On Roberts and Ruzzi's Connections over Posets

August 16, 2007

On Roberts and Ruzzi’s Connections over Posets

Posted by Urs Schreiber

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I should say more about how

J. Roberts, G. Ruzzi
A cohomological description of connections and curvature over posets
math/0604173

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

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

$\bullet$ a one-object groupoid which they call $1G$

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

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

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

(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 $G_{(2)}$ any strict 2-group (so it might in particular be just an ordinary (1-)group), there is a 3-group $\mathrm{INN}(G_{(2)})$ (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 $1 \to Z(G_{(2)}) \to \mathrm{INN}(G_{(2)}) \to \mathrm{AUT}(G_{(2)}) \to \mathrm{OUT}(G_{(2)}) \to 1 \,.$ (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 $\mathrm{INN}(G_{(2)})$ does remember the center of the 2-group.

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

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

The fact that $\mathrm{INN}(G_{(2)})$ “remembers the center” is important, since it makes $\mathrm{INN}_0(G_{(2)})$ “contractible”. This, in turn, makes it play the role of the universal $G_{(2)}$ -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 $Z_0(G_{(2)})$ from $\mathrm{INN}_0(G_{(2)})$ . I believe what one gets is exactly what John Roberts and G. Ruzzi find.

I believe that

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

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

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

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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 $\mathrm{INN}_0(\mathrm{INN}(G))$ as used by David Roberts and myself in our article:

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

Posted at August 16, 2007 10:17 AM UTC

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Re: On Roberts and Ruzzi’s Connections over Posets

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

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

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

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

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

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

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

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

Next, an important definition for Roberts&Ruzzi is that of an inflating singular $n$ -simplex in $K$ .

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

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

Re: On Roberts and Ruzzi’s Connections over Posets

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

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

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

Now, a $G$ -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 $1G$ , what I’d call $\Sigma G$ : the one-object groupoid corresponding to the group $G$ .

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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August 16, 2007