# What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

Let $$G$$ and $$H$$ be Lie groupoids. We know that there are two notions of equivalences of Lie groupoids:

1. Strongly equivalent Lie groupoids: (My terminology)

A homomorphism $$\phi:G \rightarrow H$$ of Lie groupoids is called a strong equivalence if there is a Lie groupoid homomorphism $$\psi:H \rightarrow G$$ and natural transformation of Lie groupoid homomorphism $$T: \phi \circ \psi \Rightarrow \mathrm{id}_H$$ and $$S: \psi \circ \phi \rightarrow \mathrm{id}_G$$. In this case $$G$$ and $$H$$ is said to be strongly equivalent Lie groupoids.

1. Weakly Equivalent or Morita Equivalent Lie groupoids :

A homomorphism $$\phi:G \rightarrow H$$ of Lie groupoids is called a weak equivalence if it satisfies the following two conditions where $$H_0$$, $$H_1$$ are object set and morphism set of Lie groupoid H respectively. Similar meaning holds for symbols $$G_0$$ and $$G_1$$. Here symbols $$s$$ and $$t$$ are source and target maps respectively. The notation $$pr_1$$ is the projection to the first factor from the fibre product. from t. Here the condition (ES) says about essential surjectivity and the condition (FF) says about full faithfulness.

One says that two Lie Groupoids $$G$$ and $$H$$ are weakly equivalent or Morita equivalent if there exist weak equivalences $$\phi:P \rightarrow G$$ and $$\phi':P \rightarrow H$$ for a third Lie groupoid $$P$$.

(According to https://ncatlab.org/nlab/show/Lie+groupoid#2CatOfGrpds one motivation for introducing Morita equivalence is the failure of the axiom of choice in the category of smooth manifolds )

What I am looking for:

Now let we replace $$G$$ and $$H$$ by categories $$G'$$ and $$H'$$ which are categories internal to a category of generalized smooth spaces (For example, category of Chen spaces or category of diffeological spaces... etc). For instance, our categories $$G'$$ , $$H'$$ can be path groupoids.

Analogous to the case of Lie groupoids I can easily define the notion of Strongly equivalent categories internal to a category of generalized smooth spaces.

Now if I assume that the axiom of choice fails also in the category of generalized smooth spaces then it seems reasonable to introduce a notion of weakly equivalent or some sort of Morita equivalent categories internal to a category of generalized smooth spaces.

But it seems that we cannot directly define the notion of weakly equivalent or Morita equivalent categories internal to a category of Generalized Smooth Spaces in an analogous way as we have done for Lie Groupoids. Precisely in the condition of essential surjectivity (ES) we need a notion of surjective submersion but I don't know the analogue of surjective submersion for generalized smooth spaces

I heard that Morita equivalence of Lie groupoids are actually something called "Anaequivalences" between Lie groupoids.(Though I don't have much idea about anafunctors and anaequivalences).

So my guess is that the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces has something to do with anaequivalence between categories internal to a category of generalized smooth spaces. Is it correct?

My Question is the following:

What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

EDIT:

In the comments section after the answer by David Roberts we also had a discussion on the following two questions:

1. Let $$F: G \rightarrow H$$ be a Lie groupoid Homomorphism such that $$F$$ is fully faithful and essentially surjective as a functor between the underlying categories. Let us also assume the $$G$$ and $$H$$ are not Morita Equivalent. Then what are the properties that Lie groupoids $$G$$ and $$H$$ has in common apart from the trivial fact that they have equivalent underlying categories?

2. In papers on Higher gauge theory like Principal 2 bundles and their Gauge 2 groups by Christoph Wockel https://arxiv.org/pdf/0803.3692.pdf and the paper Higher Gauge theory 2-connections by Baez and Schreiber https://arxiv.org/pdf/hep-th/0412325.pdf why strong equivalence is preferred over weak equivalence in the notion of Local triviality for Principal-2 bundles over a manifold?(Here equivalence means equivalence between categories internal to a category of generalized smooth spaces)

My deep apology for asking two sufficiently different (from the original) question in the comments section.

Thank you.

• "Now if I assume that the Axiom of choice fails also in the category of generalized smooth spaces" <-- it definitely fails, no need to assume it. – David Roberts Jul 2 '20 at 12:50
• @DavidRoberts Thanks.. I just said to be safe!! – Adittya Chaudhuri Jul 2 '20 at 13:19

In place of a detailed answer, let me point to Internal categories, anafunctors and localisations, but more specific to your case is the diffeological groupoids in Smooth loop stacks of differentiable stacks and gerbes.

