I think the paper is very nice. Here are some of my thoughts and suggestions.

Section 3 definition 1: I think it would be clearer to say “…with a chosen basis such that the action of G maps basis vectors to basis vectors.” Since any representation of G can be given a basis as a vector space.

Section 4: I feel like it would be helpful, when defining a bicategory, to mention that morphisms can be composed. Likewise in the examples.

middle of p6: Have you defined what it means for a span to be “irreducible”?

Sections 5 and 12 are very different in character from the rest of the paper, and I found them next-to-impossible to understand without a lot of background at my fingertips that I don’t have. I’m sure you don’t want to spend the time to give people the necessary background, but you might want to at least warn people that the necessary background is different in different sections.

Section 6, first paragraph: I don’t think that functor was defined in section 5. Again on p9 there is an I think misplaced reference to section 5.

Section 6, Definition 3: I would put a comma after “where $g\cdot x = x'$”.

Bottom of p9: “arbitrary spans of groupoids form a bicategory (after taking isomorphism classes)” is not really clear to me; what are you taking isomorphism classes of? Perhaps it has something to do with the following paragraph? If so, maybe a rearrangement would help.

And my only comment with mathematical content: I don’t think that jointly faithful spans are closed under composition. Let $G$ be any groupoid and consider the spans $1 \leftarrow G \to G$ and $G\leftarrow G \to 1$. They are each jointly faithful, but their composite is $1\leftarrow G \to 1$, which is not. I think you need to perform some sort of “faithful reflection” in order to define this bicategory (also see below).

Middle of p10: I would leave out the comma after “whereas.”

Bottom of p10: I don’t think “note that” is quite appropriate here; it seems to me that really you are *defining* what a “categorified vector” is.

The notion of “essential inverse image” is confusing me again; I remember talking about this with John and maybe deciding that the terminology was off? At issue is the point that the morphisms between objects of $p^{-1}(x)$ don’t need to respect the isomorphisms $p(v) \cong x$ in $\mathcal{G}$, whereas in the “essential inverse image” I would expect them to have to. Maybe something like “full inverse image”?

Thinking about this some more, it occurs to me that maybe it would be clearer to phrase the whole theory in terms of *connected components* of a groupoid, rather than *isomorphism classes of objects*. Of course, the two are in canonical bijection, but for instance, the full inverse image of an object looks kind of weird to me (since the morphisms don’t respect the isomorphisms as above), but as the *ordinary* inverse image of a connected component it makes much more sense.

I felt a little confused at the end of Section 6, and reading back I think it was because the action groupoid $X\sslash G$ was introduced at the beginning of that section as motivation for groupoids, but it never came back in that section. Maybe some more signposting would help. I think I was also expecting some comments to the effect that the operation $S\mathcal{V}$ is a linear operator, and that this in fact defines a functor.

Bottom of p12: I would leave out the comma after “Any functor $f\colon G\to H$”.

Top of p13: Maybe the words “Kan extension” should be said?

Section 7 Lemma 9: This looks like a special case of a result that $G Set / Z \simeq \hat{Z\sslash G}$ for any $G$-set $Z$.

How committed are you to the term “nice topos”? I don’t find it especially evocative myself. I also wonder whether there is really any point to observing that these categories are topoi? The notion of geometric morphism is mentioned early on, but doesn’t really seem to enter the picture at all—the morphisms you use between these categories are just the cocontinuous functors. I think it might be clearer not to even mention that they are topoi, and just give them a more precise name (something like “presheaf categories of groupoids” but maybe shorter—“groupoidified vector space”?).

Top of p15: I like the point of view that a groupoid is a categorified “basis” and its presheaf category is the corresponding categorified “vector space.” Here is something else that might be helpful to say here: a groupoid can be recovered up to equivalence from its presheaf category, just as a basis can be recovered up to isomorphism from its vector space, but in each case the equivalence/isomorphism is non-canonical.

Section 10: I was a little confused here. It seems to me that in Section 8 we already defined a functor in the *other* direction $Span\to Nice$. One presumes that the functor being defined here is the inverse of that one, but maybe this connection should be made explicitly.

Also, it might be helpful to remind the reader here that the unadorned name “$Span$” is being used to denote the bicategory of groupoids and jointly faithful spans.

I guess the notions of topos theory do enter briefly with the idea of “points” of a topos. However, I think there’s a more direct way to recover a groupoid $G$, up to equivalence, from its presheaf category $\hat{G}$: it consists of the objects of $\hat{G}$ which are “small-projective” in that $hom_{\hat{G}}(X,-)$ preserves all colimits. And once you have that, then the “free cocompletion” yoga gives you the rest of the functor, as on p20 (although I think there is also a hidden use here of the fact that any groupoid is isomorphic to its opposite). Perhaps you have some other reason to want to use topos-theoretic language that isn’t apparent from this expository paper, though.

It might also be helpful to observe that a cocontinuous functor between presheaf categories is the same thing as a profunctor, so that your bicategory $Nice$ is equivalent to the bicategory of groupoids and profunctors between them. Then the category-of-elements construction becomes the usual interpretation of a profunctor as a two-sided discrete fibration, and the “faithful reflection” used to compose spans of groupoids is a special case of the composition of such two-sided fibrations. (Noting that every functor between groupoids is a Grothendieck fibration in the loose sense of Street.)

Finally, end of Section 11, Claim 17: I think it would be helpful to spell out a bit more explicitly what that equivalence means. I.e. it means that given two permutation representations $X$ and $Y$, the vector space of intertwining operators between them can be constructed as the degroupoidification of the groupoid whose presheaf category is the category of spans of G-sets from $X$ to $Y$ — which is just the groupoid $(X\times Y)\sslash G$. Am I parsing that correctly?

This is neat stuff!

## Re: New Paper on the Hecke Bicategory

Very nice! A couple of trivialities:

p. 7 “…we are claiming that THEIR is a way…” typo.

p. 12 “One answer is that while the morphisms between categories are functors, the morphisms between topoi require specified extra structure and must satisfy extra properties. Such a morphism is called a geometric morphism”. That makes it sound as though geometric morphisms are the only choice when dealing with topoi, when in fact there are also logical functors.