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1. • CommentRowNumber1.
   • CommentAuthorBartek
   • CommentTimeDec 21st 2021
   • PermaLink
   Author: Bartek
   Format: MarkdownItexAdded the proof of the fundamental theorem of covering spaces in HoTT <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/13">v13</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   Added the proof of the fundamental theorem of covering spaces in HoTT

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 21st 2021
   • (edited Dec 21st 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor a more faithful type-theoretic reflection of traditional covering-space theory one ought to understand it as a topic in [[cohesive homotopy type theory]]: What deserves to be called the fundamental groupoid of a geometric homotopy type $X$ is not really $\Vert X \Vert_1$, but $\Vert \esh X \Vert_1$, where $\esh$ is the [[shape modality]] (see at *[[shape via cohesive path ∞-groupoid]]*). With this, the fundamental theorem of covering space theory is a reflection of the defining adjunction $\esh \dashv \flat$ with the [[flat modality]], which is such that $\flat Sets$ is the actual classifier for covering spaces over geometric homotopy types. I have appended ([here](https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces#InCohesiveHomotopyTypeTheory)) a paragraph along these lines. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/14">v14</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   For a more faithful type-theoretic reflection of traditional covering-space theory one ought to understand it as a topic in cohesive homotopy type theory:

   What deserves to be called the fundamental groupoid of a geometric homotopy type $X$ is not really $\Vert X \Vert_1$, but $\Vert \esh X \Vert_1$, where $\esh$ is the shape modality (see at shape via cohesive path ∞-groupoid). With this, the fundamental theorem of covering space theory is a reflection of the defining adjunction $\esh \dashv \flat$ with the flat modality, which is such that $\flat Sets$ is the actual classifier for covering spaces over geometric homotopy types.

   I have appended (here) a paragraph along these lines.

   diff, v14, current

3. • CommentRowNumber3.
   • CommentAuthorHurkyl
   • CommentTimeDec 21st 2021
   • PermaLink
   Author: Hurkyl
   Format: MarkdownItexAdded another paragraph to the idea section describing this as related to the equivalence $\infty Grpd_{/X} \simeq \infty Grpd^X$. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/15">v15</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   Added another paragraph to the idea section describing this as related to the equivalence $\infty Grpd_{/X} \simeq \infty Grpd^X$.

   diff, v15, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeDec 21st 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexAs before, since "$X$" denotes a topological space in the preceding sentences, I have changed it to $\esh X$ and added brief commentary. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/16">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/16">v16</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   As before, since “$X$” denotes a topological space in the preceding sentences, I have changed it to $\esh X$ and added brief commentary.

   diff, v16, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 26th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Felix Cherubini]], [[Egbert Rijke]], Thm. 8.7 in: *Modal Descent*, Mathematical Structures in Computer Science (2020) ([arXiv:2003.09713](https://arxiv.org/abs/2003.09713), [doi:10.1017/S0960129520000201](https://doi.org/10.1017/S0960129520000201)) which explicitly states and proves the cohesive HoTT-version of the theorem. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/17">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/17">v17</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   added pointer to

   • Felix Cherubini, Egbert Rijke, Thm. 8.7 in: Modal Descent, Mathematical Structures in Computer Science (2020) (arXiv:2003.09713, doi:10.1017/S0960129520000201)

   which explicitly states and proves the cohesive HoTT-version of the theorem.

   diff, v17, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 18th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Tammo tom Dieck]], §3 in: _Algebraic topology_, European Mathematical Society, Zürich (2008) &lbrack;[doi:10.4171/048](https://www.ems-ph.org/books/book.php?proj_nr=86), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf)&rbrack; and added cross-linking with the entry on *[[transport]]* <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/19">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/19">v19</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   added pointer to

   • Tammo tom Dieck, §3 in: Algebraic topology, European Mathematical Society, Zürich (2008) [doi:10.4171/048, pdf]

   and added cross-linking with the entry on transport

   diff, v19, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeFeb 14th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a graphics ([here](https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces#PathLifting)) illustrating path lifting in a covering space <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/21">v21</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   added a graphics (here) illustrating path lifting in a covering space

   diff, v21, current

8. • CommentRowNumber8.
   • CommentAuthorDmitri Pavlov
   • CommentTimeNov 17th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded: A detailed treatment is available in * [[Nicolas Bourbaki]], _Topologie Algébrique_, Chapitres 1 à 4, Springer, 2016. [doi:10.1007/978-3-662-49361-8](https://doi.org/10.1007/978-3-662-49361-8), ISBN 978-3-662-49361-8. <a href="https://ncatlab.org/nlab/revision/diff/fundamental+theorem+of+covering+spaces/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/fundamental+theorem+of+covering+spaces/24">v24</a>, <a href="https://ncatlab.org/nlab/show/fundamental+theorem+of+covering+spaces">current</a>

   Added:

   A detailed treatment is available in

   • Nicolas Bourbaki, Topologie Algébrique, Chapitres 1 à 4, Springer, 2016. doi:10.1007/978-3-662-49361-8, ISBN 978-3-662-49361-8.

   diff, v24, current