Timeline for Precise definition of the $\infty$-category of spaces, continuous maps, homotopies, homotopies between homotopies, and so on
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 3, 2021 at 18:18 | comment | added | Tim Porter | In fact if the function space $Y^X$ exists, then all we have been doing is working with the definition of the fundamental groupoid of that mapping space. | |
| Dec 3, 2021 at 18:17 | comment | added | Tim Porter | Think of a homotopy as an edge of the square. (I cannot draw this here!) .$H:X\times I\to Y$ and similarly for $H'$. Now a homotopy between homotopies is exactly a homotopy $K:(X\times I)\times I\to Y$, now do the obvious thing and note that the domain of $K$ is $X\times I^2$ so i$K$ has form $K(x,s,t)$. I wrote"That view has a square with H on the bottom H' on the top and constant homotopies down the sides, on f and g respectively." In the usual definition of the fundamental groupoid, and the case, $X=\{*\}$. A homotopy is a path in $Y$ and a homotopy of homotopies is a homotopy of paths. | |
| Dec 3, 2021 at 18:11 | history | edited | Tim Porter | CC BY-SA 4.0 |
deleted 1 character in body
|
| Dec 3, 2021 at 16:09 | comment | added | user997814 | Thanks. Your definition of "homotopy from $H$ to $H'$" doesn't refer to $H$ and $H'$ again, which confuses me. What's the domain and what's the codomain of $K\colon X\times I\times I\to Y$ (considered as a homotopy between homotopies rather than a map between spaces)? | |
| Dec 3, 2021 at 14:58 | history | edited | Tim Porter | CC BY-SA 4.0 |
added 4 characters in body
|
| Dec 3, 2021 at 14:40 | comment | added | Tim Porter | I have edited the reply above as the addition would not fit in the Comments. | |
| Dec 3, 2021 at 14:40 | history | edited | Tim Porter | CC BY-SA 4.0 |
added 1923 characters in body
|
| Dec 3, 2021 at 13:37 | comment | added | user997814 | Thanks! I can't find a definition of "homotopy between homotopies" in the links you give. (At least I can't spot a sentence saying "If $H$ and $H'$ are homotopies from $f$ to $g$, then a homotopy from $H$ to $H'$ is a ...".) | |
| Dec 3, 2021 at 13:16 | history | answered | Tim Porter | CC BY-SA 4.0 |