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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeJan 14th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexAdded a subsection > ### As a homotopy type > As a [[homotopy type]] the torus is the product of two copies of the [circle](https://ncatlab.org/nlab/show/circle#as_a_homotopy_type). > In [[homotopy type theory]] the torus can be formalized as the [[higher inductive type]] generated by a point `base`, two paths, $p$ and $q$, from `base` to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown ([Sojakova15](#Sojakova15)) that this type is equivalent to the product of two circles. For a treatment in [[cubical type theory]], see ([Licata-Brunierie](#Licata-Brunierie)). <a href="https://ncatlab.org/nlab/revision/diff/torus/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/torus/13">v13</a>, <a href="https://ncatlab.org/nlab/show/torus">current</a>

   Added a subsection

   As a homotopy type

   As a homotopy type the torus is the product of two copies of the circle.

   In homotopy type theory the torus can be formalized as the higher inductive type generated by a point base, two paths, $p$ and $q$, from base to itself, and an element $t$ of $p\cdot q = q \cdot p$. It has been formally shown (Sojakova15) that this type is equivalent to the product of two circles. For a treatment in cubical type theory, see (Licata-Brunierie).

   diff, v13, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 4th 2022
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   Author: Urs
   Format: MarkdownItexadded statement of the *stable* homotopy type of the 2-torus, and graphics illustrating this in line with the homotopy types of punctured tori ([here](https://nlab.preschema.com/nlab/show/torus#StableHomotopyType)) <a href="https://ncatlab.org/nlab/revision/diff/torus/15">diff</a>, <a href="https://ncatlab.org/nlab/revision/torus/15">v15</a>, <a href="https://ncatlab.org/nlab/show/torus">current</a>

   added statement of the stable homotopy type of the 2-torus, and graphics illustrating this in line with the homotopy types of punctured tori (here)

   diff, v15, current