ \downarrow^F \\ & \nearrow \searrow^{\mathrm{Id}} \\ C' &\Downarrow^{s'(g)}& C' \\ & \searrow \nearrow^{\gamma'(g)} }

Posted by: Urs Schreiber on August 15, 2007 3:32 PM | Permalink | Reply to this

Re: Lazaroiu on G-Flows on Categories

One can continue, in this fashion, going through Calin Lazaroiu’s component formulas and highlighting the intrinsic diagrammatics which they come from.

For instance, given the notion of morphism of categories with GG-flow just recalled, one is certainly reminded of pseudonatural transformations and might be tempted to functorially fill this tin can by 2-cells of the form (g)C γ(g) C F η(g) F C γ(g) C. (\bullet \stackrel{g}{\to} \bullet) \;\;\;\; \mapsto \;\;\;\; \array{ C &\stackrel{\gamma(g)}{\to}& C \\ \downarrow^F &\Downarrow^{\eta(g)}& \downarrow^F \\ C &\stackrel{\gamma'(g)}{\to}& C } \,. Sure enough, that’s what Calin Lazaroiu considers on p. 8, where condition (2.9) is (the component version of) the functoriality of the above assignment. This pseudonatural transformation he calls a framing.

Posted by: Urs Schreiber on August 15, 2007 6:00 PM | Permalink | Reply to this

Re: Lazaroiu on G-Flows on Categories

Furthermore, the “skew category” T[C]T[C] (bottom of p.7 of Lazaroiu’s article, apparently originally introduced in math.RA/0312214) which is obtained from a category CC with GG-action γ:GAut(C) \gamma : G \to \mathrm{Aut}(C) is nothing but, I think, the weak coequalizer of this action in Cat\mathrm{Cat} Cγ(g)idCT[C]. C \stackrel{\stackrel{\mathrm{id}}{\to}}{\stackrel{\gamma(g)}{\to}} C \to T[C] \,.

This must be a well-known construction to category theorists, though I don’t know a canonical reference off the top of my head. I discussed this in a slightly different but analogous context in Universal Transition which later became section B.1 in my article with Konrad:

This coequalizer, which I claim is the same as T[C]T[C], is generated from CC together with one new morphism as a(g)γ(g)(a) a \stackrel{s_a(g)}{\to}\gamma(g)(a) per object aaof CC and element gGg \in G, subject to the relation a f b s a(g) s b(g) γ(g)(a) γ(g)(f) γ(g)(b) \array{ a &\stackrel{f}{\to}& b \\ \downarrow^{s_a(g)} && \downarrow^{s_b(g)} \\ \gamma(g)(a) &\stackrel{\gamma(g)(f)}{\to}& \gamma(g)(b) } for all gg and ff together with the obvious respect of the s ()s_\cdot(\cdot) for the group action.

Posted by: Urs Schreiber on August 17, 2007 8:38 AM | Permalink | Reply to this
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