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 $T: phi circ psi Rightarrow id_H$ and $S: psi circ phi rightarrow id_G$. 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 Fully 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 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 Difffeological 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 Weak 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 Surjectiveness (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 Questions is the following:
What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?