hi

I'm sorry but I don't quite understand why your example means the definition I suggest is wrong when we're working over a point.

From my understanding there is no special situation of 'working over a point'. The general definition is concerned with objects acted upon by some group object, and should include that the object surjects onto the terminal object

But why is $A\to \bf 1$ being an epi enough?

hm, enough for what?

Sorry, my question didn't make sense. What I'm trying to ask is this: the nlab gives a definition using the word "inhabited".

I took this to mean globally inhabited. What should it be instead?

I think it should be 'X -> 1' is an epimorphism

which is much weaker than the existence of a global point 1 -> X (i.e. a section of X -> 1)

Right, and the nlab says that this condition is equivalent to $X$ being internally inhabited if the underlying category is a topos.

yes

But I don't want $\mathsf C$ to be just a topos. And the nlab suggests using a regular epi for regular categories.

But I don't want that assumption either

ah, so your point is that you are seeking for a suitable definition of torsor for categories more general than regular ones?

I guess it comes down to that, yep.

so what do you want to assume?

Perhaps I should ask $X\to \bf 1$ to be an effective descent morphism?

Just finite limits. That's all I need for the product functor to preserve group objects.

hm, and how do you want to judge whether your definition is suitable? do you have certain simple properties or equivalences in mind you'd like to have?

I'd like the equivalence between the conditions in my MSE question to remain valid, and to generalize to a valid equivalence over a base

but (1) is far too strong, right? being isomorphic to the multiplicative action should mean being a trivial torsor

Ah, sorry. I'm mixing things up. I'll first say what I want informally

I want a definition of a trivial torsor, and then I want to define a general torsor as an action which admits an effective descent morphism pulling it back to a trivial torsor

1 iff 2i,2ii I want to hold for trivial torsors somehow

That's why I put a global section at 2.i

because I thought that would give a good definition of a trivial torsor

If I'm too disorganized to understand, perhaps I'll write my proposed definition of a trivial torsor and you'll tell me what you think?

I think a trivial $G$-torsor should simply be defined as an action isomorphic to multiplication. A locally trivial $G$-torsor should be an action on an invariant bundle which pulls back to a trivial $G$-torsor along an effective descent morphism.

Perhaps in that case, an invariant arrow is a locally trivial $G$-torsor iff $(\varphi,\pi_2)$ is an iso and $A\to \bf 1$ is an effective descent morphism. That sounds reasonable.

yes that sounds reasonable

Cool. Very sorry for the hassle.

What's the intuition behind $A\to \bf 1$ being an effective descent morphism?

I don't know the details very well I have to admit, but I agree that intuitively it sounds good

I guess I'll think of it as "locally inhabited", which is similar to having nonempty fibers.

Ok, thank you very very much for your time. You really helped me to organize my thoughts :)

I don't get the def of eff desc morph intuitively either - currently I can only grasp the equivalent formulation of a stable effective epimorphism

no problem, didn't contribute much

I think eff desc is the right notion in general because its completely geometric and retrieves most of the subtler classes of epis as one removes exactness assumptions on the underyling category

I'll medidate over it ;)

@Hanno :D

Ok, the contexts for effective epimorphism and effective descent morphism seem to be different at first. For effective descent A -> B, I consider objects C -> A which I somehow might think of as sheaves over A, and I require that I have some kind of transport between fibers over points of A that sit over the same point of B. Then the assertion is that such a sheaf actually comes from / descents to a sheaf C -> B

While for effecitve epimorphisms, I am concerned with describing morphisms B -> C as special morphisms A -> C. Maybe the connection is that to any morphism f: A -> C you can assign graph(f) together with the projection graph(f) -> A, thereby getting into the 'sheaf' context

just thinking out loud why intuitively an eff desc morph is necessarily effective epi

@Hanno this is way above my head. I got all my intuition from looking at stuff in Tholen and Janelidze's Beyond Barr Exactness. In particular, from theorem 3.7.

I can mail you a copy if you want.

I guess I won't have time to look at it in detail. Got to go now, have fun, bye

Bye!