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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 9th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added to [[continuum]] a paragraph titled [In cohesive homotopy type theory](http://ncatlab.org/nlab/show/continuum#InCohesiveHomotopyTypeTheory). This is a simple observation and idea that I have been carrying around for a while. Several people are currently thinking about ways to axiomatize the reals in (homotopy) type theory. With _cohesive_ homotopy type theory there is what looks like an interesting option for an approach different to the other ones: one can ask more generally about [[line objects]] $\mathbb{A}^1$ that look like continua. One simple way to axiomatize this would be to say: 1. $\mathbb{A}$ is a ring object; 1. it is geometrically contractible, $\mathbf{\Pi} \mathbb{A} \simeq *$. The last condition reflects the "continuumness". For instance in the standard model [[Smooth∞Grpd]] for smooth homotopy cohesion, this distibuishes $\mathbb{A} = \mathbb{Z}, \mathbb{Q}$ from $\mathbb{A} = \mathbb{R}, \mathbb{C}$. So while this axiomatization clearly captures one aspect of "continuum" very elegantly, I don't know yet how far one can carry this in order to actually derive statements that one would want to make, say, about the real numbers.

   I have added to continuum a paragraph titled In cohesive homotopy type theory.

   This is a simple observation and idea that I have been carrying around for a while. Several people are currently thinking about ways to axiomatize the reals in (homotopy) type theory.

   With cohesive homotopy type theory there is what looks like an interesting option for an approach different to the other ones: one can ask more generally about line objects $\mathbb{A}^1$ that look like continua.

   One simple way to axiomatize this would be to say:

   1. $\mathbb{A}$ is a ring object;

   2. it is geometrically contractible, $\mathbf{\Pi} \mathbb{A} \simeq *$.

   The last condition reflects the “continuumness”. For instance in the standard model Smooth∞Grpd for smooth homotopy cohesion, this distibuishes $\mathbb{A} = \mathbb{Z}, \mathbb{Q}$ from $\mathbb{A} = \mathbb{R}, \mathbb{C}$.

   So while this axiomatization clearly captures one aspect of “continuum” very elegantly, I don’t know yet how far one can carry this in order to actually derive statements that one would want to make, say, about the real numbers.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 30th 2012
   • (edited Sep 30th 2012)
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexI just saw this page. A remark that you might like is that if $\mathbb{A}$ is a path-connected internal ring in (say) $Top$, then it is automatically contractible. Actually, all we really need is that there is a path $\alpha: I \to \mathbb{A}$ that connects the additive identity $0$ to the multiplicative identity $1$, since then $$I \times \mathbb{A} \stackrel{\alpha \times 1}{\to} \mathbb{A} \times \mathbb{A} \stackrel{mult}{\to} \mathbb{A}$$ defines a contracting homotopy. The reason I'm bringing this up is that the first section of [[continuum]] mentions connectedness but not at all contractibility. The two are brought together when we have a ring object.

   I just saw this page. A remark that you might like is that if $\mathbb{A}$ is a path-connected internal ring in (say) $Top$, then it is automatically contractible. Actually, all we really need is that there is a path $\alpha: I \to \mathbb{A}$ that connects the additive identity $0$ to the multiplicative identity $1$, since then

   $$I \times \mathbb{A} \stackrel{\alpha \times 1}{\to} \mathbb{A} \times \mathbb{A} \stackrel{mult}{\to} \mathbb{A}$$

   defines a contracting homotopy.

   The reason I’m bringing this up is that the first section of continuum mentions connectedness but not at all contractibility. The two are brought together when we have a ring object.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, good point. I have [added a remark](http://ncatlab.org/nlab/show/continuum#ContractibilityOfAFromThatOfI) on this. I have tried to state it in generality: if there is a subobject $I \hookrightarrow \mathbb{A}^1$ which is itself geometrically contractible and contains the elements 0 and 1, then this implies already that the ring object $\mathbb{A}^1$ is geometrically conctractible. (This needs of the axioms of cohesion the statement that $\Pi$ preserves products! )

   Thanks, good point. I have added a remark on this.

   I have tried to state it in generality: if there is a subobject $I \hookrightarrow \mathbb{A}^1$ which is itself geometrically contractible and contains the elements 0 and 1, then this implies already that the ring object $\mathbb{A}^1$ is geometrically conctractible.

