### 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 $n$-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

its gauge covariant curvature is the 2-form

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

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

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

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

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

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

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

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

which satisfies

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.

## Re: Synthetic Transitions

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

D