### Large Smooth Categories

#### Posted by Urs Schreiber

Behind the scenes, I am having a long email discussion with Bruce Bartlett about some puzzling subtleties in the definition of smooth categories.

As Bruce rightly emphasized, we need to be careful with comparing the following two definitions

1) a category internal to the category of Chen-smooth spaces

2) a stack over manifolds – which is the same as a category fibered over manifolds.

It gets particularly subtle when the categories in question are *large*. I am unsure about some subtle details of that. Here are some questions.

**Reminder: Chen-smooth spaces (Diffeology).**

For my purposes here, a “smooth space” is a special sort of sheaf (of sets) on manifolds, namely a sheaf $\mathbf{X}$ such that for each manifold $U$ the set $\mathbf{X}(U)$ is a subset of the set of maps-of-sets $U \to X \,,$ where $X$ is any fixed set.

Such a sheaf is known as a Chen-smooth structure on the set $X$. (See Quantization and Cohomology (Week 20)). In this context the elements of $\mathbf{X}(U)$ are called *plots* of $X$.

**A puzzling example: Internal smooth structure on $\mathrm{Vect}$?**

In our discussion, I imagined realizing the category of vector spaces as a category internal to smooth spaces by defining the following sheaf:

For each manifold $U$, the set $\mathbf{Mor(Vect)}(U)$ is that of maps $U \to \mathrm{Mor}(\mathrm{Vect})$ which arise as component maps of smooth morphisms $\array{ V_1 &&\stackrel{\phi}{\to}&& V_2 \\ &\searrow && \swarrow \\ && U }$ of smooth vector bundles $V_1 \to U$ and $V_2 \to U$ over $U$.

I did notice that this sounds like there might be a subtlety hidden. Bruce very much emphasized that we need to be careful concerning sets and classes here. I realize I don’t have this sufficiently under control, conceptually.

Does the above definition of a sheaf of plots make sense? If not, is there any chance of fixing it, say by mumbling something about Grothendieck universes? Whatever the answer is, I would very much appreciate a detailed explanation.

**A non-puzzling example: $\mathrm{Vect}$ as a smooth stack.**

Alternatively, we may equip $\mathrm{Vect}$ with a notion of smooth structure by realizing it as the value over the point of the standard stack $\mathbf{Vect}$ which sends manifolds to the category of smooth vector bundles over them.

This should be the same, dually, as a fibred category: thinking of $\mathbf{Vect}$ as the category of smooth vector bundles over all possible manifolds (morphisms are morphisms of bases spaces together with morphisms of of one bundle with the pullback of the other), together with the obvious forgetful functor $\mathbf{Vect} \to \mathrm{Manifolds} \,,$ which is a fibred category.

**Questions.**

Before I completely abandon the idea of being able to realize $\mathrm{Vect}$ internal to Chen-smooth spaces, I would like to understand precisely what is going on.

How do I decide if $\mathrm{Mor}(C)$ is a set or a class? If it is a class, are maps from sets into it a set or a class? Is there any chance to get a sheaf (of sets) of maps into $\mathrm{Mor}(\mathrm{Vect})$?

**The issue in full generality.**

What I really would like to understand one fine day (of course Bruce already made a couple of remarks on that) is how the following two concepts are related, generally, for $S$ any site:

A) categories internal to sheaves over $S$

B1) categories fibred over $S$

B2) stacks on $S$

Can anyone educate me here? Thanks!

## Re: Large Smooth Categories

I am confused. Is Vect a stack? For example, Vect(point) doesn’t look like a groupoid. Nor do I see how the descent would work. It’s true that Vect^n, the category of vector bundles of a fixed rank n is a nice Artin stack, but Vect?