# 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/abs/0803.3692 and the paper Higher Gauge theory: 2-connections by Baez and Schreiber https://arxiv.org/abs/hep-th/0412325 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. Jul 2, 2020 at 12:50
• @DavidRoberts Thanks.. I just said to be safe!! Jul 2, 2020 at 13:19
• @DavidRoberts: Why does it fails? Could you explain it bit more. Jun 13, 2021 at 22:04
• @Bumblebee the category of manifolds is a full subcategory of that of generalized smooth spaces, so it suffices to give a surjective submersion between manifolds with no smooth section. Take the squaring map $U(1) \to U(1)$. Jun 13, 2021 at 23:54

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.

1. 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. It follows that since $$G(x,y)$$ is a principal homogeneous $$G(x,x)$$-space, and $$H(Fx,Fy)$$ is a principal homogeneous $$H(Fx,Fx)$$-space, and $$G(x,y)\to H(Fx,Fy)$$ is smooth and equivariant with respect to the groups and the isomorphism between them, then $$G(x,y)\to H(Fx,Fy)$$ is a diffeomorphism (essentially because it factors through a sequence of three diffeomorphisms involving all the data so far). Picking single isomorphisms $$Fx\to x'$$ and $$Fy\to y'$$ in $$H$$, for arbitrary objects $$x',y'$$ in $$H$$, we get a diffeomorphism $$H(x',y') \simeq H(Fx,Fy)$$, and hence every hom-manifold of $$H$$ is diffeomorphic to one in $$G$$. However, the diffeomorphisms don't necessarily assemble into a smooth family of diffeomorphisms indexed by the square of the appropriate object manifold.
On the other hand, 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 arising from the Morita equivalence, not available here.

2. I don't know why the authors used strong equivalence, rather than weak equivalence, but it is the wrong notion. The weak version is correct, there are specific examples where it is clear that the weak version is in fact necessary. Wockel's paper detailing a principal 2-bundle over $$\Omega G$$, for $$G$$ a compact simple simply-connected Lie group, for example, states that it uses the strict notion of local triviality, but in fact it proves the weak version of local triviality, at the cost of some extra gymnastics of passing to a weakly equivalent Lie groupoid at one point.

• 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. Jul 2, 2020 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. Jul 2, 2020 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. Jul 3, 2020 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. Jul 3, 2020 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. Jul 3, 2020 at 2:26

Apologies for the late answer, I wish I'd found this earlier!

My MSc thesis was actually mainly devoted to developing a notion of Morita equivalence for diffeological groupoids! I don't think I'll have much to add to the answers by David Roberts and Joel Villatoro, and diffeological groupoids are clearly more specific than what you're looking for, but for what it's worth I'd like to contribute my two cents. You can find my thesis here: Diffeology, Groupoids & Morita Equivalence. I also wrote a paper about the main result (a "Morita Theorem"), which is available on the arXiv here: arXiv:2007.09901 (and which has been accepted for publication in the Cahiers)!

The approach I focussed on in my thesis is that of biprincipal bibundles. This is a slightly different point of view than your question, but it turns out that this gives a notion of Morita equivalence that is equivalent (no pun intended) to the definition using weak equivalences (see Section 5.1.3 of the thesis). The paper by Meyer and Zhu, mentioned in one of David Roberts' comments, also uses this point of view. I don't know to what extent their theory can be used for diffeology. I mainly wanted to focus on this part of your question:

"...I don't know the analogue of surjective submersion for generalized smooth spaces."

As David Roberts mentions, one sensible option is to replace surjective submersions by subductions. The entire point of my thesis is essentially to show that the Lie groupoid theory still works if you do this. (If you're looking for a reference on subductions, Section 2.6 of my thesis discusses them in some detail.) This also leads us to an important point that Joel Villatoro raises:

"By the way, for my part I would recommend taking surjective local subductions as your submersion for the diffeological category."

The motivation for this could be that the local subductions between smooth manifolds are exactly the surjective submersions (this is proved in the Diffeology textbook by Iglesias-Zemmour), hence directly generalising surjective submersions to the diffeological setting (which is what we were after!). However, as just mentioned, choosing subductions still makes everything work. Besides this, here are two more reasons to consider choosing subductions over local subductions:

1. There are naturally occurring bundle-like objects in diffeology that are subductions, but not local subductions. The main examples I have in mind are the internal tangent bundles. For example, if you consider diffeological space that is the union of the two coordinate axes in $$\mathbb{R}^2$$, its internal tangent bundle is 2-dimensional at the origin, but 1-dimensional everywhere else (see Example 3.17 in arXiv:1411.5425 by Christensen and Wu). The internal tangent bundle of this space is then a subduction, but not a local subduction (thanks to an argument by Christensen). If we want to study such objects to occur in our theory of diffeological groupoid bundles and Morita equivalence, we need to allow subductions and not just local subductions.

2. The main reason that we assume the source and target maps of a Lie groupoid $$G\rightrightarrows G_0$$ are surjective submersions is to ensure that the fibred product $$G\times_{G_0}G$$ of composable arrows is again a smooth manifold. Since the category $$\mathbf{Diffeol}$$ of diffeological spaces is (co)complete, this assumption becomes redundant. However, the source and target maps of a diffeological groupoid are always subductions!

I elaborate on the choice of subductions over local ones in Sections 4.2 and 4.4.3 in my thesis. My answer to your main question in the setting of diffeology is therefore the same as that of David Roberts: an appropriate notion of Morita equivalence for diffeological groupoids is exactly as for Lie groupoids, but with surjective submersion replaced by subduction.

As to a more structured approach to generalising surjective submersions to the setting of generalised smooth spaces (your actual main question): I believe that the subductions in the category $$\mathbf{Diffeol}$$ of diffeological spaces are exactly the strong epimorphisms, cf. Proposition 37 in arXiv:0807.1704 by Baez and Hoffnung. In a more abstract setting of generalised smooth spaces you could therefore consider trying to use the strong epimorphisms to replace the surjective submersions!

• Thanks a lot for the answer and the references! Feb 6, 2021 at 2:26
• I hope it was of some use! Feb 6, 2021 at 14:31

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. Jul 5, 2020 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) Jul 5, 2020 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. Jul 5, 2020 at 5:18
• ok, thanks, I'll think about that. Jul 5, 2020 at 9:11
• @JoelVillatoro Thanks for the answer and the link to your thesis! Jul 7, 2020 at 5:11