Synthetic Transitions

Posted by Urs Schreiber

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On the occasion of the availability of the new edition of Anders Kock’s book on synthetic differential geometry ( $\to$ ) I want to go through an exercise which I wanted to type long time ago already.

I’ll redo the derivation of the transition laws for 2-connections ( $\to$ ) using synthetic language. This greatly simplifies the derivation, to the extent that the equations in terms of differential forms become almost identical to the diagrammatic equations that we derive them from.

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The main two facts of synthetic differential geometry that I’ll need are the following.

1) A function from $n$ -simplices to some group, which sends degenerate $n$ -simplices to the identity element is a (group-valued) $n$ -form. For the additive group $\mathbb{R}$ this is an ordiary $n$ -form. More generally, this function is to be thought of as the infinitesimal exponential of a Lie algebra-valued $p$ -form.

2) Given a group-valued 1-form

(1)

(x \to y) \mapsto \exp(A_{x,y}) \,,

its gauge covariant curvature is the 2-form

(2)

\begin{aligned} (x \to y \to z) \mapsto & \exp F(x,y,z) \\ &= \exp(A(x,y))\exp(A(y,z))\exp(A(z,x)) \end{aligned} \,.

So fix some space $X$ together with a good covering by open sets $U_i$ . On each $U_i$ we have a transport 2-functor which sends infinitesimal 2-simplices

(3)

\array{ x & \to & y & \to & z & \to & x \\ \downarrow &&&&&& \downarrow \\ x & \cellopts{\colspan{5} }\overset{\space{0}{0}{110}}{\to} & x }

to elements of the 2-group $H \to G$ :

(4)

\array{\arrayopts{\colalign{right center center center center center left}} \bullet & \overset{\exp A(x,y)}{\to} & \bullet & \overset{\exp A(y,z)}{\to} & \bullet & \overset{\exp A(z,x)}{\to} & \bullet \\ \mathrm{Id}\downarrow &&&\exp B(x,y,z)&&& \downarrow\mathrm{Id} \\ \bullet &\cellopts{\colspan{5}} \underset{\space{0}{0}{110}\mathrm{Id}\space{0}{0}{110}}{\to} & \bullet } \,.

From the source-target relation in 2-groups we read off that the 1-form $A$ and 2-form $B$ satisfy the fake flatness condition

(5)

F_A + B = 0 \,.

On double intersections, two such 2-functors are related by a pseudonatural transformation. This is a 1-form

(6)

(x \to y) \; \mapsto \; \array{\arrayopts{\colalign{right center left}} \bullet &\overset{\exp A_i(x,y)}{\to}& \bullet \\ g_{ij}(x) \downarrow &\exp a_{ij}(x,y)& \downarrow g_{ij}(y) \\ \bullet &\underset{\exp A_j(x,y)}{\to}& \bullet } \,.

Its values are composed horizontally using whiskering in $(H\to G)$ . We may hence think of this as a 1-form taking values in the semidirect product group $G \ltimes H$ . I’ll denote the $H$ -part of its synthetic curvature by $\exp F_{a_{ij},A_i}$ below.

The mere existence of the 2-cell on the right above means that

(7)

t(\exp a_{ij}(x,y)) \exp A_i(x,y) g_{ij}(x) = g_{ij}(y) \exp A_j(x,y) \,,

which is equivalent to the transition law for $A_i$ .

In order to qualify as a pseudonatural transformation, this 1-form must satisfy the equation

(8)

\begin{aligned} & \array{\arrayopts{\colalign{right center center center center center left}} \bullet & \overset{\exp A_i(x,y)}{\to} & \bullet & \overset{\exp A_i(y,z)}{\to} & \bullet & \overset{\exp A_i(z,x)}{\to} &\bullet \\ \mathrm{Id}\downarrow &&&\exp B_i(x,y,z)&&&\downarrow\mathrm{Id} \\ \bullet &\cellopts{\colspan{5}} \overset{\space{1}{0}{125}\mathrm{Id}\space{1}{0}{125}}{\to} &\bullet \\ g_{ij}(x) \downarrow &&&\mathrm{Id} &&&\downarrow g_{ij}(x) \\ \bullet &\cellopts{\colspan{5}} \underset{\space{1}{0}{125}\mathrm{Id}\space{1}{0}{125}}{\to} &\bullet } \\ \phantom{M}\\ =& \array{\arrayopts{\colalign{right center right center right center left}} \bullet &\overset{A_i(x,y)}{\to}& \bullet &\overset{A_i(y,z)}{\to}& \bullet &\overset{A_i(z,x)}{\to}&\bullet \\ g_{ij}(x)\downarrow &\exp a_{ij}(x,y)& g_{ij}(y)\downarrow &\exp a_{ij}(y,z)& g_{ij}(z)\downarrow &\exp a_{ij}(z,x)& \downarrow g_{ij}(x) \\ \bullet & \overset{\exp A_j(x,y)}{\to} & \bullet & \overset{\exp A_j(y,z)}{\to} & \bullet & \overset{\exp A_j(z,x)}{\to} &\bullet \\ \mathrm{Id}\downarrow &&&\exp B_j(x,y,z)&&&\downarrow\mathrm{Id} \\ \bullet & \cellopts{\colspan{5}} \underset{\space{1}{0}{150}\mathrm{Id}\space{1}{0}{150}}{\to} &\bullet } \end{aligned}