   (This needs of the axioms of cohesion the statement that $\Pi$ preserves products! )

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 30th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOf course, $i: I \to \mathbb{A}$ doesn't have to be a _subobject_. All we need is an interval (a cospan $\ast \to I \leftarrow \ast$ such that $I$ is geometrically contractible) together with a cospan map from the interval cospan to the cospan $\ast \stackrel{0}{\to} \mathbb{A} \stackrel{1}{\leftarrow} \ast$. If you don't object, I'll add this in.

   Of course, $i: I \to \mathbb{A}$ doesn’t have to be a subobject. All we need is an interval (a cospan $\ast \to I \leftarrow \ast$ such that $I$ is geometrically contractible) together with a cospan map from the interval cospan to the cospan $\ast \stackrel{0}{\to} \mathbb{A} \stackrel{1}{\leftarrow} \ast$.

   If you don’t object, I’ll add this in.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexSure, very good.

   Sure, very good.

6. • CommentRowNumber6.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 30th 2012
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexOkay, I added something in. Not knowing the best terminology, I referred to a map from an [[interval type]] $(left, right, I)$ to the bipointed type $(0, 1, \mathbb{A}^1)$. By the way, I changed in this section "continuum" to "line continuum", to avoid conflict with the way that "continuum" was being used in the previous section. Hope that's okay with you.

   Okay, I added something in. Not knowing the best terminology, I referred to a map from an interval type $(left, right, I)$ to the bipointed type $(0, 1, \mathbb{A}^1)$. By the way, I changed in this section “continuum” to “line continuum”, to avoid conflict with the way that “continuum” was being used in the previous section. Hope that’s okay with you.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. I had entirely forgotten that we have _[[interval type]]_.

   Thanks. I had entirely forgotten that we have interval type.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2012
   • (edited Sep 30th 2012)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWait, we have to be careful here. The unit interval $[0,1] \in TopMfd \hookrightarrow ETop\inftyGrpd$ is not (the interpretation of) an interval type. Instead $\Pi([0,1]) \in \infty Grpd$ is: the difference is that an interval type is required to have its endpoints be equivalent by morphism. But in $[0,1]$ there are no non-trivial morphisms at all. I'll edit at [[continuum]] a bit more to reflect this.

   Wait, we have to be careful here. The unit interval $[0,1] \in TopMfd \hookrightarrow ETop\inftyGrpd$ is not (the interpretation of) an interval type. Instead $\Pi([0,1]) \in \infty Grpd$ is:

   the difference is that an interval type is required to have its endpoints be equivalent by morphism. But in $[0,1]$ there are no non-trivial morphisms at all.

   I’ll edit at continuum a bit more to reflect this.

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeOct 1st 2012
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexThis is nice!

   This is nice!

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 27th 2013
    • PermaLink
    Author: Urs
    Format: MarkdownItexAdded this pointer: * [[Peter Johnstone]], _Topos-theoretic models of the continuum_ ([slides](http://www.cs.ox.ac.uk/quantum/slides/clap2-peterjohnstone.pdf), [lecture video](https://www.youtube.com/watch?feature=player_embedded&v=pKWYa9sc5UI))

    Added this pointer:

    • Peter Johnstone, Topos-theoretic models of the continuum (slides, lecture video)
11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeSep 28th 2013
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexHe didn't mention the higher inductive-inductive Cauchy reals! (-: Although I suppose he also didn't mention the problem with the Cauchy reals that they solve (lack of Cauchy-completeness), while they don't solve the problem with the Cauchy reals that he did mention (undesirable behavior in models).

    He didn’t mention the higher inductive-inductive Cauchy reals! (-: Although I suppose he also didn’t mention the problem with the Cauchy reals that they solve (lack of Cauchy-completeness), while they don’t solve the problem with the Cauchy reals that he did mention (undesirable behavior in models).

