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August 16, 2007

On Roberts and Ruzzi’s Connections over Posets

Posted by Urs Schreiber

I should say more about how

J. Roberts, G. Ruzzi
A cohomological description of connections and curvature over posets

which last time I mentioned here, relates to what I keep talking about.

Given any group GG, the authors of the above paper consider (I am slightly rephrasing this)

\bullet a one-object groupoid which they call 1G1G

\bullet a one-object strict 2-groupoid which they call I(1G)=2GI(1G) = 2G

\bullet a one-object strict 3-groupoid which they call I(2G)=3GI(2G) = 3G

They then discuss how a connection on a GG-bundle is, locally, essentially a functor taking values in 1G1G, while its curvature is a 2-functor taking values in 2G2G, while the Bianchi identity is a statement in 3G3G.

(Actually, J. Roberts and G. Ruzzi don’t write about 1- and 2-functors, but about cocycles on posets with values in 1- and 2-categories. But, as roughly indicated on their p. 22, below equation (43), there is a functorial picture underlying this.)

In The inner automorphism 3-group of a strict 2-group, David Roberts and I discuss how for G (2)G_{(2)} any strict 2-group (so it might in particular be just an ordinary (1-)group), there is a 3-group INN(G (2)) \mathrm{INN}(G_{(2)}) (a Gray-group, in general, meaning that everything is strict except possibly the 2-functoriality of the product 2-functor) which fits into an exact sequence 1Z(G (2))INN(G (2))AUT(G (2))OUT(G (2))1. 1 \to Z(G_{(2)}) \to \mathrm{INN}(G_{(2)}) \to \mathrm{AUT}(G_{(2)}) \to \mathrm{OUT}(G_{(2)}) \to 1 \,. (We consider this as an exact sequence on the level of the underlying strict 2-groupoids, but in fact all morphisms are morphisms of 3-groups.)

Among other things, the capital letters here are supposed to remind us that INN(G (2))\mathrm{INN}(G_{(2)}) does remember the center of the 2-group.

I am claiming that a good way to think of this is to realize INN(G (2))=T Id ΣG (2)(Aut(ΣG (2))) \mathrm{INN}(G_{(2)}) = T_{\mathrm{Id}_{\Sigma G_{(2)}}}(\mathrm{Aut}(\Sigma G_{(2)})) as the “tangent 2-groupoid” to the identity 2-functor on ΣG (2)\Sigma G_{(2)}.

In fact, we have an embedding [proposition 3, p. 11] T ΣG (2)INN(G (2)) T_\bullet \Sigma G_{(2)} \hookrightarrow \mathrm{INN}(G_{(2)}) whose image we call INN 0(G (2))\mathrm{INN}_0(G_{(2)}), and which is the really interesting part.

The fact that INN(G (2))\mathrm{INN}(G_{(2)}) “remembers the center” is important, since it makes INN 0(G (2))\mathrm{INN}_0(G_{(2)}) “contractible”. This, in turn, makes it play the role of the universal G (2)G_{(2)}-bundles. But it also makes it, from my perspective, the “right” codomain for 3-curvature, as described, for instance, in section 2.4 of Arrow-theoretic differential theory.)

Of course one might ask what one gets when we divide out the center Z 0(G (2))Z_0(G_{(2)}) from INN 0(G (2))\mathrm{INN}_0(G_{(2)}). I believe what one gets is exactly what John Roberts and G. Ruzzi find.

I believe that

\bullet the groupoid 1G1G is what I call ΣG\Sigma G, namely the one-object groupoid whose set of morphisms is the group GG (that much is obvious)

\bullet the 2-groupoid 2G2G is the quotient of (the suspension of) INN 0(G)\mathrm{INN}_0(G) by the categorical center Z 0(G)Z_0(G) 2G=Σ(INN(G)/Z(G)). 2 G = \Sigma( \mathrm{INN}(G)/Z(G) ) \,.

\bullet the 3-groupoid 3G3G is the quotient of (the suspension of) INN 0(INN(G)/Z(G))\mathrm{INN}_0(\mathrm{INN}(G)/Z(G)) by the center Z 0(INN(G))Z_0(\mathrm{INN}(G)) 3G=Σ(INN 0(2G)/Z 0(2G)). 3 G = \Sigma( \mathrm{INN}_0(2G) / Z_0(2G)) \,.

The first statement is obvious. The second statement amounts to just unwrapping all definitions. In principle the same is true for the third, though it’s becoming more involved to keep track of the details.

