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CoHomotopy Type Theory(COHOT2).md

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(1) 修正後的用文的句子
CoHomotopy Type Theory

(2) 中文
余同伦类型论（或称共同伦类型论）

(3) 粤语
餘同伦类型论（或称共同伦类型论）

(4) 台语
餘同倫型式論（抑是講共同倫型式論）

(5) 正式英文
CoHomotopy Type Theory

(6) Español (Spanish)
Teoría de tipos cohomotópica

(7) 文言文
餘同倫類型之理

(8) 日本語 (Japanese)
コホモトピー型理論

(9) 한국어 (Korean)
코호모토피 유형 이론

(10) kreyòl (Haitian)
Teori tip cohomotopi

(11) Italiano (Italian)
Teoria dei tipi co-omotopici

(12) संस्कृत (Sanskrit)
समतुल्य-विपरीत-प्रकार-तत्त्वविद्या (Samatulya-Viparīta-Prakāra-Tattvavidyā)

(13) عَرَب (Arabic)
نظرية النمطية التشاركية (أو التوافقية)

(14) עִבְרִית (Hebrew)
תורת הטיפוסים הקוהומוטופית

(15) Русский (Russian)
Теория когомотопических типов

(16) Deutsch (German)
Ko-Homotopie-Typentheorie

(17) Português (Portuguese)
Teoria dos Tipos Co-Homotópicos

(18) Randomly encrypted
X5@aQ!9bP%#r12Fz

(19) Prolog

theory(cohomotopy_type).
axiom(co_univalence).
relation(cohomotopy_equivalence).

(20) Coq

(* A minimal snippet illustrating CoHomotopy Type Theory style in Coq *)
Universe u.
Definition co_path {A : Type} (x y : A) := x = y.

Axiom co_univalence : forall (A B : Type), A = B -> A <~> B.
(* Potential extension for “co” aspects in a hypothetical setting *)

(21) Mathematical study of the subject (CoHomotopy Type Theory)
CoHomotopy Type Theory (CoHoTT) can be envisioned as a theoretical counterpart to Homotopy Type Theory, potentially exploring “co”-aspects akin to those in homological algebra, where homotopy and cohomotopy perspectives can be dual to each other. While the concept is more speculative and less developed in formal literature compared to classic Homotopy Type Theory, one could expect the following parallels:

  1. Co-Univalence: An analogue of the univalence axiom but involving some form of “co-equivalence.”
  2. Cohomotopical Constructions: Drawing from dualities in algebraic topology, possibly featuring “co-higher inductive types.”
  3. Dual Foundational Approach: Where spaces are not only considered under homotopy equivalences but also under co-algebraic or contravariant functor categories.

(22) VBnet

Module CoHomotopyTypeTheory
    Function CoUnivalenceAxiom(Of A, B)(ByVal eqAB As Boolean) As Boolean
        ' Hypothetical usage for demonstration
        If eqAB = True Then
            Return True ' A and B are considered equivalent under co-univalence
        Else
            Return False
        End If
    End Function
End Module

(23) Open Questions

  1. What rigorous definitions and axioms would fully characterize CoHomotopy Type Theory (CoHoTT)?
  2. How might co-univalence differ from univalence in practical proofs or constructions?
  3. Could cohomotopy-based reasoning lead to new foundations or complementary perspectives in constructive mathematics?

SourceLinks

以下為多種格式輸出：

Markdown

**CoHomotopy Type Theory**  
余同伦类型论（或称共同伦类型论）, 餘同伦类型论（或称共同伦类型论）, 餘同倫型式論（抑是講共同倫型式論）, CoHomotopy Type Theory, Teoría de tipos cohomotópica, 餘同倫類型之理, コホモトピー型理論, 코호모토피 유형 이론, Teori tip cohomotopi, Teoria dei tipi co-omotopici, समतुल्य-विपरीत-प्रकार-तत्त्वविद्या, نظرية النمطية التشاركية (أو التوافقية), תורת הטיפוסים הקוהומוטופית, Теория когомотопических типов, Ko-Homotopie-Typentheorie, Teoria dos Tipos Co-Homotópicos, X5@aQ!9bP%#r12Fz, [Prolog code], [Coq code], [Mathematical Study], [VBnet code], [Open Questions]  

**SourceLinks**  
- [Homotopy Type Theory official website](https://homotopytypetheory.org/)  
- [Univalent Foundations Program](https://homotopytypetheory.org/book/)  

