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.(T*hough 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?*