But using the two SDG facts stated above, together with the rules for composition in the 2-group $(H\to G)$ , this immediately says that

(9)

B_i = \alpha(g_ij)(B_j) + F_{a_{ij},A_{i}} \,.

Next, on triple intersections the pseudonatural transformations $\exp a_{ij}$ are related by an isomodification. This says that there is a 0-form

(10)

x \;\;\; \mapsto \;\;\; \array{\arrayopts{\colalign{right center left}} \bullet & \overset{\mathrm{Id}}{\to} &\bullet \\ \left. g_{ij}(x)\right\downarrow&& \cellopts{\rowspan{3}\rowalign{top}} \left\downarrow \phantom{\space{0}{40}{0}}g_{ik}(x)\right. \\ \bullet & f_{ijk}(x) \\ \left. g_{jk}(x)\right\downarrow & \\ \bullet & \underset{\mathrm{Id}}{\to} &\bullet }

which satisfies

(11)

\begin{aligned} &\array{\arrayopts{\colalign{right center left}} \bullet & \overset{\exp A_i(x,y)}{\to} &\bullet \\ g_{ij}(x)\downarrow &\exp a_{ij}(x,y)&| g_{ij}(y) \\ \bullet & \overset{\exp A_j(x,y)}{\to} &\bullet \\ g_{jk}(x)\downarrow &\exp a_{jk}(x,y)&\downarrow g_{jk}(y) \\ \bullet & \underset{\exp A_k(x,y)}{\to} &\bullet } \\ \phantom{M}\\ = & \array{\arrayopts{\colalign{right center center center center center left}} \bullet & \overset{\mathrm{Id}}{\to} & \bullet & \overset{\exp A_i(x,y)}{\to} & \bullet & \overset{\mathrm{Id}}{\to} &\bullet \\ \left. g_{ij}(x)\right\downarrow &&\cellopts{\rowspan{3}\rowalign{top}}\left\vert \space{0}{40}{0}\right.&&\cellopts{\rowspan{3}\rowalign{top}}\left\vert \space{0}{40}{0}\right.&&\left\vert g_{ij}(y)\right. \\ \bullet & f_{ijk}(x) & \exp a_{ik}(x,y) & f_{ijk}^{-1}(y) &\bullet \\ \left. g_{jk}(x)\right\downarrow &&&&\left\downarrow g_{jk}(y)\right. \\ \bullet & \overset{\mathrm{Id}}{\to} & \bullet & \overset{\exp A_k(x,y)}{\to} & \bullet & \overset{\mathrm{Id}}{\to} &\bullet } \end{aligned}

Here we just need to collect exponents to get the expected transition law.

Finally, there is the tetrahedron law on quadruple intersections. This only involves 0-forms and SDG does not tell us anything here that we did no know before.

In conclusion, SDG here mainly helps handling the otherwise somewhat subtle curvature $F_{a_{ij},A_{i}}$ in the transition on double intersections, and makes reading off the law on triple intersections a little more systematic.

Posted at July 28, 2006 7:58 PM UTC

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Re: Synthetic Transitions

So, Urs, when are the pseudofunctor versions of the above going to be sprung on us?

Posted by: DM Roberts on July 31, 2006 1:20 AM | Permalink | Reply to this

when are the pseudofunctor versions of the above going to be sprung on us

I think the above formulation seamlessly generalizes to this case in an obvious way.

I am still sort of on vacation. I’ll type this stuff cleanly when I am back in my office.

With respect to what we talked about in Vienna, however, I must say that I don’t think that any weakening will completely remove the “fake flatness” constraint. The weak 2-functors which we looked at in Vienna have a weaker fake flatness constraint, but different from the Breen-Messing degrees of freedom.

Posted by: urs on August 7, 2006 10:00 AM | Permalink | Reply to this

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July 28, 2006