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1. We have the following inclusions of full subcategories $$Mfd \subset Diff \subset Sh \subset PSh$$ where $Mfd$ is the category of smooth finite dim manifolds, $Diff$ are diffeological spaces (i.e. concrete sheaves on cartesian spaces), Sh are sheaves on cartesian spaces and $PSh$ are presheaves on cartesian spaces. The last two inclusions are reflexive.

Lets us first have a closer look at the inclusion $Sh \subset PSh$. Following the same vein of argument as above, there is a priori no reason to work with $Sh$ instead of $PSh$ since both categories are equally nice (topoi) and the definition of a presheaf is clearly simpler than that of a sheaf. But there are some colimits in $Mfd$ that we really like, namely the coequalizer diagram correspoding to an open cover $(U_\alpha)$ of a manifold $M$. Under the inclusion of $Mfd$ into $PSh$ this is not a coequalizer anymore, in other words: If we glue open sets in $PSh$ together we do not get the same thing that we get when glueing together as manifolds. This defect is exactly cured by the sheaf property. That means restricting to the smaller subcategory $Sh \subset PSh$ the colimits change such that gluing of open sets behaves as nice as in manifolds. The punchline is that the restriction to $Sh$ provides the category with the "right" coequalizers of open sets.

Now lets turn towards the inclusion $Diff \subset Sh$. The situation is exactly the same as before. Limits in $Diff$ are computed as Limits in $Sh$ (and hence also $PSh$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.

2. If we want to "make" geometry over diffeological spaces it turns out that there are two possible definition of principal bundles:

The first topology has an obvious analogue on the category $Sh$ of all sheaves but the second crucially uses the underlying topological space of a diffeological space.

First, we have the following inclusions of full subcategories $$\mathit{Mfd} \subset \mathit{Diff} \subset \mathit{Sh} \subset \mathit{PSh}$$ where $\mathit{Mfd}$ is the category of smooth finite dimensional manifolds, $\mathit{Diff}$ are diffeological spaces (i.e. concrete sheaves on cartesian spaces), $\mathit{Sh}$ are sheaves on cartesian spaces and $\mathit{PSh}$ are presheaves on cartesian spaces. The last two inclusions are reflexive.

Lets us first have a closer look at the inclusion $\mathit{Sh} \subset \mathit{PSh}$. Following the same vein of argument as above, there is a priori no reason to work with $\mathit{Sh}$ instead of $\mathit{PSh}$ since both categories are equally nice (topoi) and the definition of a presheaf is clearly simpler than that of a sheaf. But there are some colimits in $\mathit{Mfd}$ that we really like, namely the coequalizer diagram correspoding to an open cover $(U_\alpha)$ of a manifold $M$. Under the inclusion of $\mathit{Mfd}$ into $\mathit{PSh}$ this is not a coequalizer anymore, in other words: If we glue open sets in $\mathit{PSh}$ together we do not get the same thing that we get when glueing together as manifolds. This defect is exactly cured by the sheaf property. That means restricting to the smaller subcategory $Sh \subset \mathit{PSh}$ the colimits change such that gluing of open sets behaves as nice as in manifolds. The punchline is that the restriction to $\mathit{Sh}$ provides the category with the "right" coequalizers of open sets.

Now lets turn towards the inclusion $\mathit{Diff} \subset \mathit{Sh}$. The situation is exactly the same as before. Limits in $\mathit{Diff}$ are computed as Limits in $\mathit{Sh}$ (and hence also $\mathit{PSh}$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.

Second, if we want to "make" geometry over diffeological spaces it turns out that there are two possible definition of principal bundles:

The first topology has an obvious analogue on the category $\mathit{Sh}$ of all sheaves but the second crucially uses the underlying topological space of a diffeological space.

• a bundle over a diffeological space $M$ is a morphism to the stack of bundles over finite dimensional manifolds. This means that we have a family of bundles over each plot together with coherent isomorphisms. Note that this type of bundle is determined by its pullback to finite dimensional spaces. This is equivalent to have a diffeological space $P \to M$ together with a free transitive on fibers action such that the quotient map $P \to M$ is a surjective subduction (i.e. becomes a submersion on each plot). To get those type of bundles we have to equip diffeological spaces with the Grothendieck Topology of subductions.

• a bundle over a diffeological space $M$ is a space $P \to M$ with a free transitive action such that it is locally trivial in the underlying topology. This is the type of bundle which people consider in the world of $\infty$-dimensional manifolds. To get this we have to take the grothendieck topology of morphisms that are surjective and admits local (in the topology) sections. Hence therefore we really need the underlying topological space.

• a bundle over a diffeological space $M$ is a morphism to the stack of bundles over finite dimensional manifolds. This means that we have a family of bundles over each plot together with coherent isomorphisms. Note that this type of bundle is determined by its pullback to finite dimensional spaces. This is equivalent to have a diffeological space $P \to M$ together with a free transitive on fibers action such that the quotient map $P \to M$ is a surjective subduction (i.e. becomes a submersion on each plot). To get those type of bundles we have to equip diffeological spaces with the Grothendieck Topology of subductions.

• a bundle over a diffeological space $M$ is a space $P \to M$ with a free, transitive on fibers, action such that it is locally trivial, where locally refers to the underlying topological space of $M$. This is the type of bundle which people consider in the world of $\infty$-dimensional manifolds. To get this we have to take the grothendieck topology of morphisms that are surjective and admits local (in the topology) sections. Hence therefore we really need the underlying topological space.

Now lets turn towards the inclusion $Diff \subset PSh$. The situation is exactly the same as before. Limits in $Diff$ are computed as Limits in $Sh$ (and hence also $PSh$) but colimits are different in general (one has to apply the concretization functor). I would say this is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. I hope that I remember correctly but from abstract reasoning it is clear that the colimits in the two categories have to differ.

Now lets turn towards the inclusion $Diff \subset Sh$. The situation is exactly the same as before. Limits in $Diff$ are computed as Limits in $Sh$ (and hence also $PSh$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.