**Thanks** for this paper!
This is a remarkable piece of exposition.

I need to take a closer look at chapter 4, and I’m looking forward to chapter 5, but lest I forget I’ll post what caught my eye so far:

A group is a category with one object: When I first heard this statement I asked myself, what role does the object play? If you identify the objects with their identiy morphism, the answer is obvious, but since you encourage your readers to distinguish the two (objects are states, morphisms are processes), a little hint that the object is inserted only to provide the morphisms = group elements with a domain and a codomain could turn out to be helpful.

A groupoid is not the first example of a category where arrows aren’t functions in the usual sense that comes to my mind. Probably because my mind makes group elements act on the group by left-/right multiplication or conjugation automatically. The very first example I come up with are cobordisms.

lazy paths: that is a particularly good example of how names of new notions should be chosen: short, intuitive, easy and unique.

p.25:

But for a finite-sized surface, this formula is no good, since it involves adding up B at different points, which is not a gauge-invariant thing to do.

Since we are talking about principal bundles, gauge-invariant means invariant with respect to the action of the gauge group on the fibers, right? But then the addition of “B at different points” is simply not defined, it has nothing to do with gauge-invariance, or has it?

p.30:

Suppose we have a representation of a Lie group G on a vector space H. We can regard H as an abelian Lie group…

Filling in some dots: H is supposed to be finite dimensional, we equip it with the unique topology that turns it into a *topological* vector space, which turns it into a Lie group with regard to vector addition, correct?

p.31:

In short, it appears that the 2-category of representations of the Poincare 2-group gives a background-free description of quantum field theory on 4d Minkowski spacetime.

I have to admit that I do not understand the preceding paragraph, so let’s simply say that I’m surprised to read about a *background-free* description on a *specific background*. (I’m aware that we are in immediate danger of opening a bag of worms here).

some typos:

superfluous “but”, p.9: “Any Lie group is a smooth groupoid, but and so is the path groupoid of any smooth manifold.”

missing article “a”, p.10: “Conversely, suppose we have smooth functor hol.”

same, p.13: “In a 2-category, we visualize the 2-morphisms as little pieces of 2-dimensional surface:”

same, p.20: “it is map between 2-categories that preserves everything in sight.”

should be “paper”, p.29: “see the papery by Gotay”

[**John Baez:** Thanks for catching all these typos! I also added a sentence about the role played by the one object in a group, at the place where I first raise this idea. There is a lot to say about this, but I’m trying to keep the presentation very brisk, so I confined myself to the most elementary observation, as you suggest: “The morphisms of this category are the elements of the group. The object is there just to provide them with a source and target.”

Your comment about how we regard a finite-dimensional vector space as an abelian Lie group is correct. Inserting the finite-dimensionality assumption led me to spot a more serious mistake, which I also fixed.

Since at this point in the exposition we are dealing with a *trivial* principal *G*-bundle, it would make sense to add elements lying in different fibers of an associated vector bundle. But this is not a gauge-invariant thing to do: it depends on the choice of trivialization. It would cease to make sense as soon as we move to nontrivial bundles. Since it gets tiresome to explain precisely why an idea is dumb, I will change the sentence in question to: “But for a finite-sized surface, this formula is no good, since it involves adding up *B* at different points, which is not a smart thing to do.”

I may release a number of drafts, but the latest up-to-date version will always be here.]

## Re: An Invitation to Higher Gauge Theory

Thanksfor this paper! This is a remarkable piece of exposition.I need to take a closer look at chapter 4, and I’m looking forward to chapter 5, but lest I forget I’ll post what caught my eye so far:

A group is a category with one object: When I first heard this statement I asked myself, what role does the object play? If you identify the objects with their identiy morphism, the answer is obvious, but since you encourage your readers to distinguish the two (objects are states, morphisms are processes), a little hint that the object is inserted only to provide the morphisms = group elements with a domain and a codomain could turn out to be helpful.

A groupoid is not the first example of a category where arrows aren’t functions in the usual sense that comes to my mind. Probably because my mind makes group elements act on the group by left-/right multiplication or conjugation automatically. The very first example I come up with are cobordisms.

lazy paths: that is a particularly good example of how names of new notions should be chosen: short, intuitive, easy and unique.

p.25:

Since we are talking about principal bundles, gauge-invariant means invariant with respect to the action of the gauge group on the fibers, right? But then the addition of “B at different points” is simply not defined, it has nothing to do with gauge-invariance, or has it?

p.30:

Filling in some dots: H is supposed to be finite dimensional, we equip it with the unique topology that turns it into a

topologicalvector space, which turns it into a Lie group with regard to vector addition, correct?p.31:

I have to admit that I do not understand the preceding paragraph, so let’s simply say that I’m surprised to read about a

background-freedescription on aspecific background. (I’m aware that we are in immediate danger of opening a bag of worms here).some typos:

superfluous “but”, p.9: “Any Lie group is a smooth groupoid, but and so is the path groupoid of any smooth manifold.”

missing article “a”, p.10: “Conversely, suppose we have smooth functor hol.”

same, p.13: “In a 2-category, we visualize the 2-morphisms as little pieces of 2-dimensional surface:”

same, p.20: “it is map between 2-categories that preserves everything in sight.”

should be “paper”, p.29: “see the papery by Gotay”

[

John Baez:Thanks for catching all these typos! I also added a sentence about the role played by the one object in a group, at the place where I first raise this idea. There is a lot to say about this, but I’m trying to keep the presentation very brisk, so I confined myself to the most elementary observation, as you suggest: “The morphisms of this category are the elements of the group. The object is there just to provide them with a source and target.”Your comment about how we regard a finite-dimensional vector space as an abelian Lie group is correct. Inserting the finite-dimensionality assumption led me to spot a more serious mistake, which I also fixed.

Since at this point in the exposition we are dealing with a

trivialprincipalG-bundle, it would make sense to add elements lying in different fibers of an associated vector bundle. But this is not a gauge-invariant thing to do: it depends on the choice of trivialization. It would cease to make sense as soon as we move to nontrivial bundles. Since it gets tiresome to explain precisely why an idea is dumb, I will change the sentence in question to: “But for a finite-sized surface, this formula is no good, since it involves adding upBat different points, which is not a smart thing to do.”I may release a number of drafts, but the latest up-to-date version will always be here.]