Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 9th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexCreated [[connectology]].

   Created connectology.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 10th 2015
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIncidentally, Mike: what motivated you to ask about this at MO?

   Incidentally, Mike: what motivated you to ask about this at MO?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 10th 2015
   • (edited Mar 10th 2015)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex@Todd the quote "[Now, however, note that Mink is also connected.](https://golem.ph.utexas.edu/category/2015/03/hits_and_the_erlangen_program.html#more)" makes me think it might be in relation to thinking of types as being connected, or at least Mike's book chapter.

   @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.

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeMar 10th 2015
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI'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.

   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.

5. • CommentRowNumber5.
   • CommentAuthorZhen Lin
   • CommentTimeMar 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexAh, 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?)

   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?)

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2015
   • (edited Mar 10th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexInteresting. 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?

   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?

7. • CommentRowNumber7.
   • CommentAuthorZhen Lin
   • CommentTimeMar 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexWe don't even have a 1-topos yet – as I said, we should choose an interesting Grothendieck topology first.

   We don’t even have a 1-topos yet – as I said, we should choose an interesting Grothendieck topology first.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeMar 10th 2015
   • (edited Mar 10th 2015)
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYes, 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorZhen Lin
   • CommentTimeMar 10th 2015
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexThere'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 $\tilde{\mathcal{C}} (A, B)$: * The objects are pairs $(E, f)$, where $E$ is a connected object in $\mathcal{C}$ and $f : E \times A \to B$ is a morphism in $\mathcal{C}$. * The morphisms $(E, f) \to (E', f')$ are morphisms $h : E \to E'$ in $\mathcal{C}$ such that $f' \circ (h \times id_A) = f$. * Composition and identities are inherited from $\mathcal{C}$. 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.

   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 $\tilde{\mathcal{C}} (A, B)$:

   • The objects are pairs $(E, f)$, where $E$ is a connected object in $\mathcal{C}$ and $f : E \times A \to B$ is a morphism in $\mathcal{C}$.
   • The morphisms $(E, f) \to (E', f')$ are morphisms $h : E \to E'$ in $\mathcal{C}$ such that $f' \circ (h \times id_A) = f$.
   • Composition and identities are inherited from $\mathcal{C}$.

   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.

10. • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMar 11th 2015
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexI was thinking more along the Artin-Mazur-Verdier lines.

    I was thinking more along the Artin-Mazur-Verdier lines.