The n-Category Café

A 3-Category of Twisted Bimodules

Posted by Urs Schreiber

Those readers not yet bored to death by my posts might recall the following:

I was arguing that the 3-group controlling Chern-Simons theory (and maybe also the gauge structure of the Green-Schwarz mechanism #) is a sub-3-group of the inner automorphism 3-group #

of the String 2-group - for $G$ an ordinary Lie group (here assumed to be compact, simple and simply connected).

Part of the evidence (I, II) I presented was the observation that the canonical 2-representation # of $\mathrm{String}_G$ on

apparently extends to a representation of $\mathrm{INN}(\mathrm{String}_G)$ on “twisted” bimodules, and that this representation seems to exhibit the expected structures #.

Like $\mathrm{Bim}(C)$ can be thought of as coming from lax functors into $\Sigma(C)$, for $C$ a 2-monoid (an abelian monoidal category), twisted bimodules

can be thought of as coming from lax functors into the endormorphism 3-monoid

of $C$ - in a way that is analogous to the step from the 2-group $\mathrm{String}_G$ to its automorphism 3-group $\mathrm{AUT}(\mathrm{String}_G)$.

3-morphisms in $\mathrm{TwBim}(C)$ look a little like the fundamental disk correlator with one bulk insertion in rational CFT #: a disk, bounded by bimodules, with a ribbon colored in $C$ running perpendicular through the disk’s center. (And this is not supposed to be a coincidence #.)

This picture suggests an obvious 3-category structure. That however is slightly oversimplified. On the other hand, the description in terms of lax functors into $\mathrm{END}(C)$ is a little too unwieldy.

Hence my goal here is to write down precisely and explicitly what $\mathrm{TwBim}(C)$ looks like and how compositions are defined. Diagrams can be found in these notes:

$\;\;\;$a 3-category of twisted bimodules

The hard part is to check coherent weak properties, like the exchange law. I have checked what looked nontrivial - and am hoping that I haven’t overlooked anything. But if anyone has seen before anything like the 3-category $\mathrm{TwBim}(C)$ that I am trying to describe here, please drop me a note.

A crucial point in all these constructions is the restriction to inner automorphisms and inner endomorphisms.

Analogous to the restriction to inner automorphisms of the structure 2-group

(1)$$\mathrm{INN}(G_2) \subset \mathrm{AUT}(G_2) \,,$$

the endomorphisms of $C$ that are used to obtain $\mathrm{TwBim}(C)$ are all “inner”.

Recall how this can be motivated # by applying Schreier theory to the Atiyah groupoid sequence.

An ordinary principal G-bundle

(2)$$\array{ P \\ \downarrow \\ X }$$

gives rise to the Atiyah sequence of groupoids (the “exponentiated Atiyah sequence” #)

(3)$$\array{ \mathrm{Ad} P &\to& \mathrm{Trans}(P) &\to& P_1(X) \\ = & & = & & = \\ P \times_G G &\to& P \times_G P &\to& X \times X } \,.$$

Crucially, the $G$ action on $G$ itself used in forming $P \times_G G$ is the adjoint action by $G$ on itself. This is where the inner automorphisms enter the game.

Schreier theory then suggests (as emphasized by Danny Stevenson # ) that a connection on $P$ is a pseudofunctor

(4)$$(\mathrm{tra},\mathrm{curv}_\mathrm{tra}) : X \times X \to \mathrm{AUT}(P \times_G G) \,,$$

where the target is the 2-groupoid whose objects are points in $X$, whose morphisms are fiber isomorphisms

(5)$$(P \times_G G)_x \to (P \times_G G)_y$$

and whose 2-morphisms are natural isomorphisms of these (where we regard the fibers - which are groups - as categories with a single object).

But this implies that an ordinary connection

(6)$$\mathrm{tra} : P_1(X) \to \mathrm{Trans}(X) \,,$$

acting by isomorphisms of fibers of $P$, will act by inner automorphisms on the $\times_G G$ factor.

More precisely, if we locally trivialize the pseudofunctor $(\mathrm{tra},\mathrm{curv}_\mathrm{tra})$, it indeed takes values in

(7)$$\Sigma(\mathrm{INN}(G))$$

instead of in all of

(8)$$\Sigma(\mathrm{AUT}(G)) \,.$$

Similar comments apply to the categorified setup, where $P$ is replaced by a principal 2-bundle and $G$ by a 2-group.

The 3-category of twisted bimodules that I am talking about # is supposed to be relevant for the associated 2-vector description of this principal setup.