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July 28, 2006

Synthetic Transitions

Posted by Urs Schreiber

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.

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

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

2) Given a group-valued 1-form

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

its gauge covariant curvature is the 2-form

(2)(xyz) expF(x,y,z) =exp(A(x,y))exp(A(y,z))exp(A(z,x)). \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 XX together with a good covering by open sets U iU_i. On each U iU_i we have a transport 2-functor which sends infinitesimal 2-simplices

(3)x y z x x x \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 HGH \to G:

(4) expA(x,y) expA(y,z) expA(z,x) Id expB(x,y,z) Id Id . \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 AA and 2-form BB satisfy the fake flatness condition

(5)F A+B=0. F_A + B = 0 \,.

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

(6)(xy) expA i(x,y) g ij(x) expa ij(x,y) g ij(y) expA j(x,y) . (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 (HG)(H\to G). We may hence think of this as a 1-form taking values in the semidirect product group GHG \ltimes H. I’ll denote the HH-part of its synthetic curvature by expF a ij,A i\exp F_{a_{ij},A_i} below.

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

(7)t(expa ij(x,y))expA i(x,y)g ij(x)=g ij(y)expA j(x,y), 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 iA_i.

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

(8) expA i(x,y) expA i(y,z) expA i(z,x) Id expB i(x,y,z) Id Id g ij(x) Id g ij(x) Id M = A i(x,y) A i(y,z) A i(z,x) g ij(x) expa ij(x,y) g ij(y) expa ij(y,z) g ij(z) expa ij(z,x) g ij(x) expA j(x,y) expA j(y,z) expA j(z,x) Id expB j(x,y,z) Id Id \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 (HG)(H\to G), this immediately says that

(9)B i=α(g ij)(B j)+F a ij,A i. B_i = \alpha(g_ij)(B_j) + F_{a_{ij},A_{i}} \,.

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

(10)x Id g ij(x) g ik(x) f ijk(x) g jk(x) Id 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) expA i(x,y) g ij(x) expa ij(x,y) |g ij(y) expA j(x,y) g jk(x) expa jk(x,y) g jk(y) expA k(x,y) M = Id expA i(x,y) Id g ij(x) | | |g ij(y) f ijk(x) expa ik(x,y) f ijk 1(y) g jk(x) g jk(y) Id expA k(x,y) Id \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 iF_{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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2 Comments & 2 Trackbacks

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

Re: Synthetic Transitions

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
Read the post On n-Transport, Part II
Weblog: The n-Category Café
Excerpt: Smooth transport, differentials of transport, and nonabelian differential cocycles.
Tracked: August 21, 2006 8:42 PM
Read the post Kock on 1-Transport
Weblog: The n-Category Café
Excerpt: A new preprint by Anders Kock on the synthetic formulation of the notion of parallel transport.
Tracked: September 8, 2006 5:50 PM

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