It helps to take equation (16) from p. 11 of J. Roberts’ and G. Ruzzi’s article and write the group elements considered there into the diagram for a 2-morphism in INN 0(INN(G))\mathrm{INN}_0(\mathrm{INN}(G)) as used by David Roberts and myself in our article:

τ γ Id g \array{ && \bullet \\ & {}^{\tau}\nearrow & \downarrow^{\gamma} & \searrow \\ \bullet &\Downarrow^{\mathrm{Id}}& \;\;\downarrow \stackrel{g}{\Rightarrow} & \downarrow \\ & \searrow & \downarrow & \swarrow \\ && \bullet }

Posted at August 16, 2007 10:17 AM UTC

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Re: On Roberts and Ruzzi’s Connections over Posets

By the way, I find it helpful to draw some diagrams which illustrate the notion of singular nn-simplex being used in that article.

We fix some poset KK, i.e. some category with at most one morphism from any object to any other.

Then a singular nn-simplex ff in KK is a functor of posets f:P{0,1,,n}K f : P\{0,1,\cdots,n\} \to K from the poset of subsets of {0,1,,n}\{0,1,\cdots,n\} to KK.

So, for example, a singular 0-simplex in KK is just an object x x in KK, while a singular 1-simplex bb in KK is not (as one might first think) a morphism xy x \to y in KK, but rather it’s a co-span in KK b 01 b 0 b 1. \array{ && b_{01} \\ & \nearrow && \nwarrow \\ b_0 &&&& b_1 } \,. Here b 01b_{01} is the object in KK corresponding to {0,1}\{0,1\} itself, while b 0b_0 and b 1b_1 are the objects corresponding to the subsets {0}\{0\} and {1}\{1\}, respectively. The two morphisms in KK need to exist in order to mimic the inclusions {0}{0,1}\{0\} \subset \{0,1\} and {1}{0,1}\{1\} \subset \{0,1\}.

A singular 3-simplex in KK in this sense is already almost impossible to draw using the means I have here. It is a “triple span of ordinary spans” in KK, in a way.

Now, since on the ordinals [n]:={0,1,,n} [n] := \{0,1,\cdots,n\} we have the usual degeneracy maps d i:[n1][n] d_i : [n-1] \to [n] and face maps s i:[n+1][n], s_i : [n+1] \to [n] \,, we can use these (or rather the pullback along them) to define face and degeneracy maps on our singular nn-simplices in KK.

We get for the incoming and the outgoing boundary of our singular 1-simplex bb given by the co-span b 01 b 0 b 1 \array{ && b_{01} \\ & \nearrow && \nwarrow \\ b_0 &&&& b_1 } the result 0b=b 1 \partial_0 b = b_1 and 1b=b 0. \partial_1 b = b_0 \,.

(So maybe I should have written b 1b_1 instead of b 0b_0 and vice versa. Or maybe not.)

Next, an important definition for Roberts&Ruzzi is that of an inflating singular nn-simplex in KK.

A singular 1-simplex bb, as above, is called inflating if there is a morphism 1b 0b \partial_1 b \to \partial_0 b in KK, i.e. if, in my above notation b 0b 1 b_0 \to b_1 exists, so if the “feet” of the co-span are connected b 01 b 0 b 1 \array{ && b_{01} \\ & \nearrow && \nwarrow \\ b_0 &&\to&& b_1 }

Posted by: Urs Schreiber on August 16, 2007 2:38 PM | Permalink | Reply to this

Re: On Roberts and Ruzzi’s Connections over Posets

Above I reviewed Roberts&Ruzzi’s definition of singular nn-simplices in some poset KK and drew some pictures.

In partiular, I said that a singular 1-simplex in KK is a cospan in KK.

Next, they define a path. This is nothing but a composble sequence of co-spans b 01 b 01 b 0 b 1=b 0 b 1 . \array{ && b_{01} &&&& b'_{01} \\ & \nearrow && \nwarrow && \nearrow && \nwarrow \\ b_0 &&&& b_1 = b'_0 &&&& b'_1 & \cdots } \,. There is the obvious notion of composing such paths.

Now, a GG-valued connection in the sense of Robberts&Ruzzi, as defined in their section 4, is essentially a functor from paths of “inflating” one simplices to what they call 1G1G, what I’d call ΣG\Sigma G: the one-object groupoid corresponding to the group GG.

But only essentially so. It is slightly more subtle than that, apparently.

Posted by: Urs Schreiber on August 16, 2007 3:08 PM | Permalink | Reply to this

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