TITLE: What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?
QUESTION [5 upvotes]: Let $G$ and $H$ be Lie groupoids. We know that there are two notions of equivalences of Lie groupoids:

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.

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:

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?

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.

REPLY [6 votes]: 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:

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.

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!