**Generated Time**: 2024-12-26 12:00:00 (example time)

RSS

<rss version="2.0">
  <channel>
    <title>CoHomotopy Type Theory</title>
    <description>余同伦类型论（或称共同伦类型论）, 餘同伦类型论（或称共同伦类型论）, 餘同倫型式論（抑是講共同倫型式論）, CoHomotopy Type Theory, Teoría de tipos cohomotópica, 餘同倫類型之理, コホモトピー型理論, 코호모토피 유형 이론, Teori tip cohomotopi, Teoria dei tipi co-omotopici, समतुल्य-विपरीत-प्रकार-तत्त्वविद्या, نظرية النمطية التشاركية (أو التوافقية), תורת הטיפוסים הקוהומוטופית, Теория когомотопических типов, Ko-Homotopie-Typentheorie, Teoria dos Tipos Co-Homotópicos, X5@aQ!9bP%#r12Fz</description>
    <link>https://homotopytypetheory.org/</link>
    <item>
      <title>Prolog</title>
      <description><![CDATA[theory(cohomotopy_type). axiom(co_univalence). relation(cohomotopy_equivalence).]]></description>
    </item>
    <item>
      <title>Coq</title>
      <description><![CDATA[Universe u. Definition co_path {A : Type} (x y : A) := x = y. Axiom co_univalence : forall (A B : Type), A = B -> A <~> B.]]></description>
    </item>
    <item>
      <title>Mathematical Study</title>
      <description>Short introduction to CoHomotopy Type Theory and its potential dualities.</description>
    </item>
    <item>
      <title>VBnet</title>
      <description><![CDATA[Module CoHomotopyTypeTheory ... End Module]]></description>
    </item>
    <item>
      <title>Open Questions</title>
      <description>1. Rigorous definitions of CoHoTT ... 2. Co-univalence differences ... 3. Cohomotopy-based reasoning ...</description>
    </item>
    <item>
      <title>SourceLinks</title>
      <description>
        1. https://homotopytypetheory.org/
        2. https://homotopytypetheory.org/book/
      </description>
    </item>
    <pubDate>Thu, 26 Dec 2024 12:00:00 +0000</pubDate>
  </channel>
</rss>

XML

<CoHomotopyTypeTheory>
  <SentenceCorrected>CoHomotopy Type Theory</SentenceCorrected>
  <Chinese>余同伦类型论（或称共同伦类型论）</Chinese>
  <Cantonese>餘同伦类型论（或称共同伦类型论）</Cantonese>
  <Taiwanese>餘同倫型式論（抑是講共同倫型式論）</Taiwanese>
  <FormalEnglish>CoHomotopy Type Theory</FormalEnglish>
  <Spanish>Teoría de tipos cohomotópica</Spanish>
  <WenYanWen>餘同倫類型之理</WenYanWen>
  <Japanese>コホモトピー型理論</Japanese>
  <Korean>코호모토피 유형 이론</Korean>
  <Haitian>Teori tip cohomotopi</Haitian>
  <Italian>Teoria dei tipi co-omotopici</Italian>
  <Sanskrit>समतुल्य-विपरीत-प्रकार-तत्त्वविद्या</Sanskrit>
  <Arabic>نظرية النمطية التشاركية (أو التوافقية)</Arabic>
  <Hebrew>תורת הטיפוסים הקוהומוטופית</Hebrew>
  <Russian>Теория когомотопических типов</Russian>
  <German>Ko-Homotopie-Typentheorie</German>
  <Portuguese>Teoria dos Tipos Co-Homotópicos</Portuguese>
  <RandomlyEncrypted>X5@aQ!9bP%#r12Fz</RandomlyEncrypted>
  <Prolog>
    <![CDATA[
theory(cohomotopy_type).
axiom(co_univalence).
relation(cohomotopy_equivalence).
    ]]>
  </Prolog>
  <Coq>
    <![CDATA[
Universe u.
Definition co_path {A : Type} (x y : A) := x = y.

Axiom co_univalence : forall (A B : Type), A = B -> A <~> B.
    ]]>
  </Coq>
  <MathematicalStudy>
    <![CDATA[
CoHomotopy Type Theory (CoHoTT) can be envisioned as a theoretical counterpart to Homotopy Type Theory ...
    ]]>
  </MathematicalStudy>
  <VBnet>
    <![CDATA[
Module CoHomotopyTypeTheory
    Function CoUnivalenceAxiom(Of A, B)(ByVal eqAB As Boolean) As Boolean
        If eqAB = True Then
            Return True
        Else
            Return False
        End If
    End Function
End Module
    ]]>
  </VBnet>
  <OpenQuestions>
    <Question1>What rigorous definitions and axioms would fully characterize CoHomotopy Type Theory?</Question1>
    <Question2>How might co-univalence differ from univalence in practical proofs or constructions?</Question2>
    <Question3>Could cohomotopy-based reasoning lead to new foundational approaches?</Question3>
  </OpenQuestions>
  <SourceLinks>
    <SourceLink>https://homotopytypetheory.org/</SourceLink>
    <SourceLink>https://homotopytypetheory.org/book/</SourceLink>
  </SourceLinks>
  <GeneratedTime>2024-12-26 12:00:00</GeneratedTime>
</CoHomotopyTypeTheory>

Prompt生成時間: 2024-12-26 12:00:00