I feel that in order to make progress with identifying the right way of looking at the axiom pieces have points / puncual connectedness in the ordinary and the higher categorical context we need to have a better idea of what such a kind of axiom can achieve or is supposed to be good for, apart from sounding plausible as an axiom characterizing cohesive -groupoids.
I’ll point out what I think is a major property that one would want to ask of a cohesive -topos, which looks like it ought to be controled by such a kind of axiom.
As you know, I am working on a document, titled Differential cohomology in a cohesive -topos, that is meant to show that when applied to -toposes instead of just 1-toposes, the axioms on strongly connected and local -toposes – in the form of a quadruple of adjoint -functors with and full and faithful and preserving finite products – imply a fairly comprehensive list of abstract structures in differential cohomology on the objects in the -topos.
Among the most basic is the fact that models geometric realizaton of cohesive -groupoids. In particular, for a connected object – denoted – in and any object, has the interpretation of the -groupoid of cohesive -principal -bundles on and we may think of the action of on this
with , as sending such cohesive bundles to their underlying topological bundles (or rather to the classifiying maps of these).
Indeed, using a theorem by Danny Stevenson and David Roberts on simplicial topological bundles, applied in a model category presentation of cohesive -groupoids with underlying topological cohesion, we find that in a large class of examples the functor is indeed an isomorphism on : this means that is indeed the classifying space for cohesive -principal -bundles.
This is clearly an important statement, and one might hope that it is not only true in the specific class of examples that I have checked, but that general abstract axioms on cohesive -toposes guarantee this property. It looks like a decisive property of the interaction of the notion of cohesion with the homotopy-theory of -toposes.
Now, this cannot be true without further qualification. There are plenty of objects in a cohesive -topos that have nontrivial categorical homotopy types (towers of homotopy sheaves) but trivial geometric homotopy type: they map under to the point. But these contain intuitively and in classes of examples non-concrete objects . And intuitively one expects the functor to be an iso on for concrete objects: notably for a cohesive -group, an -group equipped with cohesive structure. Indeed, in the classes of examples that are understood, is presented by a simplicial topological group or a simplicial Lie group, etc.
So therefore I am wondering if
there is a good notion of concrete objects in a cohesive -topos;
such that restricted to morphisms between these with the codomain connected, becomes an isomorphism on .
The reason why I think this might be relevant to the discussion here is that when unwinding this a little, it begins too look like a condition very similar to that of punctual connectedness.
As arguments by you and Peter Johnstone emphasized, in a cohesive 1-topos we find that the property that is a monomorphism on morphisms between concrete objects may be deduced from looking at the naturality square of the -unit
and using that by definition of concrete objects, the vertical morphism are monomorphisms.
Now for the problem that I am talking about, the question is whether
is an isomorphism on . And by the analogous reasoning, one finds again that this is the operation that forms the composites
So the situation seems to be similar, even though it’s different (we are not asking faithfulness and the units here are not monos).
So I am trying to see what one can do in the spirit of “punctual connectedness”-axioms to possibly find a characterization of this situation.
For instance since we want to assume that and here are concrete, we probably need to combine these squares with
and possibly demand that the vertical morphisms are something like monos on all homotopy sheaves or the like (this is where this connects to the discussion at Concrete -categories).
So this begins to look like we’d need to consider natural morphisms between the composite adjoint triple
(which I usually write ).
So maybe one needs to consider the squares
where on the right and on the bottom we have the morphisms considered in punctual connectedness, and then consider the cube obtained from all these (which I won’t try to typeset here) and then argue with some of these morphsims being epi in some sense, others mono in some sense, etc.
I do not currently know how to proceed at this point. I only observe that it seems like higher analogs of the conditions pieces have points and punctual connectedness should play a role, and that if they do, the property on that I am hoping can be implied would be a genuinely interesting and desrireble one, one that would clearly justify the extra axiom.
But I don’t know yet.
Re: Punctual Local Connectedness
There was a discussion of punctual local connectedness — a.k.a. the Nullstellensatz — at MathOverflow recently (including a long message from Lawvere).