12. • CommentRowNumber12.
    • CommentAuthornLab edit announcer
    • CommentTimeNov 15th 2022
    • PermaLink
    Author: nLab edit announcer
    Format: MarkdownItexadded redirects for [[metric continuum]] and [[metric continua]] Anonymous <a href="https://ncatlab.org/nlab/revision/diff/continuum/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/continuum/15">v15</a>, <a href="https://ncatlab.org/nlab/show/continuum">current</a>

    added redirects for metric continuum and metric continua

    Anonymous

    diff, v15, current

13. • CommentRowNumber13.
    • CommentAuthorMatt Insall
    • CommentTimeDec 8th 2024
    • (edited Dec 8th 2024)
    • PermaLink
    Author: Matt Insall
    Format: MarkdownItexContinuum theorists do not refer to a non-metrizable compact connected Hausdorff space as a continuum; they call such a thing a`Hausdorff continuum’. This isn’t my preference, due to a conflict with standard English grammar, but this practice of adding a modifying adjective in order to {\bf reduce} speciation is not uncommon in mathematics or the sciences, and it is more common in everyday English usage than in academe. The cognitive dissonance caused by this in mathematics, specifically in topology, is that it leads to three following {\bf proper} containment: $$ \{C | C \text{is a continuum}\}\subset \{H | H \text{is a Hausdorff continuum} \}. $$ Another class of topological spaces studied by continuum theorists is the class of generalized continua. These are sometimes not continua (again, there’s a bit of cognitive dissonance introduced, akin to that above), but arise from dropping the requirement of compactness from the definition of a continuum. Most uses of modifiers cause a restriction of the class of objects under study. For example, a planar continuum is a specific kind of continuum, namely one that is topologically embeddable in a Euclidean plane. They (practicing continuum theorists) commonly define a continuum as stated here for a [[metric continuum]] but they immediately forget the metric, because they only want to study the topological properties. That is, the more accurate definition of a continuum, as they are studied by continuum theorists, is as follows: a [[continuum]] is a [[compact]] [[connected]] [[metrizable]] [[topological space]]. This is more amenable to a category theoretic understanding of the discipline, and my preference (not caught on (yet?, lol), probably because I’m considered “too pedantic”) is to build the terminology as follows, so that the current praxis gets terminologically aligned with a categorical organization of the information: 1. A [[metric space]] is an ordered pair (X,d) such that d is a [[metric]] on the set X. 2. A metrized topological space is a pair ((X,t),d), where (X,t) is a [[metrizable]] [[topological space]] and d is a [[metric]] on the set X that is [[compatible]] with the [[topology]] t. 3. A metric continuum is a metric space (X,d) for which the metric d induces a compact connected topology on the set X (the result of applying a forgetful functor to a metrized topological space). Since I’ve not been a regular contributor to nCatlab, I prefer to run this discussion by you here in the forum before I make any edits to this page or add any other pages associated with this particular topic.

    Continuum theorists do not refer to a non-metrizable compact connected Hausdorff space as a continuum; they call such a thing a‘Hausdorff continuum’. This isn’t my preference, due to a conflict with standard English grammar, but this practice of adding a modifying adjective in order to {\bf reduce} speciation is not uncommon in mathematics or the sciences, and it is more common in everyday English usage than in academe. The cognitive dissonance caused by this in mathematics, specifically in topology, is that it leads to three following {\bf proper} containment:

    $$\{C | C \text{is a continuum}\}\subset \{H | H \text{is a Hausdorff continuum} \}.$$

    Another class of topological spaces studied by continuum theorists is the class of generalized continua. These are sometimes not continua (again, there’s a bit of cognitive dissonance introduced, akin to that above), but arise from dropping the requirement of compactness from the definition of a continuum. Most uses of modifiers cause a restriction of the class of objects under study. For example, a planar continuum is a specific kind of continuum, namely one that is topologically embeddable in a Euclidean plane.

    They (practicing continuum theorists) commonly define a continuum as stated here for a metric continuum but they immediately forget the metric, because they only want to study the topological properties. That is, the more accurate definition of a continuum, as they are studied by continuum theorists, is as follows: a continuum is a compact connected metrizable topological space. This is more amenable to a category theoretic understanding of the discipline, and my preference (not caught on (yet?, lol), probably because I’m considered “too pedantic”) is to build the terminology as follows, so that the current praxis gets terminologically aligned with a categorical organization of the information: 1. A metric space is an ordered pair (X,d) such that d is a metric on the set X. 2. A metrized topological space is a pair ((X,t),d), where (X,t) is a metrizable topological space and d is a metric on the set X that is compatible with the topology t. 3. A metric continuum is a metric space (X,d) for which the metric d induces a compact connected topology on the set X (the result of applying a forgetful functor to a metrized topological space).

    Since I’ve not been a regular contributor to nCatlab, I prefer to run this discussion by you here in the forum before I make any edits to this page or add any other pages associated with this particular topic.