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nLab > Latest Changes: circle type

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  1. Page created, but author did not leave any comments.

    Anonymous

    v1, current

  2. added info about the circle type

    Anonymous

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 8th 2022

    fixed some grammar and touched the wording in the Definition-section

    diff, v6, current

  4. Adding link to Jordan curve

    Anonymous

    diff, v8, current

  5. Adding coequalizer definition of the circle type

    Anonymous

    diff, v9, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 30th 2023

    tried to bring some logical order back into the list of references.

    diff, v11, current

    • CommentRowNumber7.
    • CommentAuthorGuest
    • CommentTimeMar 1st 2023

    This page says:

    Its induction principle says that for any P:S 1→TypeP:S^1\to Type equipped with a point base′:P(base)base' : P(base) and a dependent path loop′:base′=base′loop':base'= base', there is f:∏ (x:S 1)P(x)f:\prod_{(x:S^1)} P(x) such that:

    Should it instead say loop′:tr P loop(base′)=base′loop' : tr^{loop}_P(base') = base', or loop′:base′= P loopbase′loop' : base' =^{loop}_{P} base', or “dependent path loop′:base′=base′loop':base'= base' over looploop”, or something like that? Does it matter?

    Adrian

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2023
    • (edited Mar 2nd 2023)

    I think that’s right, there is a transport involved. That’s what the link dependent path is referring to. For the moment i have added (here) the missing = loop=_{loop}-subscript and a pointer to UFP13, p. 177.

    Ideally I would like to polish up the whole presentation (which is due to the notorious “Anonymous”, in revision 5) but not now.

    diff, v12, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2023

    On whether it matters: I expected it does. While I haven’t tried to write out a formal argument, a quick idea goes as follows:

    In the higher induction principle of the suspension type SS 0\mathrm{S}S^0 there is certainly such dependent identifications involved, namely along the two “meridian” paths from the “north pole” to the “south pole”: by the general rules of higher inductive types, but also because here it would not even type-check otherwise. But then for the evident map S 1→SS 0S^1 \to \mathrm{S}S^0 to be an equivalence, there must be a corresponding dependent identification also in the induction principle of S 1S^1.

  10. Assuming that the computation rule for the basepoint of S 1S^1 is propositional, the propositional computation rule for the loop of S 1S^1 listed on the page in the section titled “In natural deduction” is wrong. The application of the dependent function ind S 1(c *,c 𝓁)\mathrm{ind}_{S^1}(c_*, c_\mathcal{l}) defined in the elimination rule of S 1S^1 to the loop identification 𝓁:*= S 1*\mathcal{l}:* =_{S^1} * is actually in the heterogeneous identity type

    ind S 1(c *,c 𝓁)(*)= ̲.C 𝓁ind S 1(c *,c 𝓁)(*)\mathrm{ind}_{S^1}(c_*, c_\mathcal{l})(*) =_{\underline{ }.C}^{\mathcal{l}} \mathrm{ind}_{S^1}(c_*, c_\mathcal{l})(*)

    rather than the heterogeneous identity type c *= ̲.C 𝓁c *c_* =_{\underline{ }.C}^{\mathcal{l}} c_*. One needs to transport the application across the identification β S 1 *(c *,c 𝓁):ind S 1(c *,c 𝓁)(*)= C(*)c *\beta_{S^1}^{*}(c_*, c_\mathcal{l}):\mathrm{ind}_{S^1}(c_*, c_\mathcal{l})(*) =_{C(*)} c_* found in the propositional computation rule for the basepoint *:S 1*:S^1 in order for everything to type check in the propositional computation rule for the loop, or otherwise use a heterogeneous identity type instead of a homogeneous identity type, like

    β S 1 𝓁(c *,c 𝓁):apd ind S 1(c *,c 𝓁)(𝓁)= ̲.(−)= ̲.C 𝓁(−) β S 1 *(c *,c 𝓁)c 𝓁\beta_{S^1}^{\mathcal{l}}(c_*, c_\mathcal{l}):\mathrm{apd}_{\mathrm{ind}_{S^1}(c_*, c_\mathcal{l})}(\mathcal{l}) =_{\underline{ }.(-) =_{\underline{ }.C}^{\mathcal{l}} (-)}^{\beta_{S^1}^{*}(c_*, c_\mathcal{l})} c_\mathcal{l}

  11. fixed computation rules, and also switched notation for the identity types over to the usual a= Aba =_A b for consistency with the rest of the article and with references i.e. the HoTT book

    diff, v22, current

  12. Similarly with using base\mathrm{base} and loop\mathrm{loop} for the path and point constructors of the circle type throughout the article, for consistency with the rest of the article and with the references.

    diff, v22, current

  13. Also switched all mentions of “identification type” over to “identity type” for consistency.

    diff, v23, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2023

    The term “identification type” is the more logically consistent one, though.

  15. The term “identification type” is the more logically consistent one, though.

    If you prefer, I could use “identification type” throughout the article instead of “identity type”. The problem is that originally the article used both “identification type” and “identity type”.

  16. I added an explanation for the inference rules for the circle type, similar to the explanation for the inference rules for identity types over in that article.

    diff, v24, current

  17. By suggestion of Urs on the nForum, I have switched identity type to identification type.

    diff, v24, current

  18. I have merged the section on the induction and recursion principle under the properties section into the definitions section - where the induction and recursion principles are defined via the elimination and computation rules.

    diff, v26, current

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