Excuse my lack of understanding of monadicity, but I have been looking at toposes and monads. I see Lambek showed that the category of Toposes are monadic over the category of categories. I see the 2-category of groupoids as possibly another candidate for which there may be a category of toposes that are monadic over it. Is this the case?

A question has been asked about the 2-morphisms in my category of toposes that I think is monadic over the 2-category of groupoids. From my limited understanding, the 2-category of toposes will have functors as morphisms. The 2-morphisms would thus be natural transformations. I have limited understanding of functors between toposes, knowing only one special kind which is the geometric morphism. This means I have even less insight as to what the two morphisms would be. Perhaps natural transformations of geometric morphisms? I have to leave this question open in the sense that I am looking for any 2-category of toposes that is monadic over the 2-cat of groupoids. Does this mean there are many 2-cats of toposes that will be monadic?

The background as to why I am asking this: It has been asked a few times what my intuition is for asking this question. Unfortunately, like most times I ask a broad question like this, my intuitions are unclear and might be from some physical intuition or vague maths. This is mainly coming from the work I am doing to find polynomial monads for different types of containers. In particular, the multi-set monad is only polynomial on the 2-category of groupoids.

Why do I care about this? This comes from physics. I am working on a branch of physics where data (not information), like lists, multisets(bags), trees, are at the foundation alongside the experiments which produce them.

We have seen physics taken apart to probe our logical frameworks due to Isham and Doering. They were looking at picking a topos, other than SET, that suited modern physics, being more from a quantum origin. I am also interested in finding a topos. To that end, I asked if the 2-cat of groupoids itself is a topos. Then I did a quick search and found Lambek's paper and then was interested in a Topos that was monadic over 2-groupoids.

So, multisets are very important data structures for physics, and they are only polynomial on the 2-cat of groupoids. The polynomial nature is required for us to have a theory of Containers that encapsulates them. Given that I seem to be restricting myself to groupoids, I began to wonder where my topos might be. Spivak has recently been writing about the topos of sheaves over the interval domain. His work uses a domain of intervals to mimic a temporal background. My work sheds the idea of a background and instead focuses just on what happens in a lab, and how this relates to the systems the lab is trying to probe (think telescopes and distant galaxies). Since I seem to be stuck with the 2-cat of groupoids as a kind of interface to systems, naturally I am wondering where my topos is.

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    $\begingroup$ What are the 2-arrows in your 2-category of toposes? $\endgroup$ – David Roberts Feb 20 at 23:18
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    $\begingroup$ If the category of topoi is to be monadic over the category of groupoids, then in particular there will have to be a forgetful functor from the former category to the latter. I don't see a plausible candidate for such a functor. What underlying-groupoid functor do you have in mind? (Pardon me for omitting all the 2's that should precede words like "category" and "functor" in this situation.) $\endgroup$ – Andreas Blass Feb 21 at 18:57
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    $\begingroup$ I think everyone answering you is probably under the impression that no nontrivial categories of toposes could possibly be monadic over groupoids. I certainly can't think of any way such a category could be. Remember that when a category $C$ is monadic over $D$, that means the objects of $C$ are objects of $D$ equipped with "extra structure" in some way. But a topos is not a groupoid with structure; it is a category with structure. Yes, of course a groupoid is a particular kind of category, but the underlying category of a nontrivial topos is never a groupoid. $\endgroup$ – Mike Shulman Feb 22 at 18:24
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    $\begingroup$ @BenSprott I would say the takeaway is that when someone asks a question like this, people who think it is very unlikely to be true will sometimes express that feeling by suggesting to the asker some investigations they could do on their own that might leave them with the same feeling. Of course the asker is not expected to know the answer, but I would say there's some expectation that they've thought a little about the question, and perhaps that they have some reason to believe that it might be true. $\endgroup$ – Mike Shulman Feb 23 at 13:45
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    $\begingroup$ It might also help if you explained your motivation. There are innumerable questions one might ask of the form "is X monadic over Y?". Are topological spaces monadic over abelian groups? Are commutative rings monadic over symmetric spectra? For most of them the answer is probably no. What is it that leads you to ask whether toposes are monadic over groupoids? Most interesting questions about monadicity (which have a chance of being true) only arise once you already have a forgetful functor in mind, which is part of why everyone was asking you what forgetful functor you have in mind. $\endgroup$ – Mike Shulman Feb 23 at 13:50

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