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 $n$-groupoids.

I’ll point out what I think is a major property that one would want to ask of a cohesive $\infty$-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 $\infty$-topos, that is meant to show that when applied to $\infty$-toposes instead of just 1-toposes, the axioms on strongly connected and local $\infty$-toposes – in the form of a quadruple of adjoint $\infty$-functors $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc) : \mathbf{H} \to \infty Grpd$ with $Disc$ and $coDisc$ full and faithful and $\Pi$ preserving finite products – imply a fairly comprehensive list of abstract structures in differential cohomology on the objects in the $\infty$-topos.

Among the most basic is the fact that $|-| : \mathbf{H} \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top$ models geometric realizaton of cohesive $\infty$-groupoids. In particular, for a connected object – denoted $\mathbf{B}G$ – in $\mathbf{H}$ and $X$ any object, $\mathbf{H}(X,\mathbf{B}G)$ has the interpretation of the $\infty$-groupoid of cohesive $G$-principal $\infty$-bundles on $X$ and we may think of the action of $|-|$ on this

$|-|_{X,\mathbf{B}G} : \mathbf{H}(X, \mathbf{B}G)
\to Top(|X|, B G)$

with $B G := |\mathbf{B}G|$, 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 $\infty$-groupoids with underlying topological cohesion, we find that in a large class of examples the functor $|-|_{X,\mathbf{B}G}$ is indeed an isomorphism on $\pi_0$: this means that $|\mathbf{B}G|$ is indeed the classifying space for cohesive $G$-principal $\infty$-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 $\infty$-toposes *guarantee* this property. It looks like a decisive property of the interaction of the notion of cohesion with the homotopy-theory of $\infty$-toposes.

Now, this cannot be true without further qualification. There are plenty of objects in a cohesive $\infty$-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 $|-|_{X,\mathbf{B}G}$ to be an iso on $\pi_0$ for concrete objects: notably for $G$ a cohesive $\infty$-group, an $\infty$-group equipped with cohesive structure. Indeed, in the classes of examples that are understood, $G$ 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 $\infty$-topos;

such that restricted to morphisms between these with the codomain connected, $|-|$ becomes an isomorphism on $\pi_0$.

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 $\Gamma$ is a monomorphism on morphisms $X \to Y$ between concrete objects may be deduced from looking at the naturality square of the $(\Gamma \dashv coDisc)$-unit

$\array{
X &\to& Y
\\
\downarrow && \downarrow
\\
coDisc \Gamma X &\to& coDisc \Gamma Y
}$

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

$\Pi_{X,\mathbf{B}G} :
\mathbf{H}(X, \mathbf{B}G)
\to
\infty Grpd(\Pi(X), \Pi(\mathbf{B}G))
\stackrel{\simeq}{\to}
\mathbf{H}(X, Disc \Pi \mathbf{B}G)$

is an isomorphism on $\pi_0$. And by the analogous reasoning, one finds again that this is the operation that forms the composites

$\array{
X &\to& \mathbf{B}G
\\
\downarrow && \downarrow
\\
Disc \Pi X &\to& Disc \Pi \mathbf{B}G
}
\,.$

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 $X$ and $\mathbf{B}G$ here are concrete, we probably need to combine these squares with

$\array{
X &\to& \mathbf{B}G
\\
\downarrow && \downarrow
\\
coDisc \Gamma X &\to& coDisc \Gamma \mathbf{B}G
}$

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 $\infty$-categories).

So this begins to look like we’d need to consider natural morphisms between the composite adjoint triple

$(Disc \Pi \dashv Disc \Gamma \dashv coDisc \Gamma)$

(which I usually write $(\mathbf{\Pi} \dashv \mathbf{\flat} \dashv \mathbf{\Gamma})$).

So maybe one needs to consider the squares

$\array{
X &\to& coDisc \Gamma X
\\
\downarrow && \downarrow
\\
Disc \Pi X &\to& coDisc \Pi X
}
\,,$

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).