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Created connectology.
Incidentally, Mike: what motivated you to ask about this at MO?
@Todd
the quote “Now, however, note that Mink is also connected.” makes me think it might be in relation to thinking of types as being connected, or at least Mike’s book chapter.
I’ve been wondering whether spaces of this sort could be used to build a cohesive topos. In general, I’ve been wondering whether cohesion actually demands continuity, or whether something weaker suffices. Connectivity seemed like a reasonable thing to try, since the shape modality is built out of local connectedness.
Also, I’ve been exploring what sorts of nonclassical principles follow from real-cohesion, and while I think I can prove that $R$ is connected, I haven’t been able to prove that every endofunction of it is continuous; so I wondered whether there was a countermodel instead.
Ah, I wondered if you might be secretly thinking about cohesion there.
Let’s see. Let $\mathcal{C}$ be the category of connectological spaces. The forgetful functor $\Gamma : \mathcal{C} \to Set$ has left adjoint $\Delta : Set \to \mathcal{C}$ and a right adjoint $\nabla : Set \to \mathcal{C}$, both of which are fully faithful. The fact that maximal connected subsets exist and are disjoint gives us $\pi_0 : \mathcal{C} \to Set$ left adjoint to $\Delta : Set \to \mathcal{C}$. The canonical natural transformation $\Gamma \Rightarrow \pi_0$ is componentwise surjective. It appears to me that $\pi_0$ preserves small products. So $\mathcal{C}$ is indeed cohesive over $Set$.
Also, the subcategory of non-empty spaces is a cohesive site with respect to the “chaotic” topology, for more or less trivial reasons. But we should probably put an interesting Grothendieck topology on it. (Perhaps we should be restricting to the subcategory of non-empty connected spaces?)
Interesting. But notice that it is cohesion of 1-toposes which is about (local) connectedness, clearly, while it is cohesion of $(\infty,1)$-toposes which is about (local) contractibility, a “more continuous” thing. So: does the extra left adjoint $\pi_0$ that you identify on the 1-topos refine to an extra left adjoint $\Pi$ on the corresponding $\infty$-topos?
We don’t even have a 1-topos yet – as I said, we should choose an interesting Grothendieck topology first.
Yes, I was thinking of putting a topology on a small category of connectological spaces (nice word there) and seeing if it were cohesive. For an $\infty$-cohesive site we would probably, as Urs says, need a notion of “locally contractible” connectological space. Maybe we could define the “shape” of a connectological space in terms of its connected subsets.
I’m not sure that the notion of connectological space is really necessary either; we could just look at a category whose objects are some nice ordinary locally contractible topological spaces, but whose morphisms are connectomorphisms rather than continuous maps.
There’s no problem defining a notion of homotopy once we have a well-behaved notion of connected object. Indeed, given a pair $(A, B)$ of objects in $\mathcal{C}$, define the following category $\tilde{\mathcal{C}} (A, B)$:
Finite products of connected objects are connected, so we can define a composition à la Kleisli $\tilde{\mathcal{C}} (B, C) \times \tilde{\mathcal{C}} (A, B) \to \tilde{\mathcal{C}} (A, C)$, yielding a bicategory $\tilde{\mathcal{C}}$. We then take connected components to get the homotopy category $\mathcal{C}_h$. In particular we get a notion of contractibility. There are conditions on $\pi_0$ which ensure that the quotient functor $\mathcal{C} \to \mathcal{C}_h$ preserves finite products, but I haven’t checked if they are satisfied for the category of connectological spaces.
I don’t have any ideas about defining local contractibility, though. If we had that, we could take the Artin–Mazur–Verdier approach to defining the shape of an object in terms of hypercovers by contractible objects.
I was thinking more along the Artin-Mazur-Verdier lines.
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