To answer a more specific question here:

Precisely in the condition of essential surjectivity (ES) we need a notion of surjective submersion but I don't know the analogue of surjective submersion for generalized smooth spaces

For diffeological spaces, and I would imagine any generalised smooth spaces that can be considered as perhaps special sheaves on the category of manifolds, the type of map you want is subduction. I don't have a good canonical (nLab!) reference, but there some discussion in this answer, and such maps appear in Konrad Waldorf's work on gerbes. Subductions is also discussed (briefly) in the second linked paper above.

• I lost my comment I made. Basically, weak equivalence is the right one, strong equivalence is too strong. I don't know why the papers you give use strong equivalence, apart from they fact they are reasonably early in the study of higher geometry. Toby Bartels, in his thesis with John Baez on 2-bundles, uses the weak notion. This is roughly contemporary, but separate from, the technical development in the Baez–Schreiber paper. – David Roberts Jul 2 '20 at 14:57
• There are examples of 2-bundles that need weak equivalence to get local triviality, for instance in A topological fibrewise fundamental groupoid I show how one can take a homotopy local trivial fibration and get a topological 2-bundle whose fibres are the fundamental groupoids of the original fibration. This would fail if strong equivalence was used. – David Roberts Jul 2 '20 at 14:58
• For your first question, they have homeomorphic topological spaces of orbits (and this is induced by the given functor), and diffeomorphic hom-manifolds $G(x,y) \xrightarrow{\simeq} H(Fx,Fy)$. In particular the automorphism groups of $x$ and $Fx$ are isomorphic as Lie groups. But it doesn't make sense to ask what $G$ and $H$ can have in common by virtue of the fact they are Morita equivalent _via $F$, but $F$ doesn't preserve it. For your second question: nothing, basically. – David Roberts Jul 3 '20 at 0:48
• Actually, I misread your second question. Since you have group homomorphisms $G(x,x) \to H(Fx,Fx)$ that are smooth and bijective, and assuming you are dealing with finite-dimensional Lie groupoids, these are isomorphisms on the associated Lie algebras, hence a bijective local diffeomorphism, hence a diffeomorphism. But while you should get a bijective map between the topological quotient spaces of orbits, it doesn't necessarily follow it's a homeomorphism, since the construction of the inverse uses continuous local sections, not available here. – David Roberts Jul 3 '20 at 2:24
• But new/additional questions you should not leave in the comments. In future, it's worth editing your original question, or asking a new one, if it's sufficiently different. – David Roberts Jul 3 '20 at 2:26

I know this is a little late but I discuss this in the first two chapters of my thesis here:

https://arxiv.org/abs/1806.01939

Basically, as you mentioned, what you need is a notion of surjective submersion which generalizes surjective submersions of smooth manifolds. Once you have that, the definition falls out of it by the usual theory. In my thesis, I talk about the case where we are given a site, equipped with a distinguished set of morphisms which are the 'submersions'. That distinguished set of morphisms has to have a few properties which you can find in the definition of good site in the first chapter of my thesis.

The short version is that your category needs to be reasonably compatible with the grothendiek topology (i.e. morphisms are characterized by locally) and your notion of surjective submersions should generate the Grothendiek topology.

The other main property is that if you have a bunch of submersions $$s_i \colon P_i \to B$$ with images covering $$B$$ and some coherent transition maps, you should be able to glue the $$P_i$$ into a single submersion $$P \to B$$. Lastly, you need that if $$f \circ g$$ is a submersion then $$f$$ is a submersion.

The main difference between my thesis and the paper of Roberts and Vozzo is that they focus on when the category can be localized by a category of fractions method. My thesis is mainly concerned with constructing a 2-categorical equivalence between bibundles of internal groupoids and presentable sheaves of groupoids.

By the way, for my part I would recommend taking surjective local subductions as your submersion for the diffeological category. That's my two-cents anyway.

• Thanks for linking to your thesis! Meyer and Zhu did a lot of this type of thing in tac.mta.ca/tac/volumes/30/55/30-55abs.html, if that useful for you. Might I point out Definition 1.3.2 (GS5) goes back to a paper of Bénabou from 1975? your category needs to be reasonably compatible with the grothendiek topology (i.e. morphisms are characterized by locally) <-- looks like asking for the (pre)topology to be subcanonical, to me. – David Roberts Jul 5 '20 at 0:02
• The other main property is that if you have a bunch of submersions ... <- I can't find this in your thesis. It seems like asking for a superextensive site (see eg the appendix of arxiv.org/abs/1101.2363) – David Roberts Jul 5 '20 at 0:06
• You're right about the sub-canonical thing. At the time I wrote the thesis I had misunderstood the definition of sub-canonical and thought that it implied that arbitrary gluing of objects was possible, which seemed to exclude the category of Hausdorff manifolds. The second property is encoded in the condition that the functor F:Sub to C is a stack. I dont think it's the same as being superextensive. – Joel Villatoro Jul 5 '20 at 5:18
• ok, thanks, I'll think about that. – David Roberts Jul 5 '20 at 9:11